Memoirs on Differential Equations and Mathematical Physics

Volume 3. 1994

Roman Koplatadze

ON OSCILLATORY PROPERTIES OF SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS

Abstract. A functional differential equation

u^{(n)}(t) + F (u)(t) = 0,  (1)

is considered with continuous F : C(R_{+}; R) → L_{loc}(R_{+}; R).

Oscillatory properties of proper solutions of (1) are studied. In particular sufficient conditions are given for equation (1) to have the property A or B (Ã or B̃) which are optimal in a certain sense. Sufficient conditions for every solution of (1) to be oscillatory are obtained as well as existence conditions for an oscillatory solution.

Chapter 6 is dedicated to boundary value problem (16.1)-(16.2). Sufficient conditions are established for the existence of a unique solution, a unique oscillatory solution and a unique bounded oscillatory solution of this problem.

Mathematics Subject Classification. 34K15

Key words and Phrases. Functional differential equation, proper solution, equations with properties A, B, Ã, B̃, Kneser-type solution, oscillatory solution, bounded solution, boundary value problem.

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PREFACE

0.1. Subject of Investigation and a Brief Survey of Obtained Results. Let τ, σ ∈ C(ℝ_{+}; ℝ_{+}),^{1} lim_{t→+∞} τ(t) = +∞ and τ(t) ≤ σ(t) for t ∈ ℝ_{+}. Denote by V (τ) (V (τ, σ)) the set of continuous mappings F : C(ℝ_{+}; ℝ) → L_{loc}(ℝ_{+}; ℝ) satisfying the condition: F (x)(t) = F (y)(t) holds for any t ∈ ℝ_{+} and x, y ∈ C(ℝ_{+}; ℝ) provided that x(s) = y(s) for s ≥ τ(t) (τ(t) ≤ s ≤ σ(t)). Obviously V (τ, σ) ⊂ V (τ).

This work is dedicated to the study of oscillatory properties of solutions of a functional differential equation of the form

u^{(n)}(t) + F (u)(t) = 0,  (0.1)

where n ≥ 1 and F ∈ V (τ) (F ∈ V (τ, σ)).

For any t_{0} ∈ ℝ_{+} we denote by H_{t_{0},τ} the set of all functions u ∈ C(ℝ_{+}; ℝ) satisfying u(t) ≠ 0 for t ≥ t_{∗}, where t_{∗} = min{t_{0}, τ_{∗}(t_{0})}, τ_{∗}(t) = inf(τ(s) : s ≥ t}.

Throughout the work whenever the notation V (τ), V (τ, σ) and H_{t_{0},τ} occurs, it will be understood unless specified otherwise that the functions τ and σ satisfy the conditions stated above.

It will always be assumed that either the condition

F (u)(t) u(t) ≥ 0 for t ≥ t_{0}, u ∈ H_{t_{0},τ}  (0.2)

or the condition

F (u)(t) u(t) ≤ 0 for t ≥ t_{0}, u ∈ H_{t_{0},τ}  (0.3)

is fulfilled.

Let t_{0} ∈ ℝ_{+}. A function u : [t_{0}, +∞[→ ℝ is said to be a proper solution of equation (0.1) if it is locally absolutely continuous together with its derivatives up to order n − 1 inclusive,

sup{|u(s)| : s ∈ [t, +∞[} > 0 for t ≥ t_{0}

and there exists a function \overline{u} ∈ C(ℝ_{+}; ℝ) such that \overline{u}(t) ≡ u(t) on [t_{0}, +∞[ and the equality \overline{u}^{(n)}(t) + F (\overline{u}(t)) = 0 holds for t ∈ [t_{0}, +∞[. A proper solution u : [t_{0}, +∞[→ ℝ of equation (0.1) is said to be oscillatory if it has a sequence of zeros tending to +∞. Otherwise the solution u is said to be nonoscillatory.

Definition 0.1 ([40]). We say that equation (0.1) has property A if any of its proper solutions is oscillatory when n is even and either is oscillatory or satisfies

|u^{(i)}(t)| ↓ 0 as t ↑ +∞ (i = 0, …, n − 1)  (0.4)

when n is odd.

1For the notation see Subsection 0.2.

Definition 0.2 ([33]). We say that equation (0.1) has property B if any of its proper solutions either is oscillatory or satisfies either (0.4) or

|u^{i}(t)| ↑ +∞ for t ↑ +∞ (i = 0, …, n − 1)  (0.5)

when n is even, and either is oscillatory or satisfies (0.5) when n is odd.

Definition 0.3 ([50]). We say that equation (0.1) has property Ã if any of its proper solutions is oscillatory when n is odd, and either is oscillatory or satisfies (0.5) when n is even.

Definition 0.4 ([50]). We say that equation (0.1) has property B̃ if any of its proper solutions is oscillatory when n is odd, and either is oscillatory or satisfies (0.4) when n is even.

At the close of the last centure A. Kneser [38] posed the problem of finding conditions for an equation

u^{(n)}(t) + p(t)u(t) = 0  (0.6)

to have properties similar to either u^{(n)} + u = 0 or u^{(n)} − u = 0. Using the well-known comparison theorem of Strum, he proved that if n = 2 and \underline{lim}_{t→+∞}t^{2}p(t) > 1/4, then (0.6) has property A, i.e. any of its solutions is

oscillatory. This result for second order linear differential equations was further improved and generalized by quite a number of authors (see the survey of M. Ráb [94]).

Sufficient conditions for an ordinary differential equation to have property A or B can be found in A. Kneser [38], W.B. Fite [22], I. Mikusinski [84], F.W. Atkinson [4], V.A. Kondratyev [39, 40], G.V. Ananyeva and V.I. Balaganskii [2], Š. Belohorec [6], I.T. Kiguradze [31 – 37], L. Ličko and M. Švec [81], D.V. Izyumova [28], T.A. Chanturia [8 – 15], M. Bartušek [5] and in other papers. The monographs by O. Boru^{∘}vka [7] and F. Neuman [90] should also be mentioned as they deal with the problem of global (in particular, oscillatory) behaviour of solutions of second [7] and n-th order [90] linear differential equations.

Analogous results for differential equations with deviating arguments and functional-differential equations are obtained in [20, 23-26, 29, 41-50, 53, 61-64, 69, 71, 73, 79, 83, 85, 91, 92, 96-101, 103].

If p ∈ L_{loc}(ℝ_{+}; ℝ_{+}) (n ≥ 3 and p ∈ L_{loc}(ℝ_{+}; ℝ_{−})) and (0.6) has property A(B), then it has solutions of all the types mentioned in Definitions 0.1 and 0.2 [32, 40]. However for a differential equation with a delayed argument

u^{(n)}(t) + p(t)u(t − Δ) = 0

(Δ = const > 0, (−1)^{n+1}p(t) ≥ 0 for t ∈ ℝ_{+}) properties A and B do not guarantee the existence of proper solutions satisfying (0.4), while for a differential equation with an advanced argument

u^{(n)}(t) + p(t)u(t + Δ) = 0

(Δ = const > 0, p(t) ≤ 0 for t ∈ ℝ_{+}) property B does not guarantee the existence of proper solutions satisfying (0.5). As for a functional differential equation of the form

u^{(n)}(t) + p(t) ∫_{t−Δ}^{t+Δ} u(s)ds = 0

(Δ = const > 0, p(t) ≤ 0 for t ∈ ℝ_{+}) property B does nor guarantee the existence of proper solutions of forms (0.4) and (0.5).

The above examples show that the set of nonoscillatory solutions of functional differential equations has a structure differing from that of the set of solutions of ordinary differential equations.

Oscillation criteria specific of differential equations with delay were for the first time suggested by A. Myshkis [86]. Subsequently analogous problems were studied in [3, 19, 27, 42, 43, 54, 55, 70, 78, 87, 106] for first and second order linear and nonlinear differential equations.

Sufficient conditions for higher order differential equations with deviating arguments to have property Ã or B̃ can be found in [16, 17, 48, 50, 65-68, 79, 80, 89, 97, 104, 105, 107]. Sufficient (necessary and sufficient) conditions for every proper solution of a higher order functional differential equation to be oscillatory are given in [72].

Chapter 1 of this work is concerned with equations having property A or B. Some basic definitions and auxiliary statements are formulated in §1. Comparison theorems are proved in §2, thereby making it possible to derive property A or B of the considered equations. Based on these theorems, sufficient (necessary and sufficient) conditions are established in §3 (in §4) for an essentially nonlinear functional differential equation to have property A or B.

Chapter 2 deals with analogous problems for equation (0.1) with the operator F admitting a linear minorant. In §5 we prove some auxiliary lemmas for linear differential inequalities with deviating arguments, which in §§6 and 7 are used to derive sufficient conditions for a functional differential equation with a linear minorant to have property A or B.^{2}

Not only the results obtained in §§6 and 7 are the new ones for equation (0.1). They also improve some of the previous well-known results for equation (0.6). Chapter 2 concludes with some sufficient conditions for equation (0.1) not to have property A (B), thereby illustrating to what extent the results of §§6 and 7 are precise.

Chapters 3 and 4 are concerned with solutions satisfying conditions (0.4) and (0.5). In §8 auxiliary lemmas are formulated which enable one to establish the asymptotic behaviour near +∞ of solutions satisfying (0.4) of differential equations and inequalities with a delayed argument. These lemmas are used in §§9 and 11 to find the sufficient or necessary and sufficient conditions for equation (0.1) not to have a solution satisfying (0.4) or (0.5).

2Alongside with the problem of equation (0.1) having property A or B, there arises a need to study whether all solutions of (0.1) are or are not oscillatory. Interesing results in this direction were obtained by N.V. Azbelev [1].

In §§12 and 13 of Chapter 5 the previously obtained results are used to find the sufficient or necessary and sufficient conditions for equation (0.1) to have property Ã or B̃, while in §14 the sufficient or necessary and sufficient conditions are established for any solution of (0.1) to be oscillatory. The results presented in this chapter are specific of functional differential equations and have no analogues for ordinary differential equations.

Chapter 6 is dedicated to second order differential equations with a delayed argument. The question of the existence and uniqueness of a solution of the linear problem (16.1), (16.2) is studied in §16, while in §17 we discuss the boundedness of all oscillatory solutions of both linear and nonlinear equations. In §18 sufficient conditions are given for the unique solution of problem (16.1), (16.2) to be oscillatory or bounded oscillatory.

The results of this work make it possible to extend a number of the earlier results concerning the oscillatory behaviour of differential equations with deviating arguments to the case of general functional differential equations. Some of them improve well-known results not only for differential equations with deviating arguments but for ordinary differential equations as well. Besides, the work presents essentially new results specific of functional differential equations.

0.2. Basic Notation. The following notation will be used throughout the work:

N is the set of natural numbers;

ℝ =] − ∞, +∞[; ℝ_{+} = [0, +∞[; ℝ_{−} =] − ∞, 0]; ℝ^{k} = ℝ \underbrace{× ⋯ ×}_{k times} ℝ.

If I ⊂ ℝ is an interval and E ⊂ ℝ is a set, then:

C(I; E) denotes the space of all continuous functions with the topology of uniform convergence on any finite subsegment of I. L_{loc}(I; E) denotes the space of all locally integrable functions u : I → E with the topology of convergence in mean on any finite subsegment of I.

C̃_{loc}^{k}(I; E) denotes the space of all functions u : I → E which are absolutely continuous on any finite subsegment of I together with their derivatives up to order k inclusive.

K_{loc}(I × ℝ^{k}; E) denotes the set of functions f : I × ℝ^{k} → E satisfying the local Carathéodory conditions, i.e. on any finite subsegment of I the function f(·, x) is measurable for every x ∈ ℝ^{k}, f(t, ·) is continuous for almost every t ∈ I and for any r > 0

sup{|f(t, x)| : ∥x∥ ≤ r} ∈ L_{loc}(ℝ_{+}; ℝ_{+}).

CHAPTER I

ESSENTIALLY NONLINEAR EQUATIONS WITH

PROPERTIES A AND B

§ 1. Some Auxiliary Statements

1.1. On Some Classes of Nonoscillatory Functions.

Lemma 1.1. Let u ∈ C̃_{loc}^{n−1}([t_{0}, +∞[; ℝ) and

u(t) > 0, u^{(n)}(t) ≤ 0 (u^{(n)}(t) ≥ 0) for t ≥ t_{0}.  (1.1)

Then there exist t_{1} ∈ [t_{0}, +∞[ and l ∈ {0, …, n} such that l + n is odd (even) and

\begin{matrix}u^{(i)}(t) > 0 for t ≥ t_{0} (i = 0, …, l − 1), \\ (−1)^{i+l}u^{(i)}(t) ≥ 0 for t ≥ t_{0} (i = l, …, n). \end{matrix}  (1.2)

The proof of this lemma can be found in [34] (Lemmas 14.1 and 14.2).

Lemma 1.2. Let u ∈ C̃_{loc}^{n−1}([t_{0}, +∞[; ℝ) and

(−1)^{i}u^{(i)}(t) > 0 (i = 0, …, n − 1), (−1)^{n}u^{(n)}(t) ≥ 0 for t ≥ t_{0}.

Then

∫_{t_{0}}^{+∞} t^{n−1}|u^{(n)}(t)|dt < +∞,  (1.3)
|u^{(i)}(t)| ≥ \frac{1}{(n − i − 1)!} ∫_{t}^{+∞}(s − t)^{n−i−1}|u^{(n)}(s)|ds  (1.4)
for t ≥ t_{0} (i = 0, …, n − 1)

and

u(t) ≥ ∑_{i=0}^{n−1} \frac{|u^{(i)}(s)|(s − t)^{i}}{i!} for s ≥ t ≥ t_{0}.  (1.5)

Proof. Using the signs of the derivatives of u, we can readily obtain (1.3) and (1.5) from the identity

u^{(i)}(t) = ∑_{j=i}^{k−1} \frac{u^{(j)}(s)}{(j − i)!}(t − s)^{j−i} +

+\frac{1}{(k − i − 1)!} ∫_{s}^{t}(t − ξ)^{k−i−1}u^{(k)}(ξ)dξ  (1.6_{ik})

with i = 0, k = n, s ≥ t. As to (1.4), it is an immediate consequence of (1.6_{in}). ■

Lemma 1.3. Let u ∈ C̃_{loc}^{n−1}([t_{0}, +∞[) and (1.2) be fulfilled for some l ∈ {1, …, n − 1}. Then

∫_{t_{0}}^{+∞} t^{n−l−1}|u^{(n)}(t)|dt < +∞,  (1.7)

u^{(i)}(t) ≥ u^{(i)}(t_{0}) + \frac{1}{(l − i − 1)!(n − l − 1)!} ∫_{t_{0}}^{t}(t − ξ)^{l−i−1} ×

× ∫_{ξ}^{+∞}(s − ξ)^{n−l−1}|u^{(n)}(s)|ds dξ for t ≥ t_{0} (i = 0, …, l − 1), (1.8)
|u^{(i)}(t)| ≥ \frac{1}{(n − i − 1)!} ∫_{t}^{+∞}(s − t)^{n−i−1}|u^{(n)}(s)|ds  (1.9)
for t ≥ t_{0} (i = l, …, n − 1).

If, in addition,

∫_{t_{0}}^{+∞} t^{n−l}|u^{(n)}(t)|dt = +∞,  (1.10)

then

\frac{u(t)}{t^{l}} ↓, \frac{u(t)}{t^{l−1}} ↑ +∞ as t ↑ +∞,  (1.11)
u(t) ≥ \frac{t^{l}}{l!(n − l)!} ∫_{t}^{+∞} s^{n−l−1}|u^{(n)}(s)|ds for large t  (1.12)

and for any nondecreasing β ∈ C(ℝ_{+}; ℝ_{+}) satisfying

β(t) ≤ t for t ∈ ℝ_{+}, lim_{t→+∞} β(t) = +∞  (1.13)

there exists t_{∗} ∈ [t_{0}, +∞[ such that

u(β(t)) ≥ \frac{β^{l−1}(t)}{l!(n − l)!} ∫_{t_{∗}}^{t} β^{n−l}(s)|u^{(n)}(s)|ds for t ≥ t_{∗}.  (1.14)

Proof. By virtue of (1.2) condition (1.7) readily follows from the identity

∑_{j=i}^{k−1} \frac{(−1)^{j}t^{j−i}u^{(j)}(t)}{(j − i)!} = ∑_{j=i}^{k−1} \frac{(−1)^{j}t_{0}^{j−i}u^{(j)}(t_{0})}{(j − i)!} +

+\frac{(−1)^{k−1}}{(k − i − 1)!} ∫_{t_{0}}^{t} s^{k−i−1}u^{(k)}(s)ds  (1.15_{ik})

with i = l, k = n. The same identity also implies the inequality

∑_{j=l}^{n−1} \frac{t^{j−l}|u^{(j)}(t)|}{(j − l)!} ≥ \frac{1}{(n − l − 1)!} ∫_{t}^{+∞} s^{n−l−1}|u^{(n)}(s)|ds for t ≥ t_{0}. (1.16)

On account of (1.2) and (1.7), from (1.6_{in}) with s → +∞ we obtain (1.9). Analogously (1.6_{il}) with s = t_{0} gives

u^{(i)}(t) ≥ u^{(i)}(t_{0}) + \frac{1}{(l − i − 1)!} ∫_{t_{0}}^{t}(t − ξ)^{l−i−1}u^{(l)}(ξ)dξ

for t ≥ t_{0} (i = 0, …, l − 1).

Hence by (1.9) we obtain (1.8).

Assume now that (1.10) is fulfilled. Using (1.2), from (1.15_{l−1 n}) we have

lim_{t→+∞}(u^{(l−1)}(t) − tu^{(l)}(t)) = +∞  (1.17)

and

u^{(l−1)}(t) ≥ ∑_{j=l}^{n−1} \frac{t^{j−l+1}|u^{(j)}(t)|}{(j − l + 1)!} for large t.  (1.18)

For any t ≥ t_{0} and i ∈ {1, …, l} put

ρ_{i}(t) = iu^{(l−i)}(t) − tu^{(l−i+1)}(t) = −t^{i+1}(t^{−i}u^{(l−i)}(t))′,  (1.19)
r_{i}(t) = tu^{(l−i+1)}(t) − (i − 1)u^{(l−i)}(t) = t^{i}(t^{1−i}u^{(l−i)}(t))′. (1.20)

Applying (1.17) and L’Hospital’s rule, we have

lim_{t→+∞} t^{1−i}u^{(l−i)}(t) = +∞ (i = 1, …, l),  (1.21)

so that in view of (1.20) there exist t_{l} ≥ ⋯ ≥ t_{1} ≥ t_{0} such that r_{i}(t_{i}) > 0 (i = 1, …, l). Since by (1.17) ρ_{1}(t) → ∞ as t → +∞ ρ′_{i+1}(t) = ρ_{i}(t), r′_{i+1}(t) = r_{i}(t) and r_{1}(t) = tu^{(l)}(t) > 0 for t ≥ t_{0} (i = 1, …, l − 1), we find that ρ_{i}(t) → +∞ as t → +∞ and r_{i}(t) > 0 for t ≥ t_{i} (i = 1, …, l). These facts along with (1.19)–(1.21) prove (1.11). On the other hand, by (1.19) we have

iu^{(l−i)}(t) ≥ tu^{(l−i+1)}(t) for large t (i = 1, …, l)

which implies

u(t) ≥ \frac{t^{l−1}}{l!} u^{(l−1)}(t) for large t.  (1.22)

Inequalities (1.16), (1.18) and (1.22) imply (1.12).

It remains to prove (1.14). Let t_{∗} > t_{1} be such that β(t) ≥ t_{1} for t ≥ t_{∗}. From (1.15_{l−1 n−1}) we have

u^{(l−1)}(β(t)) ≥ u^{(l−1)}(β(t_{∗})) − ∑_{j=l}^{n−2} \frac{β^{j−l+1}(t_{∗})|u^{(j)}(β(t_{∗}))|}{(j − l + 1)!} +

+\frac{1}{(n − l − 1)!} ∫_{β(t_{∗})}^{β(t)} s^{n−l−1}|u^{(n−1)}(s)|ds.  (1.23)

On the other hand, changing the order of integration and taking into account (1.13) and the fact that β is monotone, we obtain

\frac{1}{(n − l − 1)!} ∫_{β(t_{∗})}^{β(t)} s^{n−l−1}|u^{(n−1)}(s)|ds ≥ \frac{1}{(n − l − 1)!} ∫_{β(t_{∗})}^{β(t)} s^{n−l−1} ×

× ∫_{s}^{t} |u^{(n)}(ξ)|dξ ds ≥ \frac{1}{(n − l − 1)!} ∫_{t_{∗}}^{t} |u^{(n)}(ξ)| ×

× ∫_{β(t_{∗})}^{β(ξ)} s^{n−l−1}ds dξ ≥ \frac{1}{(n − l)!} ∫_{t_{∗}}^{t} β^{n−l}(ξ)|u^{(n)}(ξ)|dξ −

−\frac{1}{(n − l)!}|u^{(n−1)}(β(t_{∗}))|β^{n−l}(t_{∗}) for t ≥ t_{∗},

so that, applying (1.20), we arrive at

u^{(l−1)}(β(t)) ≥ c_{0} + \frac{1}{(n − l)!} ∫_{t_{0}}^{t} β^{n−l}(s)|u^{(n)}(s)|ds for t ≥ t_{0}, (1.24)

where

c_{0} = u^{(l−1)}(β(t_{∗})) − ∑_{j=l}^{n−1} \frac{β^{j−l+1}(t_{∗})}{(j − l + 1)!}|u^{(j)}(β(t_{∗}))|.

By (1.18) t_{∗} can be assumed to be large enough for the inequality c_{0} ≥ 0 to hold. Therefore (1.22) and (1.24) immediately imply (1.14). ■

1.2. On Some Classes of Mappings from C(ℝ_{+}; ℝ) into L_{loc}(ℝ_{+}; ℝ). Let τ, σ : ℝ_{+} → ℝ_{+} satisfy the condions given at the begining of Subsection 0.1.

Definition 1.1. Denote by M(τ) (M(τ, σ)) the set of all φ ∈ V (τ) (φ ∈ V (τ, σ)) satisfying the condition: for any t_{0} ∈ ℝ_{+}, t ∈ [t_{0}, +∞[ and x, y ∈ H_{t_{0},τ} such that x(s)y(s) > 0, |x(s)| ≥ |y(s)| for s ≥ τ(t) (τ(t) ≤ s ≤ σ(t)) one has φ(x)(t) sign x(t) ≥ φ(y)(t) sign y(t) ≥ 0.

Definition 1.2. Denote by M_{1}(τ) (M_{1}(τ, σ)) the set of all φ ∈ M(τ) (φ ∈ M(τ, σ)) satisfying the condition: for any t_{0} ∈ ℝ_{+} and m ∈ {1, 2} the integral inequality

y(t) ≥ ∫_{t}^{+∞} |φ((−1)^{m}y)(s)|ds  (1.25)

has no nonincreasing positive solution y : [t_{0}, +∞[→]0, +∞[.

Definition 1.3. For any nondecreasing function β ∈ C(ℝ_{+}; ℝ_{+}) satisfying (1.13) denote by M_{2}^{β}(τ) (M_{2}^{β}(τ, σ)) the set of all φ ∈ M(τ) (φ ∈ M(τ, σ)) satisfying the condition: for any t_{0} ∈ ℝ_{+} and m ∈ {1, 2} the integral inequality

y(β(t)) ≥ ∫_{t_{0}}^{t} |φ((−1)^{m}y)(s)|ds  (1.26)

has no nondecreasing positive solution y : [t_{0}, +∞[→]0, +∞[.

Remark 1.1. As in the case of equation (0.1), a function y ∈ C([t_{0}, +∞[; ℝ) is said to be a solution of (1.25) ((1.26)) if there exists \overline{y} ∈ C(ℝ_{+}; ℝ) which satisfies (1.25) ((1.26)) for any t ∈ ℝ_{+} and whose restriction to [t_{0}, +∞[ coincides with y.

Remark 1.2. As in the case of V (τ) (V (τ, σ)), whenever in the sequel the notations M(τ), M_{1}(τ) and M_{2}^{β}(τ) (M(τ, σ), M_{1}(τ, σ) and M_{2}^{β}(τ, σ)) are used, τ and σ will be assumed, unless stated otherwise, to satisfy the conditions given at the beginning of Subsection 0.1 and β will be assumed to be nondecreasing and to satisfy (1.13).

Obviously, M(τ, σ) ⊂ M(τ), M_{1}(τ, σ) ⊂ M_{1}(τ) and M_{2}^{β}(τ, σ) ⊂ M_{2}^{β}(τ). On account of the definitions of the sets M_{1}(τ) and M_{2}^{β}(τ), one can easily

ascertain that the following lemma is valid.

Lemma 1.4. If φ ∈ M_{1}(τ) (φ ∈ M_{2}^{β}(τ)), then

∫ ^{+∞} |φ(c)|(t)dt = +∞.^{3}

for any c ≠ 0.

Lemma 1.5. If α ∈ C(ℝ_{+}; ]0, +∞[) is nondecreasing and φ ∈ M_{1}(τ), then ψ ∈ M_{1}(τ) where

ψ(y)(t) = α(t)φ(y/α)(t) for y ∈ C(ℝ_{+}; ℝ), t ∈ ℝ_{+}.

.

Proof. Assume the contrary: ψ ∉ M_{1}(τ). In that case there exist t_{0} ∈ ℝ_{+} and m ∈ {1.2} such that the inequality

y(t) ≥ ∫_{t}^{+∞} α(s)|φ((−1)^{m}y/α)(s)|ds

has a nonincreasing solution y : [t_{0}, +∞[→]0, +∞[. Since α is nondecreasing, we find that the nonincreasing function z = y/α satisfies

z(t) ≥ ∫_{t}^{+∞} |φ((−1)^{m}z)(s)|ds for t ≥ t_{0}.

Therefore φ ∉ M_{1}(τ) which is a contradiction. ■

3We usually do not distinguish between the notations of a constant and a function identically equal to this constant.

Lemma 1.6. Let φ ∈ M(τ) and there exist t_{0} ∈ ℝ_{+} such that for any m ∈ {1, 2} we have

|φ((−1)^{m}y)(t)| ≥ p(t)ω(y(t)) for y ∈ C(ℝ_{+}; ]0, +∞[), t ≥ t_{0}, (1.27)

where p ∈ L_{loc}([t_{0}, +∞[; ℝ_{+}), ω ∈ C(ℝ_{+}; ℝ_{+}) is nondecreasing, ω(s) > 0 for s > 0 and

∫_{0}^{1} \frac{ds}{ω(s)} < +∞, ∫ ^{+∞} p(t)dt = +∞.  (1.28)

Then φ ∈ M_{1}(τ).

Proof. Assume the contrary: φ ∉ M_{1}(τ). Then there exist t_{1} ∈ [t_{0}, +∞[ and m ∈ {1, 2} such that (1.25) has a nonincreasing solution y : [t_{1}, +∞[→ ]0, +∞[. According to (1.27) we have

y(t) ≥ ∫_{t}^{+∞} p(s)ω(y(s))ds for t ≥ t_{1}.

Therefore, since ω is nondecreasing,

p(t)ω(y(t)) ≥ p(t)ω( ∫_{t}^{+∞} p(s)ω(y(s))ds) for t ≥ t_{1}.

This inequality implies

∫_{x(t)}^{x(t_{1})} \frac{ds}{ω(s)} ≥ ∫_{t_{1}}^{t} p(s)ds for t ≥ t_{1}

with x(t) = ∫_{t}^{+∞} p(s)ω(y(s))ds, which contradicts (1.28). ■

Corollary 1.1. Let φ ∈ M(τ) there exists and t_{0} ∈ ℝ_{+} such that for any m ∈ {1, 2} we have

|φ((−1)^{m}y)(t)| ≥ p(t)y^{λ}(t) for y ∈ C(ℝ_{+}; ]0, +∞[), t ≥ t_{0},

where λ ∈]0, 1[ and p ∈ L_{loc}(ℝ_{+}; ℝ_{+}) satisfies the second condition of (1.28). Then φ ∈ M_{1}(τ).

Lemma 1.7. Let φ ∈ M_{2}^{β}(τ) and α ∈ C(ℝ_{+}; ]0, +∞[) be nonincreasing. Then ψ ∈ M_{2}^{β}(τ), where

ψ(y)(t) = α(β(t))φ(y/α)(t) for y ∈ C(ℝ_{+}; ℝ), t ∈ ℝ_{+}.

Proof. Assume the contrary: ψ ∉ M_{2}^{β}(τ). Then there exist t_{0} ∈ ℝ_{+} and m ∈ {1, 2} such that the inequality

y(β(t)) ≥ ∫_{t_{0}}^{t} α(β(s))|φ((−1)^{m}y/α)(s)|ds

has a nondecreasing solution y : [t_{0}, +∞[→]0, +∞[. Since α is nonincreasing, we find that the nondecreasing function z = y/α satisfies

z(β(t)) ≥ ∫_{t_{0}}^{t} |φ((−1)^{m}z)(s)|ds for t ≥ t_{0}.

Therefore φ ∉ M_{2}^{β}(τ) which is a contradiction. ■

Lemma 1.8. Let φ ∈ M(τ), a nondecreasing function β ∈ C(ℝ_{+}; ℝ_{+}) satisfy (1.13) and there exist t_{0} ∈ ℝ_{+} such that

|φ((−1)^{m}y)(t)|≥p(t)ω(y(β(t))) for y ∈C(ℝ_{+}; ]0, +∞[), t ≥ t_{0}, (1.29)

where p ∈ L_{loc}(ℝ_{+}; ℝ_{+}) satisfies the second condition of (1.28), ω ∈ C(ℝ_{+}; ℝ_{+}) is nondecreasing, ω(s) > 0 for s > 0 and

∫_{1}^{+∞} \frac{ds}{ω(s)} < +∞.  (1.30)

Then φ ∈ M_{2}^{β}(τ).

Proof. Assume the contrary: φ ∉ M_{2}^{β}(τ). Then there exist t_{1} ∈ [t_{0}, +∞[ and m ∈ {1, 2} such that (1.26) has a nondecreasing solution y : [t_{1}, +∞[→ ]0, +∞[. By (1.29)

z(t) ≥ ∫_{t_{1}}^{t} p(s)ω(z(s))ds for t ≥ t_{1},

where z(t) = y(β(t)). Therefore, since ω is nondecreasing, we have

p(t)ω(z(t)) ≥ p(t)ω( ∫_{t_{1}}^{t} p(s)ω(z(s))ds) for t ≥ t_{1}.  (1.31)

Choose t_{2} ∈]t_{1}, +∞[ such that ∫_{t_{1}}^{t_{2}} p(s)ω(z(s))ds > 0. Then (1.31) implies

∫_{x(t_{2})}^{x(t)} \frac{ds}{ω(s)} ≥ ∫_{t_{2}}^{t} p(s)ds for t ≥ t_{0},

where x(t) = ∫_{t_{1}}^{t} p(s)ω(z(s))ds. But this contradicts the second condition of (1.28) and (1.30). ■

Corollary 1.2. Let φ ∈ M(τ), a nondecreasing function β ∈ C(ℝ_{+}; ℝ_{+}) satisfy (1.13) and there exist t_{0} ∈ ℝ_{+} such that for any m ∈ {1, 2} we have

|φ((−1)^{m}y)(t)|≥p(t)y^{λ}(β(t)) for y ∈C(ℝ_{+}; ]0, +∞[), t ≥ t_{0},

where λ ∈]1, +∞[ and p ∈ L_{loc}(ℝ_{+}; ℝ_{+}) satisfies the second condition of (1.28). Then φ ∈ M_{2}^{β}(τ).

§ 2. Comparison Theorems

In this section we shall prove the comparison theorems for functional differential equations and inequalities. The essence of the comparison principle consists in the following: φ, Φ : C(ℝ_{+}; ℝ) → L_{loc}(ℝ_{+}; ℝ_{+}) being somehow related continuous operators, the fact that the equation

v^{(n)}(t) + φ(v)(t) = 0  (2.1)
(v^{(n)}(t) − φ(v)(t) = 0)  (2.2)

has property A (property B) implies that the inequality

[u^{(n)}(t) + Φ(u)(t)] sign u(t) ≤ 0  (2.3)
([u^{(n)}(t) − Φ(u)(t)] sign u(t) ≥ 0)  (2.4)

also possesses the same property.

We shall consider here relations of two types existing between φ and Φ:

1) φ is a minorant of Φ (2.1. Minorant Case); 2) φ is a superposition of the form

φ(u)(t) = Φ(ψ_{t}(u))(t) for u ∈ C(ℝ_{+}; ℝ), t ∈ ℝ_{+},  (2.5)

where {ψ_{t} : C(ℝ_{+}; ℝ) → C(ℝ_{+}; ℝ)}_{t∈ℝ_{+}} is a family of operators of either type (2.2. Superposition Case).

The results obtained will enable us in §§3–4 to derive the sufficient and necessary conditions for equation (0.1) to possess property A (property B).

We begin by considering a lemma which is a special case of the Schauder-Tikhonoff theorem (see, for example, [18, p. 227]).

Lemma 2.1. Let t_{0} ∈ ℝ, U be a closed bounded convex subset of C([t_{0}, +∞[; ℝ), and let T : U → U be a continuous mapping such that the set T (U) is equicontinuous on every finite subsegment of [t_{0}, +∞[. Then T has a fixed point.

2.1. Minorant Case.

Theorem 2.1. Let Φ ∈ V (τ), t_{0} ∈ ℝ_{+}, and assume that for any u, v ∈ H_{t_{0},τ} satisfying |u(t)| ≥ |v(t)|, u(t)v(t) > 0 for t ≥ t_{0} the inequality

Φ(u)(t) sign u(t) ≥ φ(v)(t) sign v(t) ≥ 0 for t ≥ t_{0}  (2.6)

holds, where φ ∈ V (τ). Let, moreover, equation (2.1) have property A. Then inequality (2.3) also has the property A.

Proof. Let u : [t_{0}, +∞[→ ℝ be a proper nonoscillatory solution of (2.3). It can be assumed without loss of generality that

u(t) > 0, u^{(i)}(t) ≠ 0 for t ≥ t_{0} (i = 1, …, n − 1).  (2.7)

By (2.6) and Lemma 1.1 there exist t_{1} ∈ [t_{0}, +∞[ and l ∈ {0, …, n − 1} such that l + n is odd and

u^{(i)}(t) > 0 (i = 0, …, l − 1), (−1)^{i+l}u^{(i)}(t) > 0  (2.8_{l})
(i = l, …, n − 1) for t ≥ t_{1}.

First we assume that l ∈ {1, …, n − 1}. In that case (2.8_{l}) and Lemma 1.3 imply

u(t) ≥ u(t_{2}) + \frac{1}{(l − 1)!(n − l − 1)!} ∫_{t_{2}}^{t}(t − ξ)^{l−1} ×

× ∫_{ξ}^{+∞}(s − ξ)^{n−l−1}Φ(u)(s)ds dξ for t ≥ t_{2},  (2.9)

where t_{2} ∈ [t_{1}, +∞[ is a sufficiently large number to be chosen such that t_{∗} = min(t_{2}, inf{τ(t) : t ≥ t_{2}}) ≥ t_{1}.

Let U be the set of all v ∈ C([t_{∗}, +∞[; ℝ) satisfying u(t_{1}) ≤ v(t) ≤ u(t) for t ≥ t_{2}, v(t) = u(t) for t_{∗} ≤ t ≤ t_{2}. Define T : u → C([t_{∗}, +∞[; ℝ) by

T (v)(t) = \begin{cases}u(t_{2}) + \frac{1}{(l−1)!(n−l−1)!} ∫_{t_{2}}^{t}(t − ξ)^{l−1} ∫_{ξ}^{+∞}(s − ξ)^{n−l−1}× \\ ×φ(v)(ξ)ds dξ for t ≥ t_{2} \\ u(t) for t_{∗} ≤ t ≤ t_{2}. \end{cases}(2.10)

By virtue of (2.6) and (2.8_{l})–(2.10) we have T (U) ⊂ U.

Let v_{k} ∈ U (k = 1, 2, …) and lim_{k→∞} v_{k} = v_{0} uniformly on every finite subsegment of [t_{∗}, +∞[. Take arbitrarily ε > 0 and t^{∗} > t_{2}, and choose t_{3} ∈]t^{∗}, +∞[ and k_{0} ∈ ℕ such that

2(t^{∗} − t_{2})^{l} ∫_{t_{3}}^{+∞} s^{n−l−1}Φ(u)(s)ds < ε,

(t_{3} − t_{2})^{n−1} ∫_{t_{2}}^{t_{3}} |φ(v_{k})(s) − φ(v_{0})(s)|ds < ε for k = k_{0}, k_{0} + 1, ….

Then since

|T (v_{k})(t) − T (v_{0})(t)| ≤ ∫_{t_{2}}^{t^{∗}}(t^{∗} − ξ)^{l−1} ∫_{ξ}^{t_{3}}(s − ξ)^{n−l−1} ×

×|φ(v_{k})(s) − φ(v_{0})(s)|ds dξ + 2 ∫_{t_{2}}^{t^{∗}}(t^{∗} − ξ)^{l−1} ×
× ∫_{t_{3}}^{+∞}(s − ξ)^{n−l−1}Φ(u)(s)ds dξ < 2ε for t_{∗} ≤ t ≤ t^{∗},

we find that T (v_{k})(t) → T (v_{0})(t) uniformly on [t_{∗}, t^{∗}]. As t^{∗} is arbitrary, this implies the continuity of T.

Let v ∈ U, t′, t′′ ∈ [t_{2}, t^{∗}] and t′′ > t′. Then

|T (v)(t′′) − T (v)(t′)| ≤ ∫_{t_{2}}^{t′}[(t′′ − ξ)^{l−1} − (t′ − ξ)^{l−1}] ∫_{ξ}^{+∞} s^{n−l−1} ×

×Φ(u)(s)ds dξ + ∫_{t′}^{t′′}(t′′ − ξ)^{l−1} ∫_{ξ}^{+∞} s^{n−l−1}Φ(u)(s)ds dξ.

Thus the set T (U) is equicontinuous on every finite subsegment of [t_{∗}, +∞[. Since U is closed, bounded and convex, by Lemma 2.1 there exists v ∈ U such that v = T (v). The function v is obviously a solution of (2.1) on [t_{0}, +∞[, satisfying u(t_{1}) ≤ v(t) ≤ u(t) for t ≥ t_{2}. This however contradicts the fact that equation (2.1) possesses property A. The obtained contradiction proves that l ∉ {1, …, n−1}. If n is even, then l = 0 cannot take place either, which proves the theorem in this case.

Let now n be odd and l = 0. Then (2.8_{l}) implies

lim_{t→+∞} u^{(i)}(t) = 0 (i = 1, …, n − 1), lim_{t→+∞} u(t) = c_{0} ≥ 0.

Assume that c_{0} > 0. Let U be the set of all v ∈ C([t_{∗}, +∞[; ℝ) satisfying c_{0} ≤ v(t) ≤ u(t) for t ≥ t_{∗}. Using the above reasoning, we can show that the operator T : U → U defined by

T (v)(t) = \begin{cases}c_{0} + \frac{1}{(n−1)!} ∫_{t}^{+∞}(s − t)^{n−1}φ(v)(s)ds for t ≥ t_{2} \\ T (v)(t_{2}) for t_{∗} ≤ t < t_{2} \end{cases}

has a fixed point v which is a solution of (2.1) on [t_{2}, +∞[, satisfying v(t) ≥ c_{0} for t ≥ t_{2}. But this contradicts property A of (2.1). Therefore (0.4) is fulfilled when n is odd and l = 0. ■

Theorem 2.2. Let Φ ∈ V (τ), t_{0} ∈ ℝ and assume that for any u, v ∈ H_{t_{0},τ} satisfying |u(t)| ≥ |v(t)|, u(t)v(t) > 0 for t ≥ t_{0} inequality (2.6) hold, where φ ∈ V (τ). Let, moreover, equation (2.2) have property B. Then inequality (2.4) also has property B.

Proof. Let u : [t_{0}, +∞[→ ℝ be a proper nonoscillatory solution of (2.4). Assume that (2.7) is fulfilled. Then (2.6) and Lemma 1.1 imply that there exist t_{1} ∈ [t_{0}, +∞[ and l ∈ {0, …, n} such that l + n is even and (2.8_{l}) holds. Like in proving Theorem 2.1, it can be shown that l ∉ {1, …, n − 2} and (0.4) is fulfilled when n is even and l = 0. To complete the proof it suffices to show that (0.5) is valid when l = n.

Thus assuming that l = n, from (2.8_{n}) and (2.4) we obtain

u^{(i−1)}(t) ≥ \frac{(t − t_{1})^{n−i}u^{(n−1)}(t_{1})}{(n − i)!} + \frac{1}{(n − i)!} ∫_{t_{1}}^{t}(t − s)^{n−i} ×

×Φ(u)(s)ds for t ≥ t_{1} (i = 1, …, n),  (2.11)

where t_{1} ∈ [t_{0}, +∞[ is such that t_{∗} = min(t_{1}, inf{τ(t) : t ≥ t_{1}}) ≥ t_{0}.

Let U be the set of all v ∈ C([t_{∗}, +∞[; ℝ) satisfying

\frac{u^{(n−1)}(t_{1})(t − t_{1})^{n−1}}{(n − 1)!} ≤ v(t) ≤ u(t).  (2.12)

As in proving Theorem 1.2, by (2.6), (2.11), (2.12) and Lemma 2.1 we find that the operator T : U → U defined by

T (v)(t) = \begin{cases}\frac{u^{(n−1)}(t_{1})(t−t_{1})^{n−1}}{(n−1)!} + \frac{1}{(n−1)!} ∫_{t_{1}}^{t}(t − s)^{n−1}× \\ ×φ(v)(s)ds for t ≥ t_{1} \\ 0 for t_{∗} ≤ t ≤ t_{1}. \end{cases}

has a fixed point v which is a solution of (2.2) satisfying

\frac{u^{(n−1)}(t_{1})(t − t_{1})^{n−i}}{(n − i)!} ≤ v^{(i−1)}(t) ≤ u^{(i−1)}(t) for t ≥ t_{1} (i = 1, …, n).

Since (2.2) has property B, v^{(i)}(t) ↑ +∞ as t ↑ +∞ (i = 0, …, n − 1) and therefore (0.5) is fulfilled. ■

Remark 2.1. By Theorem 2.1 (Theorem 2.2) it is obvious that if φ ∈ M(τ) and equation (2.1) (equation (2.2)) has property A (property B), then the inequality

[u^{(n)}(t) + φ(u)(t)] sign u(t) ≤ 0
([u^{(n)}(t) − φ(u)(t)] sign u(t) ≥ 0)

(2.13)

also has property A (property B).

Remark 2.2. If φ ∈ M(τ) and inequality (2.13) has a solution u : [t_{0}, +∞[ → ℝ satisfying

u^{(i)}(t)u(t) > 0 (i = 0, …, l − 1), (−1)^{i+l}u^{(i)}(t)u(t) ≥ 0 (2.14_{l})
(i = l, …, n), for t ≥ t_{0},

where l = {1, …, n−1} and l+n is odd (even), then equation (2.1) (equation (2.2)) also has a solution of the same type.

If condition (0.2) ((0.3)) is fulfilled and u : [t_{0}, +∞[→ ℝ is a proper solution of equation (0.1) not satisfying (0.4) (satisfying neither (0.4) nor (0.5)), then there exists c ∈]0, +∞[ such that 1/c ≤ u(t) ≤ ct^{n−1} for t ≥ t_{1} with t_{1} ∈ [t_{0}, +∞[ sufficiently large. Taking this fact into account and repeating the arguments we used in proving Theorems 2.1, 2.2, we can easily ascertain that the following result is valid.

Theorem 2.3. Let F ∈ V (τ), condition (0.2) ((0.3)) be fulfilled, and for any sufficiently large c ∈]0, +∞[ let there exist φ_{c} ∈ M(τ) and t_{c} ∈ ℝ_{+} such that for any u ∈ H_{t_{c},τ} satisfying

\frac{1}{c} ≤ |u(t)| ≤ ct^{n−1}, for t ≥ t_{c}  (2.15)

we have the inequality

F (u)(t) sign u(t) ≥ φ_{c}(u)(t) sign u(t) for t ≥ t_{c}  (2.16)
(F (u)(t) sign u(t) ≤ −φ_{c}(u)(t) sign u(t) for t ≥ t_{c})  (2.17)

Then if the equation

v^{(n)} + φ_{c}(v)(t) sign v(t) = 0 (v^{(n)} − φ_{c}(v)(t) sign v(t) = 0) (2.18)

has property A (property B), equation (0.1) will also have property A (property B).

2.2. Superposition Case.

Lemma 2.2. Assume that Φ ∈ M(τ), l ∈ {1, …, n − 1}, l + n is odd

\begin{matrix}μ ∈]0, 1], τ_{0} ∈ C(ℝ_{+}; ℝ_{+}), τ_{0}(t) ≤ τ(t) for t ∈ ℝ_{+}, \\ lim_{t→+∞} τ_{0}(t) = +∞ \end{matrix}  (2.19)

and φ is defined by (2.5), where for any t ∈ ℝ_{+} a function ψ_{t} : C(ℝ_{+}; ℝ) → C(ℝ_{+}; ℝ) is given by

ψ_{t}(v)(s) = [τ_{0}(t)]^{−μ(l−1)}|v(τ_{0}(t))|^{μ} sign v(τ_{0}(t))s^{l−1} for s ∈ ℝ_{+}.(2.20_{l})

Then if inequality (2.3) has a proper nonoscillatory solution u : [t_{0}, +∞[→ ℝ satisfying (2.14_{l}), then equation (2.1) also has a solution of the same type.

Proof. It can be assumed without loss of generality that u(t) > 0 for t ≥ t_{0}. Then either

∫_{t_{0}}^{+∞} s^{n−l}|u^{(n)}(s)|ds < +∞ and u(t) ≥ c_{0}t^{l−1} for t ≥ t_{1} (2.21)

with t_{1} ∈ [t_{0}, +∞[ and c_{0} ∈]0, 1], or by Lemma 1.3

\frac{u(t)}{t^{l−1}} ↑ +∞ as t ↑ +∞.  (2.22)

Let (2.21) hold. Then by (2.3) there exists t_{2} ∈ [t_{1}, +∞[ such that t_{∗} = min({t_{2}, inf{τ(t) : t ≥ t_{2}}) ≥ t_{1} and

∫_{t_{2}}^{+∞} t^{n−l}Φ(θ)(t)dt < \frac{c_{0}^{1/μ}}{2} ,  (2.23)

where θ(s) = c_{0}s^{l−1}.

Let further U be the set of all v ∈ C([t_{∗}, +∞[; ℝ) satisfying

\frac{c_{0}^{1/μ}}{2} t^{l−1} ≤ v(t) ≤ c_{0}^{1/μ}t^{l−1} for t ≥ t_{∗}.  (2.24)

Define T : U → C([t_{∗}, +∞[; ℝ) by

T (v)(t) = \begin{cases}c_{0}^{1/μ}t^{l−1} − \frac{1}{(l−2)!(n−l)!} ∫_{t_{2}}^{t}(t − s)^{l−2}× \\ × ∫_{s}^{+∞}(ξ − s)^{n−l}φ(v)(ξ)dξ ds for t ≥ t_{2} \\ c_{0}^{1/μ}t^{l−1} for t_{∗} ≤ t ≤ t_{2} \end{cases}

if l > 1 and by

T (v)(t) = \begin{cases}c_{0}^{1/μ} − \frac{1}{(n−1)!} & ∫_{t}^{+∞}(s − t)^{n−1}φ(v)(s)ds for t ≥ t_{2} \\ T (v)(t_{2}) & for t_{∗} ≤ t ≤ t_{2} \end{cases}

if l = 1. According to (2.24) \frac{c_{0}}{2^{μ}} ≤ [τ_{0}(t)]^{−μ(l−1)}[v(τ_{0}(t))]^{μ} ≤ c_{0} for t ≥ t_{2}, so that T (U) ⊂ U by (2.5), (2.20_{l}), (2.23). As in proving Theorem 1.2, we find that T has a fixed point which is a solution of (2.1) on [t_{2}, +∞[ satisfying (2.14_{l}).

If (2.22) is fulfilled, then for any sufficiently large t we have u^{(n)}(t) + φ(u)(t) ≤ 0. By Remark 2.2 we conclude that (2.1) has a solution satisfying (2.14_{l}). ■

Similar arguments can be used to prove

Lemma 2.3. Assume that Φ ∈ M(τ), l ∈ {1, …, n − 2}, l + n, is even, μ and τ_{0} satisfy (2.19) and φ is defined by (2.5), where ψ_{t} : C(ℝ_{+}; ℝ) → C(ℝ_{+}; ℝ) is given by (2.20_{l}) for any t ∈ ℝ_{+}. Then if inequality (2.4) has a proper nonoscillatory solution u : [t_{0}, +∞[→ ℝ satisfying (2.14_{l}), then equation (2.2) also has a solution of the same type.

Lemma 2.4. Assume that Φ ∈ M(τ, σ), l ∈ {1, …, n − 1}, l + n is odd,

\begin{matrix}μ ∈ [1, +∞[, τ_{0} ∈ C(ℝ_{+}; ℝ_{+}), τ_{0}(t) ≤ σ(t) \\ for t ∈ ℝ_{+}, lim_{t→+∞} τ_{0}(t) = +∞ \end{matrix}  (2.25)

and φ is defined by (2.5) where for any t ∈ ℝ_{+}

ψ_{t}(v)(s) = [σ(t)]^{−μl}|v(τ_{0}(t))|^{μ} sign v(τ_{0}(t))s^{l} for s ∈ ℝ_{+}. (2.26_{l})

Then the conclusion of Lemma 2.2 is true.

Proof. Let u : [t_{0}, +∞[→ ℝ be a proper nonoscillatory solution of (2.3) satisfying (2.14_{l}). It can be assumed that u(t) > 0 for t ≥ t_{0}. Then either (2.21) is fulfilled with t_{1} ∈ [t_{0}, +∞[ and c_{0} ∈]0, 1] or by Lemma 1.3

\frac{u(t)}{t^{l}} ↓ c_{1} ≥ 0 as t ↑ +∞.  (2.27)

Let (2.21) hold. Obviously by (2.3)

∫_{t_{0}}^{+∞} t^{n−l}Φ(θ_{t})(t)dt < +∞,  (2.28)

where θ_{t}(s) = c_{0}s^{l}/[σ(t)]^{μ}, s ∈ ℝ_{+}, for any t ≥ t_{0}. Using (2.8) and repeating arguments from the proof of Lemma 2.2, we find that the equation

v^{(n)}(t) + Φ(\overline{ξ}_{t}(v))(t) = 0,  (2.29)

where

\overline{ξ}_{t}(v)(s) = [σ(t)]^{−μl}|v(σ(t))|^{μ} sign v(σ(t))s^{l} for s ∈ ℝ_{+} (2.30)

for any t ∈ ℝ_{+}, has a solution v : [t_{2}, +∞[→]0, +∞[ satisfying (2.14_{l}), where t_{2} ∈ [t_{1}, +∞[ is sufficiently large.

On the other hand, since τ_{0}(t) ≤ σ(t) for t ∈ ℝ_{+}, by (2.5) and (2.26_{l}) v satisfies

v^{(n)}(t) + φ(v)(t) ≤ 0  (2.31)

on [t_{2}, +∞[. Following Remark 2.2 equation (2.1) has a solution of type (2.14_{l}).

Now consider the case when (2.27) is fulfilled. Assume at first that c_{1} > 0. Then there exists t_{1} ∈ [t_{0}, +∞[ such that u(t) ≥ c_{1}t^{l}/2 for t ≥ t_{1}. Therefore by (2.1) and Lemma 1.3 we have

∫_{t_{1}}^{+∞} t^{n−l−1}Φ(θ)(t)dt < +∞,

where θ(s) = c_{1}s^{l}/2 for s ∈ ℝ_{+}. Choose t_{2} ∈ [t_{1}, +∞[ such that t_{∗} = min(t_{2}, inf{τ(t) : t ≥ t_{2}}) ≥ t_{1} and

∫_{t_{2}}^{+∞} t^{n−l−1}Φ(θ)(t)dt < \frac{c_{1}^{1/μ}}{4} .  (2.32)

Let U be the set of all v ∈ C([t_{∗}, +∞[; ℝ) satisfying

\frac{c_{1}^{1/μ}}{4} t^{l} ≤ v(t) ≤ \frac{c_{1}^{1/μ}}{2} t^{l} for t ≥ t_{∗}.  (2.33)

Define T : U → C([t_{∗}, +∞[; ℝ) by

T (v)(t) = \begin{cases}\frac{c_{1}^{1/μ}}{4} t^{l} − \frac{1}{(l−1)!(n−l−1)!} ∫_{t_{2}}^{t}(t − s)^{l−1} ∫_{s}^{+∞}(ξ − s)^{n−l−1}× \\ ×φ(\overline{ψ}_{ξ}(v))(ξ)dξ ds for t ≥ t_{2} \\ \frac{c_{1}^{1/μ}}{4} t^{l} for t_{∗} ≤ t ≤ t_{2}, \end{cases}(2.34)

where for any t ∈ ℝ_{+}, \overline{ψ}_{t} is given by (2.30).

By (2.33) it is clear that for any v ∈ U we have

\frac{c_{1}}{4^{μ}} ≤ [σ(t)]^{−μl}[v(σ(t))]^{μ} ≤ \frac{c_{1}}{2^{μ}} for t ≥ t_{2},

and therefore (2.32) and (2.34) imply that T (U) ⊂ U. As we did previously in this section, by Lemma 2.1 we ascertain that T has a fixed point v which is a solution of (2.29) of type (2.14_{l}) on [t_{2}, +∞[. Thus v satisfies (2.31) on [t_{2}, +∞[. Hence on account of Remark 2.2 equation (2.1) also has a solution of type (2.14_{l}).

Now we shall consider the last remaining case c_{1} = 0. Since we have u(t)/t^{l} ≤ 1 for any sufficiently large t, u satisfies inequality (2.31) for large t. By Remark 2.2 equation (2.1) has a solution of type (2.14_{l}). ■

In a mannar similer to the above we can prove

Lemma 2.5. Assume that Φ ∈ M(τ, σ), l ∈ {1, …, n−2}, l +n is even, μ and τ_{0} satisfy (2.25) and φ is defined by (2.5), where ψ_{t} : C(ℝ_{+}; ℝ) → C(ℝ_{+}; ℝ) is given by (2.26) for any t ∈ ℝ_{+}. Then the conclusion of Lemma 2.3 is true.

Theorem 2.4. Assume that Φ ∈ M(τ), μ and τ_{0} satisfy (2.19), φ is defined by (2.5) where ψ_{t} is given by (2.20_{1}) for any t ∈ ℝ_{+}, and equation (2.1) has property A. Then inequality (2.3) also has property A.

Proof. Let u : [t_{0}, +∞[→ ℝ be a proper nonoscillatory solution of (2.3). Then by Lemma 1.1 there exists l ∈ {0, …, n − 1} such that l + n is odd and (2.14_{l}) holds. Assume that u(t) > 0 for t ≥ t_{0}.

Let l ∈ {1, …, n − 1}. Then by Lemma 2.2 equation (2.1) with φ defined by (2.5) and (2.20_{l}) has a solution v : [t_{1}, +∞[→]0, +∞[ of type (2.14_{l}), where t_{1} ∈ [t_{0}, +∞[ is sufficiently large. Since

(\frac{s}{[τ_{0}(t)]^{μ}} )^{l−1} ≥ 1 for s ≥ τ(t),

v satisfies (2.31) on [t_{1}, +∞[ with φ defined by (2.5) and (2.20_{1}). Following Remark 2.2, equation (2.1) with φ defined by (2.5) and (2.20_{1}) also has a solution of type (2.14_{l}). But this contradicts property A of this equation stated in the conditions of the theorem. Therefore we have proved that l ∉ {1, …, n − 1}.

Now assume that l = 0, n is odd and u(t) ↓ c > 0 as t ↑ +∞. By (2.3) and Lemma 1.2 we have

∫ ^{+∞} t^{n−1}Φ(\frac{c}{2})(t)dt < +∞.

Therefore by Lemma 4.1 to be proved later in §4 equation (2.1) with φ defined by (2.5) and (2.20_{1}) has a solution v of type (2.14_{0}) such that lim_{t→+∞} v(t) ≠ 0. But this again contradicts property A of this equation. The obtained contradiction proves that (0.4) holds for l = 0. ■

Theorem 2.4′. Assume that Φ ∈ M(τ, σ), μ and τ_{0} satisfy (2.25) φ is defined by (2.5), where ψ_{t} is given by (2.26_{n−1}) for any t ∈ ℝ_{+} and equation (2.1) has property A. Then inequality (2.3) also has property A.

Proof. Let u : [t_{0}, +∞[ be a proper nonoscillatory solution of (2.3). Then by Lemma 1.1 there exists l ∈ {0, …, n − 1} such that l + n is odd and (2.14_{l}) holds. It can be assumed that u(t) > 0 for t ≥ t_{0}.

Assume that l ∈ {1, …, n − 1}. By Lemma 2.4 equation (2.1) with φ defined by (2.5) and (2.26_{l}) has a solution v : [t_{1}, +∞[→]0, +∞[ of type (2.14_{l}), where t_{1} ∈ [t_{0}, +∞[ is sufficiently large. Since

(\frac{s}{[σ(t)]^{μ}} )^{n−1} ≤ (\frac{s}{[σ(t)]^{μ}} )^{l} for τ(t) ≤ s ≤ σ(t),

v satisfies (2.31) on [t_{1}, +∞[ with φ defined by (2.5) and (2.26_{n−1}). By Remark 2.2 equation (2.1) with φ defined by (2.5) and (2.26_{n−1}) also has a solution of type (2.14_{l}). But this contradicts property A of this equation stated in the conditions of the theorem.

Assuming that l = 0 and n is odd and applying the arguments from the proof of Theorem 2.4, we can show that condition (0.4) is fulfilled. ■

Theorem 2.5. Assume that Φ ∈ M(τ), μ and τ_{0} satisfy (2.19), φ is defined by (2.5) where ψ_{t} is given by (2.20_{1}) for any t ∈ ℝ_{+}, and equation (2.2) has property B. Then inequality (2.4) also has property B.

Proof. Let u : [t_{0}, +∞[→ ℝ be a proper nonoscillatory solution of (2.4). By Lemma 1.1 there exists l ∈ {0, …, n} such that l + n is even and (2.14_{l}) holds. As in the proof of Theorem 2.4, we can show that l ∉ {1, …, n − 2}, and if n is even and l = 0, then (0.4) is fulfilled.

Assume now that l = n and lim_{t→+∞} |u^{(n−1)}(t)| < +∞. By (2.14_{n}) there exist c_{0} ∈]0, +∞[ and t_{2} ∈]t_{1}, +∞[ such that |u(t)| ≥ c_{0}t^{n−1} for t ≥ t_{2}, and therefore (2.4) implies

∫ ^{+∞} |φ(θ)(t)|dt < +∞  (2.35)

with θ(s) = c_{0} sign u(t_{2})s^{n−1} for s ∈ ℝ_{+}.

On the other hand, by Lemma 4.1 and (2.35) equation (2.2) with φ defined by (2.5) and (2.20_{1}) has a solution v of type (2.14_{n}) satisfying lim_{t→+∞} |v^{(n−1)}(t)| < +∞. But this contradicts property B of this equation. ■

The proof of Theorem 2.4 has been a guide for us in proving Theorem 2.4′. In the same way we shall be guided by the proof of Theorem 2.5 to show that the next theorem is valid.

Theorem 2.5′. Assume that φ ∈ M(τ, σ), μ and τ_{0} satisfy (2.25), φ is defined by (2.5), where ψ_{t} is given by (2.26_{n−1}) for any t ∈ ℝ_{+}, and equation (2.2) has property B. Then inequality (2.4) also has property B.

§ 3. Sufficient Conditions

3.1. Ineffective Sufficient Conditions. In this subsection we shall derive the ineffective sufficient conditions for the equations

u^{(n)} + φ(u)(t) = 0  (3.1)
u^{(n)} − φ(u)(t) = 0  (3.2)

to have property A and B, respectively. These conditions will be stated in terms of the classes introduced in 1.2 and be used in 3.2 to obtain the effective sufficient conditions for (0.1).

Lemma 3.1. Let φ ∈ M(τ), l ∈ {1, …, n − 1} and l + n be odd. Then the condition

φ̃_{l} ∈ M_{1}(τ)  (3.3_{l})

is sufficient for equation (3.1) not to have a proper solution satisfying (2.14_{l}), where for any u ∈ C(ℝ_{+}; ℝ) and t ∈ ℝ_{+}

φ̃_{l}(u)(t) = \frac{t^{n−l−1}}{l!(n − l)!}φ(αu)(t) with α(s) = s^{l} for s ∈ ℝ_{+}. (3.4_{l})

Proof. Let, on the contrary, (3.1) have a solution u : [t_{0}, +∞[→ ℝ satisfying (2.14_{l}). It can be assumed without loss of generality that (2.8_{l}) is fulfilled, where t_{1} is sufficiently large. By (3.3_{l}), (3.4_{l}) and Lemmas 1.4, 1.5 for any c ≠ 0 we have

∫ ^{+∞} t^{n−l}φ(θ)(t)dt = +∞, with l; θ(s) = cs^{l−1} for s ∈ ℝ_{+}. (3.5)

(2.8_{l}) implies that there exist c_{0} ∈]0, +∞[ and t_{2} ∈ [t_{1}, +∞[ such that u(t) ≥ c_{0}t^{l−1} for t ≥ t_{2}. Therefore by (3.1), (3.5) we have

∫ ^{+∞} t^{n−l}|u^{(n)}(t)|dt = +∞.  (3.6)

Hence by Lemma 1.3 u(t)/t^{l} ↓ as t ↑ +∞ and

u(t) ≥ \frac{t^{l}}{l!(n − l)!} ∫_{t}^{+∞} s^{n−l−1}|u^{(n)}(s)|ds for t ≥ t_{∗},

where t_{∗} ∈ [t_{2}, +∞[ is sufficiently large, so that by (3.1), (3.4_{l}) we obtain

x(t) ≥ ∫_{t}^{+∞} φ̃_{l}(x)(s)ds for t ≥ t_{∗},

where x(t) = u(t)/t^{l} is nonincreasing. This means that φ̃_{l} ∉ M_{1}(τ), which contradicts (3.3_{l}). ■

In a similar way we can prove

Lemma 3.1′. Let φ ∈ M(τ), l ∈ {1, …, n − 2} and l + n be even. Then condition (3.3_{l}) with φ̃_{l} defined by (3.4_{l}) is sufficient for equation (3.2) not to have a proper solution satisfying (2.14_{l}).

Lemma 3.2. Let φ ∈ M(τ), l ∈ {1, …, n − 1} and l + n be odd. Let, moreover, β ∈ C(ℝ_{+}; ℝ_{+}) be a nondecreasing function satisfying (1.13). Then the condition

φ̃_{l} ∈ M_{2}^{β}(τ)  (3.7_{l})

is sufficient for equation (3.1) not to have a proper solution satisfying (2.14_{l}), where for any u ∈ C(ℝ_{+}; ℝ) and t ∈ ℝ_{+} we have

φ̃_{l}(u)(t) = \frac{β^{n−l}(t)}{l!(n − l)!}φ(αu)(t) with α(s) = s^{l−1} for s ∈ ℝ_{+}. (3.8_{l})

Proof. Let, on the contrary, (3.1) have a solution u : [t_{0}, +∞[→ ℝ satisfying (2.14_{l}). It can be assumed that (2.8_{l}) is fulfilled. Condition (3.6) is valid on account of (3.1), (3.7_{l}), (3.8_{l}) and Lemma 1.4, so that Lemma 1.3 implies

u(β(t)) ≥ \frac{β^{l−1}(t)}{l!(n − l)!} ∫_{t_{1}}^{t} β^{n−l}(s)|u^{(n)}(s)|ds for t ≥ t_{1},

where t_{1} ∈ [t_{0}, +∞[ is sufficiently large. Hence by (3.1) and (3.8_{l}) we have

x(β(t)) ≥ ∫_{t_{1}}^{t} φ̃_{l}(x)(s)ds for t ≥ t_{1},

where x(t) = u(t)/t^{l−1} is a nondecreasing function by Lemma 1.3. But this means that φ̃_{l} ∉ M_{2}^{β}(τ), which contradicts (3.7_{l}). ■

The next lemma can be proved similarly.

Lemma 3.2′. Let φ ∈ M(τ), l ∈ {1, …, n − 2} and l + n be even. Then condition (3.7_{l}), where φ̃_{l} is defined by (3.8_{l}), is sufficient for equation (3.2) not to have a proper solution satisfying (2.14_{l}).

Theorem 3.1. Let φ ∈ M(τ) and (3.3_{n−1}) be fulfilled, where φ̃_{n−1} is defined by (3.4_{n−1}). Then equation (3.1) has property A.

Proof. According to Lemma 1.5 condition (3.3_{l}) is fulfilled for any l ∈ {0, …, n − 1} with φ̃_{l} defined by (3.4_{l}).

Now assume that u : [t_{0}, +∞[→ ℝ is a proper nonoscillatory solution of (3.1). By Lemma 1.1 there exists l ∈ {0, …, n − 1} such that l + n is odd and (2.14_{l}) holds. On account of Lemma 3.1 we have l ∉ {1, …, n − 1}. Thus it can be assumed that l = 0 and n is odd.

Let u(t) ↓ c_{0} ≠ 0 as t ↑ +∞. Since φ̃_{0} ∈ M_{1}(τ), by Lemma 1.4 we have

∫ ^{+∞} |φ̃_{0}(c)(t)|dt = +∞ for any c ≠ 0.  (3.9)

Obviously there exists t_{1} ∈ [t_{0}, +∞[ such that |u(t)| ≥ c_{0}/2 for t ≥ t_{1} and therefore by (3.1) and (3.9)

∫_{t_{2}}^{+∞} t^{n−1}|u^{(n)}(t)|dt ≥ ∫_{t_{2}}^{+∞} |φ̃_{0}(\frac{c_{0}}{2} )(t)|dt = +∞.

This however contradicts (1.3). The obtained contradiction proves that (0.4) holds if n is odd and l = 0. ■

Theorem 3.2. Let φ ∈ M(τ) and (3.3_{n−2}) be satisfied, where for any u ∈ C(ℝ_{+}; ℝ) and t ∈ ℝ_{+}

φ̃_{n−2}(u)(t) = \frac{t}{(n − 1)!}φ(αu)(t) with α(s) = s^{n−2} for s ∈ ℝ_{+}. (3.10)

Moreover, let for any c ≠ 0

∫ ^{+∞} |φ(θ)(t)|dt = +∞ with θ(s) = cs^{n−1} for s ∈ ℝ_{+}. (3.11)

Then equation (3.2) has property B.

Proof. By Lemma 1.5, (3.3_{n−2}) and (3.10) condition (3.3_{l}) is fulfilled for any l ∈ {0, …, n − 2} with φ̃_{l} defined by (3.4_{l}).

Assume now that u : [t_{0}, +∞[→ ℝ is a proper nonoscillatory solution of (3.2). Then by Lemma 1.1 there exists l ∈ {0, …, n} such that l + n is even and (2.14_{l}) holds. By Lemma 3.1′ l ∉ {1, …, n − 2}. As while proving Theorem 3.1, it can be shown that (0.4) holds if n is even and l = 0.

To complete the proof it remains to show that (0.5) is fulfilled for l = n. Indeed, by (2.14_{n}) there exists t_{1} ∈ [t_{0}, +∞[ and c_{0} ∈]0, +∞[ such that |u(t)| ≥ c_{0}t^{n−1} for t ≥ t_{1}. Therefore from (3.2) we have

|u^{(n−1)}(t)| ≥ ∫_{t_{2}}^{t} |φ(θ)(s)|ds,

where θ(s) = c_{0} sign u(t_{1})s^{n−1} and t_{2} ∈ [t_{1}, +∞[ is chosen such that inf{τ(t) : t ≥ t_{2}} ≥ t_{1}. By (2.14_{n}) and (3.11) the latter inequality obviously implies (0.5). ■

Theorem 3.1′. Let φ ∈ M(τ), τ_{0} satisfy (2.19) and (3.3_{n−1}) be fulfilled, where for any u ∈ C(ℝ_{+}; ℝ) and t ∈ ℝ_{+}

\begin{matrix}φ̃_{n−1}(u)(t) = \frac{1}{(n − 1)!}φ(ψ_{t}(u))(t) \\ with ψ_{t}(u)(s) = τ_{0}^{n−1}(t)u(τ_{0}(t)) for s ∈ ℝ_{+}. \end{matrix}  (3.12)

Proof. By Theorem 3.1, (3.3_{n−1}) and (3.12) the equation

v^{(n)} + φ(ψ_{t}(v))(t) = 0,  (3.13)

where ψ_{t}(v)(s) = v(τ_{0}(t)), for any v ∈ C(ℝ_{+}; ℝ) and t ∈ ℝ_{+}, has property A. Therefore by Theorem 2.4 (μ = 1) equation (3.1) also has property A. ■

Theorem 3.2′. Let φ ∈ M(τ), τ_{0} satisfy (2.19), let (3.11) hold for any c ≠ 0 and (3.3_{n−2}) be fulfilled, where for any u ∈ C(ℝ_{+}; ℝ) and t ∈ ℝ_{+}

\begin{matrix}φ̃_{n−2}(u)(t) = \frac{t}{(n − 1)!}φ(ψ_{t}(u))(t) \\ with ψ_{t}(u)(s) = τ_{0}^{n−2}(t)u(τ_{0}(t)) for s ∈ ℝ_{+}. \end{matrix}  (3.14)

Then equation (3.2) has property B.

Proof. By Theorem 3.2, (3.3_{n−2}) and (3.11) the equation

v^{(n)} − φ(ψ_{t}(v))(t) = 0,  (3.15)

where ψ_{t} is defined by (2.20_{1}) (μ = 1) has property B. Therefore according to Theorem 2.5 equation (3.2) also has property B. ■

Theorem 3.1′′. Let φ ∈ M(τ, σ), τ_{0} satisfy (2.25) and

φ̃_{n−1} ∈ M_{1}(τ, σ),  (3.16)

where for any u ∈ C(ℝ_{+}; ℝ) and t ∈ ℝ_{+}

\begin{matrix}φ̃_{n−1}(u)(t) = \frac{1}{(n − 1)!}φ(ψ_{t}(u))(t) \\ with ψ_{t}(u)(s) = [σ(t)]^{1−n}[τ_{0}(t)]^{n−1}u(τ_{0}(t))s^{n−1}. \end{matrix}  (3.17)

Then equation (3.1) has property A.

Proof. By Theorem 3.1, (3.16) and (3.17) equation (3.13) with ψ_{t} defined by (2.26_{n−1}) (μ = 1) has property A. Therefore by Theorem 2.4′ equation (3.1) also has property A. ■

Theorem 3.2′′. Let φ ∈ M(τ, σ), τ_{0} satisfy (2.25) and

φ̃_{n−2} ∈ M_{1}(τ, σ),  (3.18)

where for any u ∈ C(ℝ_{+}; ℝ) and t ∈ ℝ_{+}

\begin{matrix}φ̃_{n−2}(u)(t) = \frac{t}{(n − 1)!}φ(ψ_{t}(u))(t) \\ with ψ_{t}(u)(s) = [σ(t)]^{2−n}[τ_{0}(t)]^{n−2}u(τ_{0}(t))s^{n−2} for s ∈ ℝ_{+}. \end{matrix} (3.19)

Moreover, let (3.11) hold for any c ≠ 0. Then equation (3.2) has property B.

Proof. By Theorem 3.2, (3.18) and (3.19) equation (3.15) with ψ_{t} defined by (2.26_{n−2}) (μ = 1) has property B. Therefore by Theorem 2.5′ equation (3.2) also has property B. ■

Theorem 3.3. Let φ ∈ M(τ) and (3.7_{1}) be fulfilled with φ̃_{1} defined by (3.8_{1}). Then equation (3.1) has property A.

Proof. By Lemma 1.7 and (3.7_{1}) condition (3.7_{l}) is fulfilled for any l ∈ {1, …, n − 1}, where φ̃_{l} is defined by (3.8_{l}).

Assume that u : [t_{0}, +∞[→ ℝ is a proper nonoscillatory solution of (3.1). Then by Lemma 1.1 there exists l ∈ {0, …, n−1} such that l+n is odd and (2.14_{l}) is satisfied. According to Lemma 3.2 l ∉ {1, …, n − 1}. Condition (3.7_{1}) and Lemma 1.4 imply that

∫ ^{+∞} |φ̃_{1}(c)(t)|dt = +∞  (3.20)

for any c ≠ 0. Therefore, as while proving Theorem 3.1, it can be shown that (0.4) is satisfied if n is odd and l = 0. ■

Theorem 3.4. Let φ ∈ M(τ) and (3.7_{1}) be fulfilled with φ̃_{1} defined by (3.8_{1}). Then equation (3.2) has property B.

Proof. As above, by Lemma 1.7 and (3.7_{1}) condition (3.7_{l}) is fulfilled for any l ∈ {1, …, n}, where φ̃_{l} is defined by (3.8_{l}).

Assume that u : [t_{0}, +∞[→ ℝ is a proper nonoscillatory solution of (3.2). Then by Lemma 1.1 there exists l ∈ {0, …, n} such that l + n is even and (2.14_{l}) holds. By Lemma 3.1′ we have l ∉ {1, …, n − 2}.

On the other hand, since φ̃_{l} ∈ M_{2}^{β}(τ) (l = 1, …, n), by Lemma 1.4 condition (3.20) holds and

∫ ^{+∞} |φ̃_{n}(c)(t)|dt = +∞

for any c ≠ 0.

As while proving Theorem 3.2, it can be shown that (0.4) ((0.5)) is satisfied if n is even and l = 0 (l = n). ■

Theorem 3.3′. Let φ ∈ M(τ), τ_{0} satisfy (2.19) and (3.7_{1}) be fulfilled, where

\begin{matrix}φ̃_{1}(u)(t) = \frac{[β(t)]^{n−1}}{(n − 1)!} φ(ψ_{t}(u))(t) \\ with ψ_{t}(u)(s) = u(τ_{0}(t)) for s ∈ ℝ_{+} \end{matrix}  (3.21)

for any u ∈ C(ℝ_{+}; ℝ) and t ∈ ℝ_{+}. Then equation (3.1) has property A.

Proof. By Theorem 3.3, (3.7_{1}) and (3.21) the equation v^{(n)}(t)+φ(ψ_{t}(v))(t)= 0 with ψ_{t} defined by (2.20_{1}) (μ = 1) has property A. Therefore by Theorem 2.4 equation (3.1) also has property A. ■

In a similar manner one can prove

Theorem 3.4′. Let φ ∈ M(τ), τ_{0} satsify (2.19) and condition (3.7_{1}) be fulfilled with φ_{1} defined by (3.21). Then equation (3.2) has property B.

Theorem 3.5. Let φ ∈ M(τ, σ), τ_{0} satisfy (2.25) and

φ̃_{1} ∈ M_{2}^{β}(τ, σ),  (3.22)

where for any u ∈ C(ℝ_{+}; ℝ) and t ∈ ℝ_{+}

\begin{matrix}φ̃_{1}(u)(t) = \frac{[β(t)]^{n−1}}{(n − 1)!} φ(ψ_{t}(u))(t) \\ with ψ_{t}(u)(s) = [σ(t)]^{1−n}u(τ_{0}(t))s^{n−1} for s ∈ ℝ_{+}. \end{matrix}  (3.23)

Then equation (3.1) has property A.

Proof. The equation v^{(n)}(t) + φ(ψ_{t}(u))(t) = 0 with ψ_{t} defined by (2.26_{n−1}) (μ = 1) has property A by virtue of Theorem 3.3, (3.22) and (3.23). Therefore by Theorem 2.4′ equation (3.1) also has property A. ■

Quite similarly one can prove

Theorem 3.6. Let φ ∈ M(τ, σ), τ_{0} satisfy (2.25) and let (3.22) be fulfilled, where

φ̃_{1}(u)(t) = \frac{[β(t)]^{n−1}}{(n − 1)!} φ(ψ_{t}(u))(t)

with ψ_{t}(u)(s) = [σ(t)]^{2−n}u(τ_{0}(t))s^{n−2} for s ∈ ℝ_{+}

for any u ∈ C(ℝ_{+}; ℝ) and t ∈ ℝ_{+}. Then equation (3.2) has property B.

We conclude this subsection by a general theorem concerning equation (0.1).

Theorem 3.7. Let F ∈ V (τ), condition (0.2) ((0.3)) be fulfilled, and for any sufficiently large c > 0 let there exist t_{c} ∈ ℝ_{+} and φ_{c} ∈ M(τ) such that inequality (2.16) ((2.17)) holds for any u ∈ H_{t_{c},τ} satisfying (2.15). Then for equation (0.1) to have property A (B) it is sufficient that φ_{c} satisfy the conditions of anyone of Theorems 3.1, 3.1′, 3.1′′, 3.3, 3.3′, 3.5 (3.2, 3.2′, 3.2′′, 3.4, 3.4′, 3.5).

This theorem immediately follows from Theorems 2.1 and 2.3 (2.2 and 2.3).

Corollary 3.1. Let F ∈ V (τ), condition (0.2) ((0.3)) be fulfilled, and let for any sufficiently large c > 0 there exist t_{c} ∈ ℝ_{+} and a_{c} ∈ L_{loc}(ℝ_{+}; ℝ_{+}) such that for any u ∈ H_{t_{0},τ} satisfying 1/c ≤ |u(t)| ≤ ct^{n−1} for t ≥ t_{c}

|F (u)(t)| ≥ a_{c}(t) for t ≥ t_{c}.  (3.24)

Then the condition

∫ ^{+∞} a_{c}(t)dt = +∞  (3.25)

is sufficient for equation (0.1) to have property A (B).

Proof. Assume that c > 0 is sufficiently large. Then by (0.2), (3.24) ((0.3), (3.24)) inequality (2.16) ((2.17)) holds with

φ_{c}(u)(t) = \frac{a_{c}(t)|u(t)|^{λ} sign u(t)}{c^{λ}t^{λ(n−1)}}, λ ∈]0, 1[.

By Corollary 1.1 of Lemma 1.6 and (3.25) φ_{l} satisfies the conditions of Theorem 3.1′ (Theorem 3.2′). Therefore by Theorem 3.7 equation (0.1) has property A (B). ■

3.2. Effective Sufficient Conditions. In the theorems throughout this and next subsections the following conditions will appear:

\begin{matrix}m ∈ ℕ, τ_{i}, σ_{i} ∈ C(ℝ_{+}; ℝ_{+}), τ_{i}(t) ≤ σ_{i}(t) for t ∈ ℝ_{+}, \\ lim_{t→+∞} τ_{i}(t) = +∞ (i = 1, …, m); \end{matrix} (3.26)
r_{ic} : ℝ_{+}^{2} → ℝ is measurable, r_{i}(·, t) is nondecreasing

for t ∈ ℝ_{+} (i = 1, …, m); (3.27)

ω_{i} ∈ C(ℝ_{+}; ℝ_{+}), ω_{i}(0) = 0, ω_{i}(s) > 0
for s > 0, ω_{i} is nondecreasing(i = 1, …, m).

(3.28)

Theorem 3.8. Let F ∈ V (τ), condition (0.2) ((0.3)) be fulfilled, and let for any sufficiently large c > 0 there exist t_{c} ∈ ℝ_{+} such that for any u ∈ H_{t_{c},τ} satisfying 1/c ≤ |u(t)| ≤ ct^{n−1} for t ≥ t_{c} we have

|F (u)(t)| ≥ ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} ω_{ic}(τ_{∗}^{1−n}(t)|u(s)|)d_{s}r_{ic}(s, t) for t ≥ t_{c}, (3.29)

where (3.26)–(3.28) hold, τ_{∗}(t) = min{τ_{i}(t), t (i = 1, …, m)} and

∫_{0}^{1} \frac{ds}{∏_{i=1}^{m} ω_{ic}(s)} < +∞,  (3.30)

∫ ^{+∞} ∏_{i=1}^{m}[r_{ic}(σ_{i}(t), t) − r_{ic}(τ_{i}(t), t)]dt = +∞.  (3.31)

Then equation (0.1) has property A (B).

Proof. By (0.2), (3.29) ((0.3), (3.29)) inequality (2.16) ((2.17)) holds with

φ_{c}(u)(t) = ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} ω_{ic}(τ_{∗}^{1−n}(t)|u(s)|) sign u(s)d_{s}r_{ic}(s, t).

On the other hand, by (3.30), (3.31) and Lemma 1.6 we have φ̃_{c n−1} ∈ M_{1}(τ), where

φ̃_{c n−1}(u)(t) = \frac{1}{(n − 1)!} ∏_{i=1}^{m} ω_{ic}(u(τ_{∗}(t))) sign u(τ_{∗}(t)) ×

× ∏_{i=1}^{m}[r_{ic}(σ_{i}(t), t) − r_{ic}(τ_{i}(t), t)].  (3.32)

Since φ_{c} satisfies the conditions of Theorem 3.1′ (Theorem 3.2′), equation (0.1) has property A (B) on account of Theorem 3.7. ■

Corollary 3.2. Let F ∈ V (τ), condition (0.2) ((0.3)) be fulfilled, and let for any sufficiently large c > 0 there exist t_{c} ∈ ℝ_{+} such that for any u ∈ H_{t_{c},τ} satisfying 1/c ≤ |u(t)| ≤ ct^{n−1} for t ≥ t_{c} we have

|F (u)(t)| ≥ ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} |u(s)|^{λ_{ic}}d_{s}r_{ic}(s, t)  (3.33)

where (3.26) and (3.27) hold, λ_{ic} ∈]0, 1[ (i = 1, …, m) and

\begin{matrix}λ_{c} = ∑_{i=1}^{m} λ_{ic} < 1, \\ ∫ ^{+∞} τ_{∗}^{(n−1)λ_{c}}(t) ∏_{i=1}^{m}[r_{ic}(σ_{i}(t), t) − r_{ic}(τ_{i}(t), t)]dt = +∞ \end{matrix} (3.34)

with τ_{∗}(t) = min{τ_{i}(t), t (i = 1, …, m)}. Then equation (0.1) has property A (B).

Proof. It suffices to note that by (3.33) and (3.34) all the conditions of Theorem 3.8 are satisfied with ω_{ic}(s) = s^{λ_{ic}} for s ∈ ℝ_{+} (i = 1, …, m). ■

Corollary 3.3. Let F ∈ V (σ), condition (0.2) ((0.3)) be fulfilled, and there exist t_{0} ∈ ℝ_{+} such that for any u ∈ H_{t_{0},τ} we have

|F (u)(t)| ≥ p(t)|u(τ(t))|^{λ} for t ∈ [t_{0}, +∞[,  (3.35)

where p ∈ L_{loc}(ℝ_{+}; ℝ_{+}) and λ ∈]0, 1[. Then the condition

∫ ^{+∞} τ_{∗}^{(n−1)λ}(t)p(t)dt = +∞,

where τ_{∗}(t) = min{t, τ(t)}, is sufficient for equation (0.1) to have property A (B).

Theorem 3.9. Let F ∈ V (τ), condition (0.2) be fulfilled, and let for any sufficiently large c > 0 there exist t_{c} ∈ ℝ_{+} such that for any u ∈ H_{t_{c},τ} satisfying 1/c ≤ |u(t)| ≤ ct^{n−1} for t ≥ t_{c} we have

|F (u)(t)|≥∏_{i=1}^{m}∫_{τ_{i}(t)}^{σ_{i}(t)}ω_{ic}([σ^{∗}(t)]^{n−1}[σ_{∗}(t)]^{1−n}s^{1−n}|u(s)|)d_{s}r_{ic}(s, t) (3.36)
for t ≥ t_{c},

where (3.26)–(3.28) hold, σ^{∗}(t) = max{σ_{i}(t) : i = 1, …, m}, σ_{∗}(t) = min{σ^{∗}(t), t} and (3.30), (3.31) are satisfied. Then equation (0.1) has property A.

Proof. By (0.2) and (3.36) inequality (2.16) holds with

φ_{c}(u)(t)=∏_{i=1}^{m}∫_{τ_{i}(t)}^{σ_{i}(t)}ω_{ic}([σ^{∗}(t)]^{n−1}[σ_{∗}(t)]^{1−n}s^{1−n}|u(s)|)sign u(s)d_{s}r_{ic}(s, t).

On the other hand, by (3.30), (3.31) and Lemma 1.6 we have φ̃_{c n−1} ∈ M_{1}(τ_{∗}, σ^{∗}), where τ_{∗} = min{τ_{i}(t) : i = 1, …, m} and φ̃_{c n−1} is defined by (3.32). Since φ_{c} satisfies the conditions of Theorem 3.1′′, equation (0.1) has property A on account of Theorem 3.7. ■

Corollary 3.4. Let F ∈ V (τ), condition (0.2) be satisfied, and let for any sufficiently large c > 0 there exist t_{c} ∈ ℝ_{+} such that for any u ∈ H_{t_{c},τ} satisfying 1/c ≤ |u(t)| ≤ ct^{n−1} for t ≥ t_{c} inequality (3.33) is fulfilled, where (3.26) and (3.27) hold, λ_{i} ∈]0, 1[ (i = 1, …, m) and

\begin{matrix}λ_{c} = ∑_{i=1}^{m} λ_{ic} < 1, \\ ∫ ^{+∞}[σ_{∗}(t)]^{λ_{c}(n−1)}[σ^{∗}(t)]^{λ_{c}(1−n)}∏_{i=1}^{m}∫_{τ_{i}(t)}^{σ_{i}(t)}s^{λ_{ic}(n−1)}d_{s}r_{ic}(s, t)dt=+∞, \end{matrix}(3.37)

with σ^{∗}(t) = max{σ_{i}(t) : i = 1, …, m}, σ_{∗}(t) = min{σ^{∗}(t), t}. Then equation (0.1) has property A.

Proof. By (0.2) and (3.33) inequality (2.16) holds with

φ_{c}(u)(t) = ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} |u(s)|^{λ_{ic}} sign u(s)d_{s}r_{ic}(s, t).

On the other hand, by (3.37) and Corollary 1.1 of Lemma 1.6 we have φ̃_{c n−1} ∈ M_{1}(τ_{∗}, σ^{∗}), where τ_{∗} = min{τ_{i}(t) : i = 1, …, m} and

φ̃_{c n−1}(v)(t) = [σ_{∗}(t)]^{λ_{c}(n−1)}[σ^{∗}(t)]^{λ_{c}(1−n)} ×

× ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} |v(s)|^{λ_{ic}} sign v(s)d_{s}r_{ic}(s, t).

Thus we see that φ_{c} satisfies the conditions of Theorem 3.1′′. Therefore equation (0.1) has property A on account of Theorem 3.7. ■

Theorem 3.10. Let F ∈ V (τ), condition (0.3) be fulfilled, and let for any sufficiently large c > 0 there exist t_{c} ∈ ℝ_{+} such that for any u ∈ H_{t_{c},τ} satisfying 1/c ≤ |u(t)| ≤ ct^{n−1} for t ≥ t_{c} we have

|F (u)(t)|≥∏_{i=1}^{m}∫_{τ_{i}(t)}^{σ_{i}(t)} ω_{ic}([σ^{∗}(t)]^{n−2}[σ_{∗}(t)]^{2−n}s^{2−n}|u(s)|)d_{s}r_{ic}(s, t) (3.38)
for t ∈ [t_{c}, +∞[,

where (3.26)–(3.28) hold and σ^{∗}(t) = max{σ_{i}(t) : i = 1, …, m}, σ_{∗}(t) = min{σ^{∗}(t), t}. Moreover, let conditions (3.30) and

∫ ^{+∞} t ∏_{i=1}^{m}[r_{ic}(σ_{i}(t), t) − r_{ic}(τ_{i}(t), t)]dt = +∞,  (3.39)

∫ ^{+∞} ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} ω_{ic}([σ^{∗}(t)]^{n−2}[σ_{∗}(t)]^{2−n}s)d_{s}r_{ic}(s, t)dt = +∞ (3.40)

be fulfilled. Then equation (0.1) has property B.

Proof. By (0.3) and (3.38) inequality (2.17) holds with

φ_{c}(u)(t) = ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} ω_{ic}([σ^{∗}(t)]^{n−2}[σ_{∗}(t)]^{2−n}s^{2−n}|u(s)|) sign u(s)d_{s}r_{ic}(s, t).

On the other hand, (3.30) and (3.39) imply φ̃_{c n−2} ∈ M_{1}(τ_{∗}, σ^{∗}), where τ_{∗}(t) = min{τ_{i}(t) : i = 1, …, m} and

φ̃_{c n−2}(v)(t) = \frac{t}{(n − 1)!} ∏_{i=1}^{m} ω_{ic}(v(σ_{∗}(t))) sign v(σ_{∗}(t)) ×

× ∏_{i=1}^{m}[r_{ic}(σ_{i}(t), t) − r_{ic}(τ_{i}(t), t)].

Taking into account (3.40), we see that φ_{c} satisfies the conditions of Theorem 3.2′′. Therefore by Theorem 3.7 equation (0.1) has property B. ■

Repeating the arguments given in Theorem 3.10 and Corollary 3.4, we easily ascertain that the corollary below is true.

Corollary 3.5. Let F ∈ V (τ), condition (0.3) be satisfied, and let for any sufficiently large c > 0 there exist t_{c} ∈ ℝ_{+} such that for any u ∈ H_{t_{c},τ} satisfying 1/c ≤ |u(t)| ≤ ct^{n−1} for t ≥ t_{c} we have inequality (3.33), where (3.26) and (3.27) hold and λ_{ic} ∈]0, 1[ (i = 1, …, m). Moreover, let the conditions

λ_{c} = ∑_{i=1}^{m} λ_{ic} < 1,

∫ ^{+∞} t[σ_{∗}(t)]^{λ_{c}(n−2)}[σ^{∗}(t)]^{λ_{c}(2−n)} ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} s^{λ_{ic}(n−2)}d_{s}r_{ic}(s, t)dt = +∞,

∫ ^{+∞} ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} s^{λ_{ic}(n−1)}d_{s}r_{ic}(s, t)dt = +∞

be fulfilled. Then equation (0.1) has property B.

Theorem 3.11. Let F ∈ V (τ), condition (0.2) ((0.3)) be fulfilled, and let for any sufficiently large c > 0 there exist t_{c} ∈ ℝ_{+} such that for any u ∈ H_{t_{0},τ} satisfying 1/c ≤ |u(t)| ≤ ct^{n−1} for t ≥ t_{c} we have

|F (u)(t)| ≥ ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} ω_{ic}(|u(s)|)d_{s}r_{ic}(s, t),  (3.41)

where (3.26)–(3.28) hold. Moreover, let

∫_{1}^{+∞} \frac{ds}{∏_{i=1}^{m} ω_{ic}(s)} < +∞,  (3.42)

∫ ^{+∞} β^{n−1}(t) ∏_{i=1}^{m}[r_{ic}(σ_{i}(t), t) − r_{ic}(τ_{i}(t), t)]dt = +∞,  (3.43)

where β ∈ C(ℝ_{+}; ℝ_{+}) is a nondecreasing function satisfying

β(t) ≤ min{τ_{i}(t), t (i = 1, …, m)}, lim_{t→+∞} β(t) = +∞. (3.44)

Then equation (0.1) has property A (B).

Proof. By (0.2) and (3.41) ((0.3) and (3.41)) inequality (2.16) ((2.17)) holds with

φ_{c}(u)(t) = ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} ω_{ic}(|u(s)|) sign u(s)d_{s}r_{ic}(s, t).

On the other hand, by (3.42), (3.43) and Lemma 1.8 we have φ̃_{1c} ∈ M_{2}^{β}(τ), where

φ̃_{1c}(u)(t) = \frac{β^{n−1}(t)}{(n − 1)!} ∏_{i=1}^{m} ω_{ic}(|u(β(t))|) sign u(β(t)) ×

× ∏_{i=1}^{m}[r_{ic}(σ_{i}(t), t) − r_{ic}(τ_{i}(t), t)].

Since φ_{c} satisfies all the conditions of Theorem 3.3′ (Theorem 3.4′), equation (0.1) has property A (B) by virtue of Theorem 3.7. ■

Corollary 3.6. Let F ∈ V (τ), condition (0.2) ((0.3)) be fulfilled, and let for any sufficiently large c > 0 there exist t_{c} ∈ ℝ_{+} such that for any u ∈ H_{t_{c},τ} satisfying 1/c ≤ |u(t)| ≤ ct^{n−1} for t ≥ t_{c} we have inequality (3.33) where (3.26) and (3.27) hold, λ_{ic} ∈]0, +∞[, (i = 1, …, m) and

\begin{matrix}λ_{c} = ∑_{i=1}^{m} λ_{ic} > 1, \\ ∫ ^{+∞} β^{n−1}(t) ∏_{i=1}^{m}[r_{ic}(σ_{i}(t), t) − r_{ic}(τ_{i}(t), t)]dt = +∞, \end{matrix}  (3.45)

where β ∈ C(ℝ_{+}; ℝ_{+}) is a nondecreasing function satisfying (3.44). Then equation (0.1) has property A (B).

Proof. By (0.2), ((0.3)), (3.33), (3.45) and Corollary 1.2 all the conditions of Theorem 3.11 are satisfied with ω_{ic}(s) = s^{λ_{ic}} for s ∈ ℝ_{+} (i = 1, …, m). ■

Corollary 3.7. Let F ∈ V (τ) and conditions (0.2), (3.35) ((0.3), (3.35)) be fulfilled, where p ∈ L_{loc}(ℝ_{+}; ℝ_{+}) and λ ∈]1, +∞[. Then the condition

∫ ^{+∞} β^{n−1}(t)p(t)dt = +∞

is sufficient for equation (0.1) to have property A (B), where β ∈ C(ℝ_{+}; ℝ_{+}) is a nondecreasing function satisfying

β(t) ≤ min{t, τ(t)}, lim_{t→+∞} β(t) = +∞.

Theorem 3.12. Let F ∈ V (τ), condition (0.2) be fulfilled, and let for any sufficiently large c > 0 there exist t_{c} ∈ ℝ_{+} such that for any u ∈ H_{t_{c},τ} satisfying 1/c ≤ |u(t)| ≤ ct^{n−1} for t ≥ t_{c} we have

|F (u)(t)| ≥ ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} ω_{ic}([σ^{∗}(t)]^{n−1}s^{1−n}|u(s)|)d_{s}r_{ic}(s, t), (3.46)

where (3.26)–(3.28) hold and σ^{∗}(t) = max{σ_{i}(t) : i = 1, …, m}. Moreover, let (3.42) and (3.43) hold, where β ∈ C(ℝ_{+}; ℝ_{+}) is a nondecreasing function satisfying

β(t) ≤ min{σ^{∗}(t), t}, lim_{t→+∞} β(t) = +∞,  (3.47)

Proof. By (0.2) and (3.46) inequality (2.16) holds with

φ_{c}(u)(t) = ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} ω_{ic}([σ^{∗}(t)]^{n−1}s^{1−n}|u(s)|) sign u(s)d_{s}r_{ic}(s, t).

On the other hand, according to (3.42), (3.43) and Lemma 1.8 we have φ̃_{1c} ∈ M_{2}^{β}(τ_{∗}, σ^{∗}), where τ_{∗} = min{τ_{i}(t) : i = 1, …, m} and

φ̃_{1c}(u)(t) = \frac{β^{n−1}(t)}{(n − 1)!} ∏_{i=1}^{m} ω_{ic}(|u(β(t))|) sign u(β(t)) ×

× ∏_{i=1}^{m}[r_{ic}(σ_{i}(t), t) − r_{ic}(τ_{i}(t), t)].

Since φ_{c} satisfies all the conditions of Theorem 3.5, equation (0.1) has property A by Theorem 3.7. ■

Corollary 3.8. Let F ∈ V (τ), condition (0.2) be fulfilled and let for any sufficiently large c > 0 let there exist t_{c} ∈ ℝ_{+} such that for any u ∈ H_{t_{c},τ} satisfying 1/c ≤ |u(t)| ≤ ct^{n−1} for t ≥ t_{c} we have inequality (3.33), where (3.26) and (3.27) hold and λ_{ic} ∈]0, +∞[ (i = 1, …, m). Moreover, let

\begin{matrix}λ_{c} = ∑_{i=1}^{m} λ_{ic} > 1, \\ ∫ ^{+∞} β^{n−1}(t)[σ^{∗}(t)]^{λ_{c}(1−n)} ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} s^{λ_{ic}(n−1)}d_{s}r_{ic}(s, t)dt = +∞, \end{matrix} (3.48)

where σ^{∗}(t) = max{σ_{i}(t) : i = 1, …, m} and β ∈ C(ℝ_{+}; ℝ_{+}) is a nondecreasing function satisfying (3.47). Then equation (0.1) has property A.

Proof. By (0.2), (3.33), (3.48) and Corollary 1.2 of Lemma 1.8 all the conditions of Theorem 3.12 are satisfied with ω_{ic}(s) = s^{λ_{ic}} for s ∈ ℝ_{+} (i = 1, …, m). ■

Theorem 3.13. Let F ∈ V (τ), condition (0.3) be fulfilled, and let for any sufficiently large c > 0 there exist t_{c} ∈ ℝ_{+} such that for any u ∈ H_{t_{c},τ} satisfying 1/c ≤ |u(t)| ≤ ct^{n−1} for t ≥ t_{c} we have

|F (u)(t)| ≥ ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} ω_{ic}([σ^{∗}(t)]^{n−2}s^{2−n}|u(s)|)d_{s}r_{ic}(s, t), (3.49)

where (3.26)–(3.28) hold and σ^{∗}(t) = max{σ_{i}(t) : i = 1, …, m}. Moreover, let (3.42) and (3.43) be fulfilled, where β ∈ C(ℝ_{+}; ℝ_{+}) is a nondecreasing function satisfying (3.47). Then equation (0.1) has property B.

Proof. By (0.3) and (3.49) inequality (2.17) holds with

φ_{c}(u)(t) = − ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} ω_{ic}([σ^{∗}(t)]^{n−2}s^{2−n}|u(s)|) sign u(s)d_{s}r_{ic}(s, t).

On the other hand, according to (3.42), (3.43) and Lemma 1.8 we have φ̃_{1c} ∈ M_{1}^{β}(τ_{∗}, σ^{∗}), where τ_{∗}(t) = min{τ_{i}(t) : i = 1, …, m} and

φ̃_{1c}(u)(t) = \frac{β^{n−1}(t)}{(n − 1)!} ∏_{i=1}^{m} ω_{ic}(|u(β(t))|) sign u(β(t)) ×

× ∏_{i=1}^{m}[r_{ic}(σ_{i}(t), t) − r_{ic}(τ_{i}(t), t)].

Since φ_{c} satisfies all the conditions of Theorem 3.6, equation (0.1) has property B on account of Theorem 3.7. ■

Corollary 3.9. Let F ∈ V (τ), condition (0.3) be fulfilled, and let for any sufficiently large c > 0 there exist t_{c} ∈ ℝ_{+} such that for any u ∈ H_{t_{c},τ} satisfying 1/c ≤ |u(t)| ≤ ct^{n−1} for t ≥ t_{c} we have inequality (3.33), where (3.26) and (3.27) hold and λ_{ic} ∈]0, +∞[ (i = 1, …, m). Moreover, let

\begin{matrix}λ_{c} = ∑_{i=1}^{m} λ_{ic} > 1, \\ ∫ ^{+∞} β^{n−1}(t)[σ^{∗}(t)]^{λ_{c}(2−n)} ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} s^{λ_{ic}(n−2)}d_{s}r_{ic}(s, t)dt = +∞, \end{matrix}(3.50)

where σ^{∗}(t) = max{σ_{i}(t) : i = 1, …, m} and β ∈ C(ℝ_{+}; ℝ_{+}) is a nondecreasing function satisfying (3.48). Then equation (0.1) has property B.

Proof. It suffices to note that by (0.3), (3.33), (3.50) and Corollary 1.2 all the conditions of Theorem 3.13 are satisfied with ω_{ic}(s) = s^{λ_{ic}}. ■

In Theorems 3.14–3.17 below the following condition will be imposed in F:

|F (u)(t)| ≥ ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} |u(s)|^{λ_{i}}d_{s}r_{i}(s, t), for t ≥ t_{0}, u ∈ H_{t_{0},τ} , (3.51)

where (3.26) and (3.27) hold and λ_{i} ∈]0, +∞[ (i = 1, …, m).

Lemma 3.3. Let F ∈ V (τ), (0.2), (3.51), ((0.3), (3.51)) hold, n be odd (even), and u : [t_{0}, +∞[→ ℝ be a proper nondecreasing solution of (0.1) satisfying (2.14_{0}). Then the condition

∫ ^{+∞} t^{n−1} ∏_{i=1}^{m}[r_{i}(σ_{i}(t), t) − r_{i}(τ_{i}(t), t)]dt = +∞  (3.52)

is sufficient for (0.4) to hold.

Proof. Assuming that (0.4) is not fulfilled, according to Lemma 1.2 we obtain the contradiction. ■

In a similar simple way one can prove

Lemma 3.4. Let F ∈ V (τ), (0.3) and (3.51) be fulfilled and

∫ ^{+∞} ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} s^{λ_{i}(n−1)}d_{s}r_{i}(s, t)dt = +∞.  (3.53)

Moreover, let u : [t_{0}, +∞[→ ℝ be a proper nonoscillatory solution of (0.1) satisfying (2.14_{n}). Then (0.5) holds.

Theorem 3.14. Let F ∈ V (τ), conditions (0.2), (3.51), (3.52) hold and ∑_{i=1}^{m} λ_{i} = 1. Moreover, let there exist nondecreasing functions ψ_{i} ∈ C(ℝ_{+}; ℝ_{+}) such that ψ_{i}(s) > 0 for s > 0, functions x → x^{λ_{i}}ψ_{i}(\frac{1}{x}) are nondecreasing on ]0, +∞[ (i = 1, …, m),

∫_{0}^{1} \frac{dx}{x ∏_{i=1}^{m} ψ_{i}(\frac{1}{x})} < +∞,  (3.54)

and for any l ∈ {1, …, n − 1} such that l + n is odd we have

∫ ^{+∞} t^{n−l−1}(\frac{τ_{0}(t)}{σ^{∗}(t)})^{l} ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} \frac{s^{λ_{i}l}d_{s}r_{i}(s, t)}{ψ_{i}(s[τ_{0}(t)]^{l}[σ^{∗}(t)]^{−l})}dt = +∞, (3.55)

where σ^{∗}(t) = max{σ_{i}(t) : i = 1, …, m} and τ_{0} ∈ C(ℝ_{+}; ℝ_{+}) is a function satisfying τ_{0}(t) ≤ min{t, σ^{∗}(t)} for t ∈ ℝ_{+} and lim_{t→+∞} τ_{0}(t) = +∞. Then equation (0.1) has property A.

Proof. Let u : [t_{0}, +∞[→ ℝ be a proper nonoscillatory solution of (0.1). By Lemma 1.1 there is l ∈ {0, …, n − 1} such that l + n is odd and (2.14_{l}) holds. It can be assumed without loss of generality that

u^{(i)}(t) > 0 (i = 0, …, l), (−1)^{i+l}u^{(i)}(t) > 0  (3.56)
(i = l, …, n − 1), t ≥ t_{0}.

Let l ∈ {1, …, n − 1}. Then by (3.56) there are c_{0} ∈]0, +∞[ and t_{1} ∈ [t_{0}, +∞[ such that

u(t) ≥ c_{0}t^{l−1} for t ≥ t_{1}.  (3.57)

Using (3.51) and (3.56), from (0.1) we obtain

u^{(n)}(t) + ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} u^{λ_{i}}(s)d_{s}r_{i}(s, t) ≤ 0,

which on account of the nondecreasing character of functions ψ_{i} (i = 1, …, m) implies

u^{(n)}(t) + ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} \frac{u^{λ_{i}}(s)ψ_{i}(\frac{c_{0}s^{l}}{[σ^{∗}(t)]^{l}u(s)})}{ψ_{i}([σ^{∗}(t)]^{−l}τ_{0}^{l}(t)s)}d_{s}r_{i}(s, t) ≤ 0 for t ≥ t_{∗},

where t_{∗} ∈ [t_{1}, +∞[ is sufficiently large. Therefore u satisfies the differential inequality

u^{(n)}(t) + c_{0}(\frac{τ_{0}(t)}{σ^{∗}(t)})^{l} ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} \frac{s^{lλ_{i}}ω_{i}(\frac{[σ^{∗}(t)]^{l}u(s)}{c_{0}[τ_{0}(t)]^{l}s^{l}})}{ψ_{i}((\frac{τ_{0}(t)}{σ^{∗}(t)})^{l}s)}d_{s}r_{i}(s, t) ≤ 0 (3.58)

on [t_{∗}, +∞[ with ω_{i}(x) = x^{λ_{i}}ψ_{i}(\frac{1}{x}) (i = 1, …, m). Following Remark 2.2 and Lemma 2.3 (μ = 1) the equation

v^{(n)}(t) + (\frac{τ_{0}(t)}{σ^{∗}(t)})^{l} ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} \frac{s^{lλ_{i}}d_{s}r_{i}(s, t)}{ψ_{i}([τ_{0}(t)]^{l}[σ^{∗}(t)]^{−l}s)} ×

× ∏_{i=1}^{m} ω_{i}(τ_{0}^{−l}(t)v(τ_{0}(t))) = 0  (3.59)

has a solution of type (3.56).

On the other hand, by (3.54), (3.55) and Lemma 1.6 we have

\frac{t^{n−l−1}}{l!(n − l)!}(\frac{τ_{0}(t)}{σ^{∗}(t)})^{l} ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} \frac{s^{lλ_{i}}d_{s}r_{i}(s, t)}{ψ_{i}([τ_{0}(t)]^{l}[σ^{∗}(t)]^{−l}s)} ∏_{i=1}^{m} ω_{i}(v(τ_{0}(t))) ∈ M_{1}(τ).

Therefore according to Lemma 3.1 equation (3.59) has no solution of type (3.56). The obtained contradiction proves that l ∉ {1, …, n − 1}.

If n is odd and l = 0, then condition (0.4) is satisfied by (3.52) and Lemma 3.3. ■

Corollary 3.10. Let F ∈ V (τ), σ_{i}(t) ≤ t for t ∈ ℝ_{+} (i = 1, …, m), (0.2) and (3.51) hold, ∑_{i=1}^{m} λ_{i} = 1, ε ∈]0, 1[ and

∫ ^{+∞} ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} s^{λ_{i}(n−1)−ε}d_{s}r_{i}(s, t)dt = +∞.

Then equation (0.1) has property A.

Similarly to Theorem 3.14 one can prove

Theorem 3.15. Let F ∈ V (τ), conditions (0.3), (3.51)–(3.53) be fulfilled and ∑_{i=1}^{m} λ_{i} = 1. Moreover, let there exist nondereasing functions ψ_{i} ∈ C(ℝ_{+}; ℝ_{+}) such that ψ_{i}(s) > 0 for s > 0, functions x → x^{λ_{i}}ψ_{i}(\frac{1}{x}) are nondecreasing on ]0, +∞[ (i = 1, …, m), (3.55) holds and for any l ∈ {1, …, n − 2} such that l + n is even we have (3.55), where σ^{∗}(t) = max{σ_{i}(t) : i = 1, …, m} and τ_{0} ∈ C(ℝ_{+}; ℝ_{+}) is a function satisfying τ_{0}(t) ≤ min{t, σ^{∗}(t)} for t ∈ ℝ_{+} and lim_{t→+∞} τ_{0}(t) = +∞. Then equation (0.1) has property B.

Corollary 3.11. Let F ∈ V (τ), σ_{i}(t) ≤ t for t ∈ ℝ_{+} (i = 1, …, m) and conditions (0.3), (3.51), (3.53) hold, ∑_{i=1}^{m} λ_{i} = 1 and

∫ ^{+∞} t ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} s^{λ_{i}(n−2)−ε}d_{s}r_{i}(s, t)dt = +∞.

Then equation (0.1) has property B.

Remark. One cannot take ε = 0 in Corollaries 3.10 and 3.11 because in that case equation (0.1) does not have, in general, property A (B). In this sense the corresponding theorems are the exact ones.

Theorem 3.16. Let F ∈ V (τ), conditions (0.2) and (3.51) hold, ∑_{i=1}^{m} λ_{i} > 1 and let for any l ∈ {1, …, n − 1} such that l + n is odd

∫ ^{+∞}[t^{n−l−1}[τ_{∗}(t)]^{1−ε}+[β(t)]^{n−l}]∏_{i=1}^{m}∫_{τ_{i}(t)}^{σ_{i}(t)}s^{λ_{i}(l−1)}d_{s}r_{i}(s, t)dt=+∞,(3.60)

where ε ∈]0, 1[, τ_{∗}(t) = min{t, τ_{i}(t) : i = 1, …, m} and β ∈ C(ℝ_{+}; ℝ_{+}) is a nondecreasing function satisfying β(t) ≤ τ_{∗}(t) for t ∈ ℝ_{+} and lim_{t→+∞} β(t) = +∞. Then equation (0.1) has property A.

Proof. Let u : [t_{0}, +∞[→ ℝ be a proper nonoscillatory solution of (0.1). By Lemma 1.1 there exists l ∈ {0, …, n − 1} such that l + n is odd and (2.14_{l}) is fulfilled. It is obvious that condition (3.52) holds by (3.60). Therefore if n is odd and l = 0, then (0.4) is satisfied by Lemma 3.3.

Let us now assume that l ∈ {1, …, n − 1} and

∫ ^{+∞} t^{n−l−1}τ_{∗}^{1−ε}(t) ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} s^{λ_{i}(l−1)}d_{s}r_{i}(s, t)dt = +∞.  (3.61)

By virtue of Lemma 2.2 with μ = (1 − ε)/λ the equation

v^{(n)}(t) + ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} s^{(l−1)λ_{i}}d_{s}r_{i}(s, t)[τ_{∗}(t)]^{−(1−ε)(l−1)} ×

×|v(τ_{∗}(t))|^{1−ε} sign v(τ_{∗}(t)) = 0  (3.62)

has a solution of type (2.14_{l}).

On the other hand, by (3.61) and Corollary 1.1 of Lemma 1.6 we have

\frac{t^{n−l−1}}{l!(n − l)!} ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} s^{(l−1)λ_{i}}d_{s}r_{i}(s, t)[τ_{∗}(t)]^{1−ε} ×

×|v(τ_{∗}(t))|^{1−ε} sign v(τ_{∗}(t)) ∈ M_{1}(τ).

Therefore by Lemma 3.1 equation (3.62) has no solution of type (2.14_{l}). This contradiction proves that l ∉ {1, …, n − 1}.

If the condition

∫ ^{+∞} β^{n−l}(t) ∏_{i=1}^{m} ∫_{τ(t)}^{σ_{i}(t)} s^{λ_{i}(l−1)}d_{s}r_{i}(s, t)dt = +∞

is fulfilled, then we can prove l ∉ {1, …, n − 1} using Lemmas 2.2 and 3.2. ■

Our next theorem is proved similarly.

Theorem 3.17. Let F ∈ V (τ), conditions (0.3) and (3.51) hold, ∑_{i=1}^{m} λ_{i} > 1, and for any l ∈ {1, …, n − 2} such that n + l is even let condition (3.60) be fulfilled, where ε ∈]0, 1[, τ_{∗}(t) = min{t, τ_{i}(t) : i = 1, …, m} and β ∈ C(ℝ_{+}; ℝ_{+}) is a nondecreasing function satisfying β(t) ≤ τ_{∗}(t) for t ∈ ℝ_{+} and lim_{t→+∞} β(t) = +∞. Then equation (0.1) has property B.

§ 4. Necessary and Sufficient Conditions

In this section we shall establish the classes of equations for which the sufficient conditions obtained in 3.2 turn out to be the necessary ones as well.

Lemma 4.1. Let F, φ ∈ V (τ), l ∈ {0, …, n − 1}, c_{1}, c ∈]0, +∞[, c_{1} < c and assume that for any u ∈ C(ℝ_{+}; ℝ), satisfying c_{1}t^{l} ≤ |u(t)| ≤ ct^{l} for t ≥ t_{0} we have

|F (u)(t)| ≤ φ(|u|)(t) for t ≥ t_{0}.  (4.1)

Moreover, let

∫ ^{+∞} t^{n−l−1}φ(θ)(t)dt < +∞,  (4.2)

where θ(s) = cs^{l} for s ∈ ℝ_{+} and

φ(x)(t) ≥ φ(y)(t) ≥ 0 for x(s) ≥ y(s) ≥ 0, s ∈ [τ(t), +∞[, (4.3)

Then for any c_{0} ∈ ℝ satisfying l!c_{1} < |c_{0}| < l!c equation (0.1) has a proper solution u : [t_{∗}, +∞[→ ℝ such that

lim_{t→+∞} u^{(l)}(t) = c_{0}.  (4.4)

Proof. Using (4.2) we can choose t_{∗} ∈ [t_{0}, +∞[ such that inf{τ(t) : t ≥ t_{∗}} ≥ t_{0} and

∫_{t_{∗}}^{+∞} t^{n−l−1}φ(θ)(t)dt ≤ min \{c − \frac{|c_{0}|}{l!}, \frac{|c_{0}|}{l!} − c_{1}\}.  (4.5)

Let U be the set of all u ∈ C([t_{0}, +∞[; ℝ) satisfying

c_{1}t^{l} ≤ u(t) sign c_{0} ≤ ct^{l} for t ≥ t_{1}.  (4.6)

Define T : U → C([t_{0}, +∞[; ℝ) by

T (u)(t) = \begin{cases}\frac{c_{0}}{l!} t^{l} + \frac{(−1)^{n−l−1}}{(l−1)!(n−l−1)!} ∫_{t_{∗}}^{t}(t − s)^{l−1} ∫_{s}^{+∞}(ξ − s)^{n−l−1}× \\ ×F (u)(s)dξ ds for t ≥ t_{∗} \\ \frac{c_{0}}{l!} t^{l} for t_{0} ≤ t < t_{∗} \end{cases} (4.7)

if l ∈ {1, …, n − 1}, and by

T (u)(t) = \begin{cases}c_{0} + \frac{(−1)^{n+1}}{(l−1)!} ∫_{t}^{+∞}(s − t)^{n−1}F (u)(s)ds for t ≥ t_{∗} \\ T (u)(t_{∗}) for t_{0} ≤ t < t_{∗} \end{cases} (4.8)

if l = 0. By virtue of (4.1), (4.3), (4.5)–(4.8) we have T (U) ⊂ U. It is easy to verify that the operator T satisfies all the conditions of Lemma 2.1. Therefore T has a fixed point u which is obviously a solution of (0.1) on [t_{∗}, +∞[ satisfying (4.4). ■

Theorem 4.1. Let F ∈ V (τ), condition (0.2) be fulfilled, and let for any sufficiently large c > 0 there exist t_{c} ∈ ℝ_{+} such that for any u ∈ H_{t_{c},τ} satisfying 1/c ≤ |u(t)| ≤ ct^{n−1} for t ≥ t_{c} we have

∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} ω_{ic}(s^{1−n}|u(s)|)d_{s}r_{ic}(s, t) ≤ |F (u)(t)| ≤

≤ δ ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} ω_{ic}(s^{1−n}|u(s)|)d_{s}r_{ic}(s, t),  (4.9)

where δ ∈ [1, +∞[, (3.26)–(3.28) hold and σ_{i}(t) ≤ t for t ∈ ℝ_{+} (i = 1, …, m). Moreover, let (3.30) be valid. Then condition (3.31) is necessary and sufficient for equation (0.1) to have property A.

Proof. Sufficiency. By virtue of (0.2), (3.30), (3.31) and (4.9) the conditions of Theorem 3.9 are obviously satisfied with σ_{∗}(t) = σ^{∗}(t) = max{σ_{i}(t) : i = 1, …, m} ≤ t for t ∈ ℝ_{+}. Therefore according to the same theorem equation (0.1) has property A.

Necessity. Assume that equation (0.1) has property A and for some c

∫ ^{+∞} ∏_{i=1}^{m}[r_{ic}(σ_{i}(t), t) − r_{ic}(τ_{i}(t), t)]dt < +∞.  (4.10)

Conditions (4.1) and (4.2), where l = n − 1 and

φ(|u|)(t) = δ ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} ω_{ic}(s^{1−n}|u(s)|)d_{s}r_{ic}(s, t), θ(s) = cs^{n−1}

are obviously fulfilled on account of (4.9) and (4.10). Therefore, following Lemma 4.1, there exists c_{0} ≠ 0 such that equation (0.1) has a proper solution u : [t_{∗}, +∞[→ ℝ satisfying lim_{t→+∞} u^{(n−1)}(t) = c_{0}. But this contradicts the fact that equation (0.1) has property A. ■

Corollary 4.1. Let F ∈ V (τ), condition (0.2) be fulfilled, and let for any sufficiently large c > 0 there exist t_{c} ∈ ℝ_{+} such that for any u ∈ H_{t_{c},τ} satisfying 1/c ≤ |u(t)| ≤ ct^{n−1} for t ≥ t_{c} we have

∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} |u(s)|^{λ_{ic}}d_{s}r_{ic}(s, t) ≤ |F (u)(t)| ≤

≤ δ ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} |u(s)|^{λ_{ic}}d_{s}r_{ic}(s, t),  (4.11)

where δ ∈ [1, +∞[, (3.26) and (3.27) hold, λ_{ic} ∈]0, 1[ (i = 1, …, m), ∑_{i=1}^{m} λ_{ic} = λ < 1 and σ_{i}(t) ≤ t for t ∈ ℝ_{+} (i = 1, …, m). Then the condition

∫ ^{+∞} ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} s^{λ_{ic}(n−1)}d_{s}r_{i}(s, t) = +∞  (4.12)

is necessary and sufficient for equation (0.1) to have property A.

Proof. Sufficiency. Since in the case under consideration σ_{∗}(t) = σ^{∗}(t) for t ∈ ℝ_{+}, (4.12) coincides with (3.37) and thus the sufficiency follows from Corollary 3.4.

Necessity. Assume that equation (0.1) has property A and (4.12) is not fulfilled for some c. Then by (4.1) and Lemma 4.1 there exists c_{0} ≠ 0 such that equation (0.1) has a proper solution u : [t_{∗}, +∞[→ ℝ satisfying lim_{t→+∞} u^{(n−1)}(t) = c_{0}. But this contradicts the fact that equation (0.1) has property A. ■

Theorem 4.2. Let F ∈ V (τ), condition (0.3) be fulfilled, and let for any sufficiently large c > 0 there exist t_{c} ∈ ℝ_{+} such that for any u ∈ H_{t_{c},τ} satisfying 1/c ≤ |u(t)| ≤ ct^{n−1} for t ≥ t_{c} we have

∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} ω_{ic}(s^{2−n}|u(s)|)d_{s}r_{ic}(s, t) ≤ |F (u)(t)| ≤

≤ δ ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} ω_{ic}(s^{2−n}|u(s)|)d_{s}r_{ic}(s, t),  (4.13)

where δ ∈ [1, +∞[, (3.26)–(3.28) hold and σ_{i}(t) ≤ t for t ∈ ℝ_{+} (i = 1, …, m). Moreover, let (3.30) be valid. Then the conditions

∫ ^{+∞} t ∏_{i=1}^{m}[r_{ic}(σ_{i}(t)) − r_{ic}(τ_{i}(t), t)]dt = +∞,  (4.14)

∫ ^{+∞} ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} ω_{ic}(s)d_{s}r_{ic}(s, t)dt = ∞  (4.15)

are necessary and sufficient for equation (0.1) to have property B.

Proof. Sufficiency. Since in the case under consideration σ_{∗}(t) = σ^{∗}(t) for t ∈ ℝ_{+}, (4.14), (4.15) coincide with (3.39), (3.40). Therefore the sufficiency follows from Theorem 3.10.

Necessity. Assume that equation (0.1) has property B and condition (4.14) ((4.15)) is not fulfilled for some c > 0. Then by (4.13) and Lemma 4.1 equation (0.1) has a proper solution u : [t_{∗}, +∞[→ ℝ satisfying lim_{t→+∞} u^{(n−2)}(t) = c_{0} (lim_{t→+∞} u^{(n−1)}(t) = c_{0}) where c_{0} ≠ 0. But this contradicts the fact that equation (0.1) has property B. ■

Corollary 4.2. Let all the conditions of Corollary 4.1 be fulfilled except (0.2) which is to be replaced by (0.3). Then condition (4.12) is necessary and sufficient for equation (0.1) to have property B.

Proof. Since in the case under consideration σ_{∗}(t) = σ^{∗}(t) for t ∈ ℝ_{+}, the sufficiency follows from Corollary 3.5. Assuming that equation (0.1) has property B and (4.12) is not fulfilled, we can show, as while proving Corollary 4.1, that equation (0.1) has a proper solution u : [t_{∗}, +∞[→ ℝ satisfying lim_{t→+∞} u^{(n−1)}(t) = c_{0}, where c_{0} ≠ 0, which is the contradiction. ■

Theorem 4.3. Let F ∈ V (τ), condition (0.2) ((0.3)) be fulfilled, and let for any sufficiently large c > 0 there exist t_{c} ∈ ℝ_{+} such that for any u ∈ H_{t_{c},τ} satisfying 1/c ≤ |u(t)| ≤ ct^{n−1} for t ≥ t_{c} we have

∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} ω_{ic}(|u(s)|)d_{s}r_{ic}(s, t) ≤ |F (u)(t)| ≤

≤ δ ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} ω_{ic}(|u(s)|)d_{s}r_{ic}(s, t) for t ≥ t_{c},  (4.16)

where δ ∈ [1, +∞[, (3.26)–(3.28) hold and

\underline{lim}_{t→+∞} \frac{τ_{i}(t)}{t} > 0 (i = 1, …, m).  (4.17)

Moreover, let (3.42) be valid. Then the condition

∫ ^{+∞} t^{n−1} ∏_{i=1}^{m}[r_{ic}(σ_{i}(t), t) − r_{ic}(τ_{i}(t), t)]dt = +∞  (4.18)

is necessary and sufficient for equation (0.1) to have property A (B).

Proof. Sufficiency. By (4.17) there exist α ∈]0, +∞[ and t_{0} ∈ ℝ_{+} such that

τ_{i}(t) ≥ αt for t ≥ t_{0} (i = 1, …, m).  (4.19)

It is obvious that condition (3.43), where β(t) = αt, is fulfilled by virtue of (4.18) and (4.19). Therefore all the conditions of Theorem 3.11 are satisfied by (0.2) ((0.3)) and (4.16)–(4.18), thereby implying the sufficiency of (4.18).

Necessity. Assume that equation (0.1) has property A (B) and (4.18) is not fulfilled for some c > 0. Then it can be shown by (4.16) and Lemma 4.1 that equation (0.1) has a proper solution u : [t_{∗}, +∞[→ ℝ satisfying lim_{t→+∞} u(t) = c_{0}, where c_{0} ≠ 0. But this contradicts the fact that (0.1) has property A (B). ■

Corollary 4.3. Let F ∈ V (τ), condition (0.2) ((0.3)) be fulfilled, and let for any sufficiently large c > 0 there exist t_{c} ∈ ℝ_{+} such that for any u ∈ H_{t_{c},τ} satisfying 1/c ≤ |u(t)| ≤ ct^{n−1} for t ≥ t_{c} we have inequality (4.11), where δ ∈ [1, +∞[, (3.26) and (3.27) hold, λ_{ic} ∈]0, +∞[ (i = 1, …, m) and ∑_{i=1}^{m} λ_{ic} > 1. Then condition (4.18) is necessary and sufficient for equation (0.1) to have property A (B).

Proof. The sufficiency follows from Corollary 3.6. The necessity can be proved similarly to Theorem 4.3. ■

Corollary 4.4. Let F ∈ V (τ), condition (0.2) ((0.3)) be fulfilled and let there exist t_{0} ∈ ℝ_{+} such that for any u ∈ H_{t_{0},τ} we have

p(t) ∫_{τ_{1}(t)}^{σ_{1}(t)} |u(s)|^{λ}ds ≤ |F (u)(t)| ≤

≤ δp(t) ∫_{τ_{1}(t)}^{σ_{1}(t)} |u(s)|^{λ}ds for t ≥ t_{0},  (4.20)

where p ∈ L_{loc}(ℝ_{+}; ℝ_{+}), λ > 1, δ ∈ [1, +∞[, τ_{1}, σ_{1} ∈ C(ℝ_{+}; ℝ_{+}), τ_{1}(t) ≤ σ_{1}(t) for t ∈ ℝ_{+}, lim_{t→+∞} τ_{1}(t) = +∞ and

\overline{lim}_{t→+∞} \frac{τ_{1}(t)}{σ_{1}(t)} < 1, \underline{lim}_{t→+∞} \frac{σ_{1}(t)}{t} > 0.  (4.21)

Then the condition

∫ ^{+∞} σ_{1}(t)t^{n−1}p(t)dt = +∞  (4.22)

is necessary and sufficient for equation (0.1) to have property A (B).

Proof. Sufficiency. By (4.21) there exist α ∈]0, +∞[ and t_{1} ∈ ℝ_{+} such that σ_{1}(t) ≥ αt for t ≥ t_{1}. Therefore by (4.20) and (4.21) condition (3.45) holds with m = 1, r_{1c}(s, t) = p(t)s and β(t) = αt. Due to Corollary 3.6 we easily ascertain that (4.22) is sufficient for equation (0.1) to have property A (B).

Necessity. Assume that equation (0.1) has property A (B) and (4.22) is not fulfilled. Then by Lemma 4.1 and (4.20), (4.21) we find that for any c ≠ 0 equation (0.1) has a proper solution u : [t_{0}, +∞[→ ℝ satisfying lim_{t→+∞} u(t) = c. But this contradicts the fact that (0.1) has property A (B). ■

CHAPTER II

PROPERTIES A AND B OF EQUATIONS WITH

A LINEAR MINORANT

In this chapter we derive sufficient conditions for functional differential equations with a linear minorant to have property A or B. The results obtained are new not only for equations of form (0.1) but can be regarded as improving to a certain extent the well-known earlier results for linear ordinary differential equations [11, 12].

§ 5. Linear Differential Inequalities with a Deviating Argument

5.1. Auxiliary Lemmas. Let us consider linear differential inequalities with a deviating argument

u^{(n)}(t) sign u(τ(t)) + p(t)|u(τ(t))| ≤ 0  (5.1)

and

u^{(n)}(t) sign u(τ(t)) − p(t)|u(τ(t))| ≥ 0,  (5.2)

where n ≥ 2, p ∈ L_{loc}(ℝ_{+}; ℝ_{+}), τ ∈ C(ℝ_{+}; ℝ_{+}), lim_{t→+∞} τ(t) = +∞. Throughout this section it will be assumed that the following condition is fulfilled:

∫_{0}^{+∞} τ_{0}^{n−1}(t)p(t)dt = +∞,  (5.3)

where

τ_{0}(t) = min{t, τ(t)}.  (5.4)

Lemma 5.1. Let p ∈ L_{loc}(ℝ_{+}; ℝ_{+}) and ∫_{t}^{+∞} p(s)ds > 0 for t ∈ R_{+}. Then for the equation

u^{(n)}(t) − p(t)u(t) = 0  (5.5)

to have a solution satisfying (2.14_{n−2}) it is necessary and sufficient that the equation

u^{(n)}(t) − (−1)^{n}p(t)u(t) = 0  (5.6)

have a solution satisfying (2.14_{2}).

The validity of Lemma 5.1 is proved similarly to that of Lemma 1.4 [11].

Corollary 5.1. Let n ≥ 4 and equation (5.5) have a solution satisfying (2.14_{l}) where l ∈ {2, …, n − 2} and l + n is even. Then it has a solution satisfying (2.14_{n−2}).

Proof. Let equation (5.5) have a solution satisfying (2.14_{l}), where l ∈{2, …, n − 2} and l + n is even. Assume that l ∈ {3, …, n − 4}^{4}. Then by Lemma 1.3 and (2.14_{l}) we have

|u^{(l−2)}(t)| ≥ |u^{(l−2)}(t_{∗})| + \frac{1}{(n − 3)!} ∫_{t_{∗}}^{t}(t − s) ×

× ∫_{s}^{+∞}(ξ − s)^{n−3}p(ξ)|u^{(l−2)}(ξ)|dξ ds for t ∈ [t_{∗}, +∞[,

where t_{∗} sufficiently large. Hence applying Lemmaa 2.1 it is easy to show that there exists a continuous function v : [t_{∗}, +∞[→ R such that

v(t) = |u^{(l−2)}(t_{∗})| + \frac{1}{(n − 3)!} ∫_{t_{∗}}^{t}(t − s) ∫_{s}^{+∞}(ξ − s)^{n−3}p(ξ)v(ξ)dξ ds,

|u^{(l−2)}(t_{∗})| ≤ v(t) ≤ |u^{(l−2)}(t)| for t ∈ [t_{∗}, +∞[.

It is clear that v is a solution of equation (5.6) satisfying (2.14_{2}). Thus by Lemma 5.1 equation (5.5) has a solution satisfying (2.14_{n−2}). ■

Lemma 5.2. Let τ(t) ≤ t for t ∈ R_{+}. For the differential inequality (5.1) to have property A it is necessary and sufficient that it have no solution satisfying (2.14_{n−1}).

Proof. Since the necessity is obvious, we shall prove the sufficiency. Let (5.1) have no property A and u_{0}(t) be its nonoscillatory proper solution. By Lemma 1.2 there exists l ∈ {0, …, n − 1} such that l + n is odd and (2.14_{l}) is fulfilled. When n is odd and l = 0, (0.4) holds by (5.3), (5.4). Assume that l ∈ {1, …, n − 3}. Then by Remark 2.2 the equation

u^{(n)}(t) + q(t)u(t) = 0  (5.7)

has a solution satisfying (2.14_{l}) where

q(t) = p(t)\frac{u_{0}(τ(t))}{u_{0}(t)}.

Therefore by [9, Lemmas 1.3 and 1.5] equation (5.7) has a solution u_{1}(t) satisfying (2.14_{n−1}). Thus there exists t_{∗} ∈ R_{+} such that on the interval [t_{∗}, +∞[ u_{1}(t) is a solution of the equation

u^{(n)}(t) + q_{1}(t)u(τ(t)) = 0,

where

q_{1}(t) = p(t) \frac{u_{0}(τ(t))u_{1}(t)}{u_{0}(t)u_{1}(τ(t))}.

On the other hand, by Lemma 1.3 we have

\frac{u_{0}(t)}{t^{l}} ↓ and \frac{u_{1}(t)}{t^{l}} ↑ as t ↑ +∞.

4for l = 2 the validity of this corollary follows from Lemma 5.1.

Thus it is clear that there exists t^{∗} ∈ [t_{∗}, +∞[ such that

\frac{u_{0}(τ(t))u_{1}(t)}{u_{0}(t)u_{1}(τ(t))} ≥ 1 for t ∈ [t^{∗}, +∞[.

Therefore on the interval [t^{∗}, +∞[ differential inequality (5.1) has a solution satisfying (2.14_{n−1}). The obtained contradiction proves the sufficiency. ■

Lemma 5.3. Let τ(t) ≤ t for t ∈ R_{+}. For (5.2) to have property B it is necessary and sufficient when n is even (when n is odd) that it have no solution satisfying (2.14_{n−2}) ((2.14_{1}) and (2.14_{n−2})).

Proof. The necessity is obvious. By virtue of Corollary 5.1 we shall can prove the sufficiency likewise to Lemma 5.2. ■

Similarly to Lemmas 5.2 and 5.3 one can prove

Lemma 5.4. Let τ(t) ≥ t for t ∈ R_{+}. Then for (5.1) to have property A it is necessary and sufficient when n is even (when n is odd) that it have no solution saatisfying (2.14_{1}) ((2.14_{2}) and (2.14_{n−1})).

Lemma 5.5. Let τ(t) ≥ t for t ∈ R_{+}. Then for (5.2) to have property B it is necessary and sufficient when n is even (n is odd) that it have no solution satisfying (2.14_{2}) ((2.14_{1})).

Denote

τ_{∗}(t) = inf{τ_{0}(s) : s ≥ t}, η_{1}(t) = max{s : τ_{∗}(s) ≤ t},
η_{i}(t) = η_{1}(η_{i−1}(t)) (i = 2, 3, …).

Lemma 5.6. Let l ∈ {1, …, n − 1}, l + n be odd (l + n be even) and u : [t_{0}, +∞[→ R be a nonoscillatory proper solution of (5.1) ((5.2)) satisfying (2.14_{l}). Then there exists t_{1} ∈ [t_{0}, +∞[ such that for any k ∈ ℕ we have

|u^{(l)}(s)| ≥ exp \{\frac{1}{l!(n − l − 1)!} ∫_{s}^{t} p(ξ)(ξ − s)^{n−l−1}τ_{∗}^{l}(ξ) ×

×φ_{lk}(ξ, t_{1})dξ}u_{l}(t, s) for t ≥ s ≥ η_{k}(t_{1}),  (5.8)

where

\begin{matrix}u_{l}(t, s) = ∑_{i=l}^{n−1} \frac{(t − s)^{i−l}}{(i − l)!} |u^{(i)}(t)|, φ_{l1}(t, t_{1}) = 0, \\ φ_{li}(t, t_{1}) = exp \{\frac{1}{l!(n − l − 1)!} ∫_{τ_{∗}(t)}^{t} p(ξ)(ξ − τ_{∗}(t))^{n−l−1} × \\ ×τ_{∗}(ξ)τ^{l−1}(ξ)φ_{l i−1}(ξ, t_{1})dξ\} for t ≥ η_{i}(t_{1}) (i = 2, …, k). \end{matrix} (5.9_{l})

Proof. It can be assumed without loss of generality that u(t) > 0 for t ∈ [t_{0}, +∞[. Since

\frac{du_{l}(t, s)}{dt} = −\frac{(t − s)^{n−l−1}}{(n − l − 1)!} |u^{(n)}(t)| and u_{l}(s, s) = u^{(l)}(s),

we obtain

u^{(l)}(s) = exp \{ ∫_{s}^{t} \frac{(ξ − s)^{n−l−1}}{(n − l − 1)!} \frac{|u^{(n)}(ξ)|}{u_{l}(ξ, s)}dξ\}u_{l}(t, s) for t ≥ t_{1} (5.10)

and

u^{(l)}(τ_{∗}(t))=exp\{∫_{τ_{∗}(t)}^{t} \frac{(ξ−τ_{∗}(t))^{n−l−1}}{(n−l−1)!} \frac{|u^{(n)}(ξ)|}{u_{l}(ξ, τ_{∗}(t))}dξ\}u_{l}(t, τ_{∗}(t)) for t≥t_{1},

where t_{1} = η_{1}(t_{0}). Keeping in mind that u_{l}(ξ, ·) is a nonincreasing function, the latter equality gives us

u^{(l)}(τ_{∗}(t)) ≥ exp \{ ∫_{τ_{∗}(t)}^{t} \frac{(ξ − τ_{∗}(t))^{n−l−1}}{(n − l − 1)!} \frac{|u^{(n)}(ξ)|}{u_{l}(ξ, τ_{∗}(ξ))}dξ\}u_{l}(t, τ_{∗}(t))

for t ∈ [t_{1}, +∞[.

Hence by virtue of (5.10) and (5.1) ((5.2)) we immediately obtain (5.8) where the function φ_{lk} is defined by means of (5.9_{l}). ■

Lemma 5.7. Let l ∈ {1, …, n − 1}, l + n be odd (l + n be even), and u : [t_{0}, +∞[→ R be a nonoscillatory proper solution of (5.1) ((5.2)) satisfying (2.14_{l}). Then there exists t_{1} ∈ [t_{0}, +∞[ such that for any k ∈ ℕ we have

|u^{(l−1)}(t)|≥(t+ \frac{1}{(n − l − 1)!}∫_{η_{k+1}(t_{1})}^{t}s^{n−l}ψ_{lk}(s, t_{1})p(s)ds)|u^{(l)}(t)| (5.11_{l})
for t ≥ η_{k+1}(t_{1}),

where for l = n − 1 the function ψ_{n−1 k} is defined by

ψ_{n−1 1}(t, t_{1}) = 0, ψ_{n−1 i}(t, t_{1}) = \frac{1}{(n − 2)!}∫_{η_{i}(t_{1})}^{τ_{∗}(t)}(τ_{∗}(t) − s)^{n−2} ×

× exp { ∫_{s}^{t} p(ξ)ψ_{n−1 i−1}(ξ, t_{1})dξ}ds  (5.12)
for t ≥ η_{i+1}(t_{1}) (i = 2, …, k),

while for l < n − 1 the function ψ_{lk} is defined by

ψ_{l1}(t, t_{1}) = 0, ψ_{li}(t, t_{1}) = \frac{1}{(l − 1)!} ∫_{η_{i}(t_{1})}^{τ_{∗}(t)}(τ_{∗}(t) − s)^{l−1} ×

× exp \{\frac{1}{(n − l − 2)!} ∫_{s}^{t} ψ_{l i−1}(ξ, t_{1})τ_{∗}^{1−l}(ξ) ∫_{ξ}^{+∞}(ξ_{1} − ξ)^{n−l−2} ×

×τ^{l−1}(ξ_{1})p(ξ_{1})dξ_{1} dξ}ds for t ≥ η_{i+1}(t) (i = 2, …, k). (5.13_{l})

Proof. It can be assumed without loss of generality that u(t) > 0 for t ∈ [t_{0}, +∞[. Then the equality

u^{(l)}(t) = u^{(l)}(t_{1}) exp \{ − ∫_{t_{1}}^{t} \frac{|u^{(l+1)}(s)|}{u^{(l)}(s)}ds\},  (5.14)

where t_{1} ∈ [t_{0}, +∞[ is sufficiently large, implies

u(t) ≥ \frac{u^{(l)}(t_{1})}{(l − 1)!} ∫_{t_{1}}^{t}(t − s)^{l−1} exp \{ − ∫_{t_{1}}^{s} \frac{|u^{(l+1)}(ξ)|}{u^{(l)}(ξ)}dξ\}ds (5.15)
for t ∈ [t_{1}, +∞[.

Consider the case l = n − 1. By (5.14), (5.15) and (5.1) we obtain

\frac{u(τ_{∗}(t))}{u^{(n−1)}(t)} ≥ \frac{1}{(n − 2)!} ∫_{t_{1}}^{τ_{∗}(t)}(τ_{∗}(t) − s)^{n−2} ×

× exp \{ ∫_{s}^{t} p(ξ)\frac{u(τ_{∗}(ξ))}{u^{(n−1)}(ξ)}dξ\}ds for t ≥ η(t_{1}),

from which it follows

u(τ_{∗}(t)) ≥ ψ_{n−1 k}(t, t_{1})u^{(n−1)}(t) for t ≥ η_{k}(t_{1}).  (5.16)

On the other hand, from (1.14_{n−2 n}) we obtain u^{(n−2)}(t) ≥ tu^{(n−1)}(t) +∫_{η_{k+1}(t_{1})}^{t}sp(s)u(τ_{∗}(s))ds for t ∈ [η_{k+1}(t_{1}), +∞[. Therefore (5.16) implies that inequality (5.11_{n−1}) is valid.

Now consider the case l ∈ {1, …, n − 2}. By (5.14), (5.15)

\frac{u(τ_{∗}(t))}{u^{(l)}(t)} ≥ \frac{1}{(l − 1)!} ∫_{t_{1}}^{τ_{∗}(t)}(τ_{∗}(t) − s)^{l−1} ×

× exp \{ ∫_{s}^{t} \frac{|u^{(l+1)}(ξ)|}{u^{(l)}(ξ)}dξ\}ds for t ≥ η(t_{1}).

Taking into account (1.9) and the fact that u(t)/t^{l−1} is a nondecreasing function, from the latter inequality we have

\frac{u(τ_{∗}(t))}{u^{(l)}(t)} ≥ \frac{1}{(l − 1)!}∫_{t_{1}}^{τ_{∗}(t)}(τ_{∗}(t) − s)^{l−1}exp \{\frac{1}{(n − l − 2)!}∫_{s}^{t} \frac{u(τ_{∗}(ξ))}{u^{(l)}(ξ)} ×

×τ_{∗}^{1−l}(ξ)∫_{ξ}^{+∞}(ξ_{1} − ξ)^{n−l−2}τ^{l−1}(ξ_{1})p(ξ_{1})dξ_{1} dξ}ds for t ≥ η(t_{1}).

So that

u(τ_{∗}(t)) ≥ ψ_{lk}(t, t_{1})u^{(l)}(t) for t ≥ η_{k}(t_{1}),  (5.17)

where the function ψ_{lk}(t, t_{1}) is defined by (5.13_{l}).

On the other hand, according to (1.14_{l−1 n}) and (2.14_{l}) we have

u^{(l−1)}(t) ≥ tu^{(l)}(t) + \frac{1}{(n − l − 1)!} ∫_{η_{k+1}(t_{1})}^{t} s^{n−l}p(s)u(τ_{∗}(s))ds

for t ≥ η_{k+1}(t_{1}).

which by virtue of (5.17) implies that inequality (5.11_{l}) is valid. ■

5.2. On Solutions of Differential Inequalities.

Lemma 5.8. For (5.1) ((5.2)) not to have a solution satisfying (2.14_{l}) where l ∈ {1, …, n − 1} aand l + n is odd (l + n is even), it is sufficient that for some k_{0} ∈ ℕ

\overline{lim}_{t→+∞}[ ∫_{τ_{∗}(t)}^{t}(s − τ_{∗}(t))^{n−l−1}τ_{∗}^{l}(s)φ̃_{lk_{0}}(s, t, 0)p(s)ds +

+ψ̃_{lk_{0}}(t, 0) ∫_{t}^{+∞}(s − τ_{∗}(t))^{n−l−1}τ^{l−1}(s)p(s)ds] > l!(n − l − 1)!, (5.18)

where

φ̃_{lk_{0}}(s, t, t_{1}) = exp \{\frac{1}{l!(n − l − 1)!} ∫_{τ_{∗}(s)}^{τ_{∗}(t)} p(ξ)(ξ − τ_{∗}(s))^{n−l−1} ×

×τ_{∗}^{l}(ξ)φ_{lk_{0}}(ξ, t_{1})dξ}ds,  (5.19_{l})
ψ̃_{lk_{0}}(t, t_{1})=τ_{∗}(t)+ \frac{1}{(n − l − 1)!} ∫_{η_{k_{0}+1}}(t_{1})^{τ_{∗}(t)}s^{n−l}ψ_{lk_{0}}(s, t_{1})p(s)ds, (5.20_{l})

φ_{lk_{0}}(t, t_{1}) is defined by (5.9_{l}), while ψ_{lk_{0}}(t, t_{1}) is given by (5.12) and (5.13_{l}).

Proof. Assume the contrary. Let (5.1) ((5.2)) have a solution u : [t_{0}, +∞[→ R satisfying (2.14_{l}) where l ∈ {1, …, n − 1} and l + n is odd (l + n is even). Since condition (1.10) is fulfilled due to (5.3) ((5.4)), by Lemmas 1.3, 5.6 and 5.7 there exists t_{1} ∈ [t_{0}, +∞[ such that

|u(τ(t))| ≥ \frac{τ^{l−1}(t)}{l!}|u^{(l−1)}(τ(t))| for t ∈ [t_{1}, +∞[,  (5.21)

|u^{(l−1)}(τ(t))| ≥ |u^{(l−1)}(τ_{∗}(t))| ≥ τ_{∗}(t)|u^{(l)}(τ_{∗}(t))| for t ≥ t_{1}, (5.22)
|u^{(l)}(τ_{∗}(s))| ≥ φ̃_{lk_{0}}(s, t, t_{1})|u^{(l)}(τ_{∗}(t))| for t ≥ η_{k_{0}+1}(t_{1}), (5.23)
|u^{(l−1)}(τ_{∗}(t))| ≥ ψ̃_{lk_{0}}(t, t_{1})|u^{(l)}(τ_{∗}(t))| for t ≥ η_{k_{0}+1}(t_{1}). (5.24)

By virtue of (5.18), (5.19_{l}) and (5.20_{l}) it is clear that

\overline{lim}_{t→+∞}[ ∫_{τ_{∗}(t)}^{t}(s − τ_{∗}(t))^{n−l−1}τ_{∗}^{l}(s)p(s)φ̃_{lk_{0}}(s, t, t_{1})ds + ψ̃_{lk_{0}}(t, t_{1}) ×

× ∫_{t}^{+∞}(s − τ_{∗}(t))^{n−l−1}τ^{l−1}(s)p(s)ds] > l!(n − l − 1)!. (5.25)

On the other hand, by (2.14_{l}), (5.21)–(5.24) we obtain from (5.1) ((5.2))

|u^{(l)}(τ_{∗}(t))| ≥ \frac{1}{(n − l − 1)!}( ∫_{τ_{∗}(t)}^{t}(s − τ_{∗}(t))^{n−l−1}p(s)|u(τ(s))|ds +

+ ∫_{t}^{+∞}(s − τ_{∗}(t))^{n−l−1}p(s)|u(τ(s))|ds) ≥ \frac{|u^{(l)}(τ_{∗}(t))|}{l!(n − l − 1)!} ×

×( ∫_{τ_{∗}(t)}^{t}(s − τ_{∗}(t))^{n−l−1}τ_{∗}^{l}(s)p(s)φ̃_{lk_{0}}(s, t, t_{1})ds +

+ψ̃_{lk_{0}}(t, t_{1}) ∫_{t}^{+∞}(s − τ_{∗}(t))^{n−l−1}τ^{l−1}(s)p(s)ds)

which contradicts (5.25). The obtained contradiction proves the validity of the lemma. ■

Corollary 5.2. For (5.1) ((5.2)) not to have a proper solution satisfying (2.14_{l}) where l ∈ {1, …, n−1} and l+n is odd (l+n is even), it is sufficient that

\overline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t} τ_{∗}^{l}(s)(s − τ_{∗}(t))^{n−l−1}p(s) exp \{\frac{1}{l!(n − l − 1)!} ×

× ∫_{τ_{∗}(s)}^{τ_{∗}(t)} p(s)(ξ − τ_{∗}(s))^{n−l−1}τ_{∗}^{l}(ξ)dξ}ds > l!(n − l − 1)! .

Corollary 5.3. For (5.1) ((5.2)) not to have a proper solution satisfying (2.14_{l}) where l ∈ {1, …, n−1} and l+n is odd (l+n is even), it is sufficient that

\overline{lim}_{t→+∞}(τ_{∗}(t) + \frac{1}{l!(n − l − 1)!} ∫_{0}^{τ_{∗}(t)} s^{n−l}τ_{∗}^{l}(s)p(s)ds) ×

× ∫_{t}^{+∞}(s − τ_{∗}(t))^{n−l−1}τ^{l−1}(s)p(s)ds > l!(n − l − 1)! .

Lemma 5.9. Let l ∈ {1, …, n − 1}, l + n be odd (l + n be even), τ(t) ≤ t for t ∈ R_{+} and

\underline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t} p(s)(s − τ_{∗}(t))^{n−l−1}τ_{∗}^{l}(s)ds > \frac{l!(n − l − 1)!}{e}. (5.26)

Then (5.1) ((5.2)) has no proper solution satisfying (2.14_{l}).

Proof. To prove the lemma it suffices to show that the conditions of Lemma 5.8 are fulfilled. By (5.26) there exist t_{1} ∈ R_{+} and c ∈ ]\frac{l!(n−l−1)!}{e}, +∞[ such that

∫_{τ_{∗}(t)}^{t} p(s)(s − τ_{∗}(t))^{n−l−1}τ_{∗}^{l}(s)ds ≥ c for t ≥ t_{1}.  (5.27)

Choose k_{0} ∈ ℕ such that

(\frac{ec}{l!(n − l − 1)!})^{k_{0}} > [(n − l − 1)! l!]^{2} \frac{4}{c^{2}} .  (5.28)

By virtue of (5.9_{l}), (5.27) we have

φ_{li}(t, t_{1}) ≥ (\frac{ec}{l!(n − l − 1)!})^{i} for t ≥ η_{i}(t_{1}) (i = 1, …, k_{0}).

Therefore

∫_{τ_{∗}(t)}^{t} τ_{∗}^{l}(s)s^{n−l−1}p(s) exp \{\frac{1}{l!(n − l − 1)!} ∫_{τ_{∗}(s)}^{τ_{∗}(t)} p(ξ)(ξ − τ_{∗}(s))^{n−l−1}τ_{∗}^{l}(s) ×

×φ_{lk_{0}}(ξ, t_{1})dξ\}ds ≥ [\frac{ec}{l!(n − l − 1)!}]^{k_{0}} \frac{1}{l!(n − l − 1)!} ∫_{τ_{∗}(t)}^{t} τ_{∗}^{l}(s)s^{n−l−1} ×

×p(s) ∫_{τ_{∗}(s)}^{τ_{∗}(t)} p(ξ)τ_{∗}^{l}(ξ)(ξ − τ_{∗}(s))^{n−l−1}dξ ds for t ≥ η_{k_{0}}(t_{1}). (5.29)

On the other hand, by (5.27) for any t ≥ η_{k_{0}}(t_{1}) there is t_{∗} ∈ [τ_{∗}(t), t] such that

∫_{τ_{∗}(t)}^{t_{∗}} p(s)(s − τ_{∗}(t))^{n−l−1}τ_{∗}^{l}(s)ds = \frac{c}{2},

∫_{τ_{∗}(t_{∗})}^{τ_{∗}(t)} p(s)(s − τ_{∗}(t))^{n−l−1}τ_{∗}^{l}(s)ds ≥ \frac{c}{2}.

Now using (5.28) and (5.29), we obtain

∫_{τ_{∗}(t)}^{t} τ_{∗}^{l}(s)(s − τ_{∗}(t))^{n−l−1}p(s) exp \{\frac{1}{l!(n − l − 1)!} ×

× ∫_{τ_{∗}(s)}^{τ_{∗}(t)} p(ξ)(ξ − τ_{∗}(s))^{n−l−1}τ_{∗}^{l}(ξ)φ_{lk_{0}}(ξ, t_{1})dξ}ds ≥

≥ [\frac{ec}{l!(n − l − 1)!}]^{k_{0}} \frac{c^{2}}{4l!(n − l − 1)!} > l!(n − l − 1)! for t ≥ η_{k_{0}}(t_{1}).

Therefore (5.18) is fulfilled. ■

Remark 5.1. One cannot replace (5.26) by

\underline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t} p(s)τ_{∗}^{l}(s)(s − τ_{∗}(t))^{n−l−1}ds > \frac{l!(n − l − 1)!}{e} − ε, (5.30)

where ε is an arbitrarily small positive number.

Indeed, let ε ∈ ]0, \frac{l!(n−l−1)!}{e}[. Choose β ∈ [l − 1, l] such that

|β(β−1) ⋯ (β+1−l)(1−γ_{l}(β−l))(β−l−1) ⋯ (β+1−n)|>l!(n−l−1)!−εe,

where

γ_{l} = \begin{cases}0 for l = n − 1 \\ ∑_{i=1}^{n−l−1} \frac{1}{n−l−i} for l < n − 1. \end{cases}

Clearly the equation

u^{(n)}(t) + p(t)u(τ(t)) = 0 (u^{(n)}(t) − p(t)u(τ(t)) = 0),^{5}

where τ(t) = αt, α = e^{\frac{1}{β−l}}, p(t) = |β(β − 1) … (β − (l − 1))(β − l) … (β + 1 − n)|α^{−β}t^{−n}, has a solution u(t) = t^{β} and, moreover, condition (5.30) is fulfilled.

Lemma 5.10. Let l ∈ {1, …, n − 1}, l + n be odd (l + n be even), σ(t) ≤ t for t ∈ R_{+} and

\underline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t} p(s)(s − τ_{∗}(t))^{n−l−1}τ_{∗}^{l}(s)ds = c > 0.^{6}  (5.31)

5it is assumed that l + n is odd (l + n is even).

Then for (5.1) ((5.2)) not to have a proper solution satisfying (2.14_{l}), it is sufficient that

\overline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t} τ_{∗}^{l}(s)(s − τ_{∗}(t))^{n−l−1}p(s) exp \{\frac{x_{0}}{l!(n − l − 1)!} ×

× ∫_{τ_{∗}(s)}^{τ_{∗}(t)} p(ξ)(ξ − τ_{∗}(s))^{n−l−1}τ_{∗}^{l}(ξ)dξ}ds > l!(n − l − 1)!, (5.32)

where x_{0} is the smallest root of the equation exp \{\frac{c}{l!(n−l−1)!}x\} = x.

Proof. By (5.32) there exists ε ∈]0, x_{0}[ such that

\overline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t} τ_{∗}^{l}(s)(s − τ_{∗}(t))^{n−l−1}p(s) exp \{\frac{x_{0} − ε}{l!(n − l − 1)!} ×

× ∫_{τ_{∗}(s)}^{τ_{∗}(t)} p(ξ)(ξ − τ_{∗}(s))^{n−l−1}τ_{∗}^{l}(ξ)dξ}ds > l!(n − l − 1)!.

Therefore by virtue of Lemma 5.8 it suffices for us to show that there exists k_{0} ∈ ℕ such that

\underline{lim}_{t→+∞}φ_{lk_{0}}(t, 0) > x_{0} − ε.  (5.33)

By (5.29) there are numbers c^{∗} ∈]0, c] and t_{0} ∈ R_{+} such that

∫_{τ_{∗}(t)}^{t} τ_{∗}^{l}(s)(s − τ_{∗}(t))^{n−l−1}p(s)ds ≥ c^{∗} for t ≥ t_{0}, x_{0}^{∗} > x_{0} − ε, (5.34)

where x_{0}^{∗} is the smallest root of the equation

exp \{\frac{c^{∗}x}{l!(n − l − 1)!}\} = x.  (5.35)

According to (5.9_{l}) and (5.34) we have φ_{li}(t, 0) ≥ α_{i} for t ≥ η_{i}(t_{0}) (i = 1, 2, …), where α_{1} = 0, α_{i} = exp\{\frac{c^{∗}α_{i−1}}{l!(n−l−1)!}\} (i = 2, 3, …).

Denote x^{∗} = lim_{i→+∞} α_{i}. Since x^{∗} is a solution of equation (5.35), there exists k_{0} ∈ ℕ such that (5.33) is fulfilled. ■

Lemma 5.11. Assume that

\underline{lim}_{t→+∞}τ_{∗}(t) ∫_{t}^{+∞} τ_{∗}^{n−2}(s)p(s)ds > β > 0,  (5.36)

6it is assumed that c ≤ \frac{l!(n−l−1)!}{e}. Otherwise condition (5.26) is fulfilled and condition (5.32) becomes unnecessary.

there exists ε > 0 such that

\underline{lim}_{t→+∞}τ_{∗}^{λ−(n−1)}(t) ∫_{η_{1}(1)}^{τ_{∗}(t)} \frac{(τ_{∗}(t) − s)^{n−2}}{τ_{∗}^{λ}(s)} ds > \frac{λ(n − 2)!}{β} + ε (5.37)

for all λ ∈ [\frac{β}{(n − 1)!}, 1]

and for some natural m

\overline{lim}_{t→+∞}(τ_{∗}(t) + \frac{1}{(n − 1)!} ∫_{0}^{τ_{∗}(t)} p(s)sτ_{∗}^{m}(s)ds) ×

× ∫_{t}^{+∞} τ^{n−2}(s)p(s)ds > (n − 1)!.  (5.38)

Then (5.1) has no proper solution satisfying (2.14_{n−1}).

Proof. By Lemma 5.8 it suffices to show that there is a number k_{0} ∈ ℕ such that

ψ_{n−1 k_{0}}(t, 0) ≥ τ_{∗}^{m}(t) for t ≥ t^{∗},  (5.39)

where t^{∗} is sufficiently large.

Put

\begin{matrix}\underline{lim}_{t→+∞} \frac{ψ_{n−1 i}(t, 0)}{τ_{∗}^{n−2}(t)} ∫_{t}^{+∞} τ_{∗}^{n−2}(s)p(s)ds = λ_{i} (i = 1, 2, …), \\ lim_{i→+∞} λ_{i} = λ^{∗}. \end{matrix} (5.40)

By (5.12) and (5.36) we readily find that λ^{∗} ≥ \frac{β}{(n−1)!}. Show that λ^{∗} > 1. Assume the contrary, i.e. λ^{∗} ≤ 1. By (5.36), (5.37) and (5.40) there exist t_{0} ∈ [η(1), +∞[, ε_{0} ∈]0, λ^{∗}[ and k ∈ ℕ such that

\underline{lim}_{t→+∞}τ_{∗}^{λ^{∗}−(n−1)−ε_{0}}(t) ∫_{η_{1}(1)}^{τ_{∗}(t)} \frac{(τ_{∗}(t) − s)^{n−2}}{τ_{∗}^{λ^{∗}−ε_{0}}(s)} ds > \frac{λ^{∗}(n − 2)!}{β}, (5.41)

τ_{∗}(t) ∫_{t}^{+∞} τ_{∗}^{n−2}(s)p(s)ds > β for t ≥ t_{0},
ψ_{n−1 i}(t, 0)∫_{t}^{+∞} τ_{∗}^{n−2}(s)p(s)ds≥(λ^{∗} − ε_{0})τ_{∗}^{n−2}(t)
(i = k, k + 1, …) for t ≥ t_{0}.

Therefore

ψ_{n−1 i}(t, 0) ≥ \frac{1}{(n − 2)!} ∫_{η_{i}(t_{0})}^{τ_{∗}(t)}(τ_{∗}(t) − s)^{n−2} exp \{ ∫_{s}^{t}(λ^{∗} − ε_{0})τ_{∗}^{n−2}(ξ) ×

×p(ξ)[ ∫_{ξ}^{+∞} τ_{∗}^{n−2}(ξ_{1})p(ξ_{1})dξ_{1}]^{−1}dξ\}ds ≥ \frac{β^{λ^{∗}−ε_{0}}}{(n − 2)!} ×

×( ∫_{t}^{+∞} τ_{∗}^{n−2}(s)p(s)ds)^{ε_{0}−λ^{∗}}( ∫_{η_{1}(1)}^{τ_{∗}(t)} \frac{(τ_{∗}(t) − s)^{n−2}}{τ_{∗}^{λ^{∗}−ε_{0}}(s)} ds −

− ∫_{η_{1}(1)}^{η_{i}(t_{0})} \frac{(τ_{∗}(t) − s)^{n−2}}{τ_{∗}^{λ^{∗}−ε_{0}}(s)} ds) for t ≥ t_{0} (i = k, k + 1, …).

Hence by (5.41) we obtain

\frac{ψ_{n−1 i}(t, 0)}{τ_{∗}^{n−2}(t)}∫_{t}^{+∞}τ_{∗}^{n−2}(s)p(s)ds≥ \frac{β^{λ^{∗}−ε_{0}}}{(n−2)!}(τ_{∗}(t)∫_{t}^{+∞}τ_{∗}^{n−2}(s)p(s)ds)^{1−λ^{∗}+ε_{0}} ×

×τ_{∗}^{λ^{∗}−(n−1+ε_{0})}(t)( ∫_{η_{1}(1)}^{τ_{∗}(t)} \frac{(τ_{∗}(t) − s)^{n−2}}{τ_{∗}^{λ^{∗}−ε_{0}}(s)} ds − ∫_{η_{1}(1)}^{η_{i}(t_{0})} \frac{(τ_{∗}(t) − s)^{n−2}}{τ_{∗}^{λ^{∗}−ε_{0}}(s)} ds) ≥

≥ \frac{βτ_{∗}^{λ^{∗}−(n−1+ε_{0})}(t)}{(n − 2)!}( ∫_{η_{1}(1)}^{τ_{∗}(t)} \frac{(τ_{∗}(t) − s)^{n−2}}{τ_{∗}^{λ^{∗}−ε_{0}}(s)} ds −

− ∫_{η_{1}(1)}^{η_{i}(t_{0})} \frac{(τ_{∗}(t) − s)^{n−2}}{τ^{λ^{∗}−ε_{0}}(s)} ds) for t ≥ t_{0} (i = k, k + 1, …).

Therefore by (5.41)

\underline{lim}_{t→+∞} \frac{ψ_{n−1 i}(t, 0)}{τ_{∗}^{n−2}(t)} ∫_{t}^{+∞} τ_{∗}^{n−2}(s)p(s)ds > λ^{∗}

which contradicts the definition of λ^{∗}. The obtained contradiction proves that λ^{∗} > 1. On the other hand, since we can assume that

\overline{lim}_{t→+∞}τ_{∗}(t) ∫_{t}^{+∞} τ_{∗}^{n−2}(s)p(s)ds ≤ n − 1

(otherwise we have (5.18) with l = n − 1), we easily find that there exists k_{0} ∈ ℕ such that (5.39) holds. ■

Corollary 5.4. Let

\underline{lim}_{t→+∞}t ∫_{t}^{+∞} \frac{τ_{∗}^{n−1}(s)}{s}p(s)ds > M_{n},  (5.42)

where M_{n} = max{x(1 − x) … (n − 1 − x) : x ∈ [0, 1]}. Then (5.1) has no proper solution satisfying (2.14_{n−1}).

Proof. Let (5.1) have a proper solution u : [t_{0}, +∞[→ R satisfying (2.14_{n−1}). By Lemma 1.3 there is t_{1} ∈ [t_{0}, +∞[ such that

\frac{|u(τ(t))|}{τ_{∗}^{n−1}(t)} ≥ \frac{|u(τ_{∗}(t))|}{τ_{∗}^{n−1}(t)} ≥ \frac{|u(t)|}{t^{n−1}} for t ∈ [t_{1}, +∞[.

Thus it is clear that on the interval [t_{1}, +∞[ the function u is a solution of the differential inequality

u^{(n)}(t) sign u(t) + q(t)|u(t)| ≤ 0,  (5.43)

where

q(t) = \frac{τ_{∗}^{n−1}(t)}{t^{n−1}}p(t).

On the other hand, by (5.42)

\underline{lim}_{t→+∞}t ∫_{t}^{+∞} s^{n−2}q(s)ds > β > M_{n}.  (5.44)

Thus, clearly, there exists ε > 0 such that

\underline{lim}_{t→+∞}t^{λ−(n−1)} ∫_{1}^{t} \frac{(t − s)^{n−2}}{s^{λ}} ds > \frac{λ(n − 2)!}{β} + ε,  (5.45)

for all λ ∈ [\frac{β}{(n − 1)!}, 1].

Assume that

\overline{lim}_{t→+∞}t ∫_{t}^{+∞} s^{n−2}q(s)ds ≤ (n − 1)!.

(otherwise (5.43) has no solution satisfying (2.14_{n−1})). Then we have

∫_{0}^{t} s^{n+1}q(s)ds ≥ ∫_{t^{\frac{1}{2}}}^{t} s^{2}s^{n−1}q(s)ds ≥ −t ∫_{t^{\frac{1}{2}}}^{t} s d ∫_{s}^{+∞} ξ^{n−2}q(ξ)dξ ≥

≥ t( − t ∫_{t}^{+∞} s^{n−2}q(s)ds + ∫_{t^{\frac{1}{2}}}^{t} ∫_{s}^{+∞} ξ^{n−2}q(ξ)dξ ds) ≥ t( − n! + \frac{M_{n}}{4} ln t).

Therefore by (5.44), (5.45) and Lemma 5.11 (τ_{∗}(t) = t) (5.43) has no solution satisfying (2.14_{n−1}). The obtained contradiction proves the validity of the corollary. ■

Corollary 5.5. Let α ∈]0, 1] and σ(t) ≥ αt for sufficiently large t,

\underline{lim}_{t→+∞}t ∫_{t}^{+∞} s^{n−2}p(s)ds > M_{n}(α),

where

M_{n}(α) = max{α^{x−n+1}x(1 − x) ⋯ (n − 1 − x) : x ∈ [0, 1]}. (5.46)

Then (5.1) has no solution satisfying (2.14_{n−1}).

To prove the corollary it suffices to note that all the conditions of Lemma 5.11 with τ_{∗}(t) = αt are fulfilled.

Corollary 5.6. Let

lim_{t→+∞} \frac{t}{τ_{∗}(t)} = +∞, \underline{lim}_{t→+∞}τ_{∗}(t) ∫_{t}^{+∞} τ_{∗}^{n−2}(s)p(s)ds > 0 (5.47)

and (5.38) be fulfilled for some natural m. Then (5.1) has no proper solution satisfying (2.14_{n−1}).

Proof. By (5.47) there is β ∈]0, +∞[ such that

\underline{lim}_{t→+∞}τ_{∗}(t) ∫_{t}^{+∞} τ_{∗}^{n−2}(s)p(s)ds > β > 0.

Thus due to Lemma 5.11 is suffices to show that (5.37) is fulfilled. Indeed, by (5.47) there is t_{0} ∈ [η(1), +∞[ such that

t ≥ (\frac{2}{β} max \{λ(1 − λ) ⋯ (n − 1 − λ) : λ ∈ [\frac{β}{(n − 1)!}, 1]\})^{\frac{1}{λ}} τ_{∗}(t) (5.48)
for t ≥ t_{0}.

Assuming that λ ∈ [\frac{β}{(n−1)!}, 1[, by (5.48) we obtain

\frac{βτ_{∗}^{λ−(n−1)}(t)}{(n − 2)!} ∫_{η_{1}(1)}^{τ_{∗}(t)} \frac{(τ_{∗}(t) − s)^{n−2}}{τ_{∗}^{λ}(s)} ds ≥ \frac{2}{(n − 2)!} max \{λ((1 − λ) ⋯

⋯ (n − 1 − λ) : λ ∈ [\frac{β}{(n − 1)!}, 1]\}τ_{∗}^{λ−(n−1)}(t) ∫_{t_{0}}^{τ_{∗}(t)} \frac{(τ_{∗}(t) − s)^{n−2}}{s^{λ}} ds >

> \frac{max \{λ(1 − λ) ⋯ (n − 1 − λ) : λ ∈ [\frac{β}{(n−1)!}, 1]\}}{(1 − λ)(2 − λ) ⋯ (n − 1 − λ)} + ε ≥ λ + ε,

for t ≥ t_{1},

where t_{1} ∈ [t_{0}, +∞[ is sufficiently large, while ε is a sufficiently small positive number. For λ = 1 it is easy to find that (5.37) is satisfied. ■

Corollary 5.7. Let

\underline{lim}_{t→+∞}t^{−α}τ(t) > 0, \underline{lim}_{t→+∞}t^{α} ∫_{t}^{+∞} s^{α(n−2)}p(s)ds > 0,

where α ∈]0, 1[. Then the (5.1) has no solution saatisfying (2.14_{n−1}).

Lemma 5.12. If the inequality τ(t) ≥ αt is fulfilled for large t,

\underline{lim}_{t→+∞}t ∫_{t}^{+∞} s^{n−2}p(s)ds = β > 0^{7}  (5.49)

and

\overline{lim}_{t→+∞}(αt + x_{0} ∫_{0}^{αt} s^{n−1}( ∫_{s}^{+∞} ξ^{n−2}p(ξ)dξ)^{−1} ×

×p(s)ds) ∫_{t}^{+∞} τ^{n−2}(s)p(s)ds > (n − 1)!.  (5.50)

where α ∈]0, 1] and x_{0} is the smallest root of the equation

x(1 − x) ⋯ (n − 1 − x)α^{x−(n−1)} = β,  (5.51)

then (5.1) has no proper solution satisfying (2.14_{n−1}).

Proof. By (5.49)–(5.51) there exist β^{∗} ∈]0, β[, ε ∈]0, β^{∗}[ and t_{0} ∈ R_{+} such that

t ∫_{t}^{+∞} s^{n−2}p(s)ds ≥ β^{∗} for t ∈ [t_{0}, +∞[,  (5.52)

\overline{lim}_{t→+∞}(αt + (x_{0}^{∗} − ε) ∫_{0}^{αt} s^{n−1}( ∫_{s}^{+∞} ξ^{n−2}p(ξ)dξ)^{−1} ×

×p(s)ds) ∫_{t}^{+∞} τ^{n−2}(s)p(s)ds > (n − 1)!,  (5.53)

where x_{0}^{∗} is the smallest root of the equation

α^{x−(n−1)}x(1 − x) ⋯ (n − 1 − x) = β^{∗} − ε.  (5.54)

By Lemma 5.7 and (5.53) it suffices to show that there exists k_{0} ∈ ℕ such that

ψ_{n−1k_{0}}(t, 0) ≥ t^{n−2}( ∫_{t}^{+∞} ξ^{n−2}p(ξ)dξ)^{−1}(x_{0}^{∗} − ε) for t ≥ t_{∗}, (5.55)

where t_{∗} ∈ [t_{0}, +∞[ is sufficiently large.

By (5.12) and (5.52) there exist β_{1} ∈]0, β^{∗}] and t_{1} ∈ [t_{0}, +∞[ such that

ψ_{n−1 2}(t, 0)t^{2−n} ∫_{t}^{+∞} ξ^{n−2}p(ξ)dξ ≥ β_{1} for t ∈ [t_{0}, +∞[.

Taking into account (5.52), we now obtain

ψ_{n−1 3}(t, 0) ≥ \frac{1}{(n − 2)!} ∫_{t_{1}}^{τ_{∗}(t)}(τ_{∗}(t) − s)^{n−2} exp \{β_{1} ∫_{s}^{t} ξ^{n−2}p(ξ) ×

×( ∫_{ξ}^{+∞} ξ_{1}^{n−2}p(ξ_{1})dξ_{1})^{−1}dξ\}ds = \frac{( ∫_{t}^{+∞} s^{n−2}p(s)ds)^{−β_{1}}}{(n − 2)!} ×

7it is assumed that β ∈]0, M_{n}(α)] where M_{n}(α) is defined by equality (5.46).Otherwise the conditions of Corollary 5.5 are fulfilled and condition (5.50) becomes unnecessary.

× ∫_{t_{1}}^{τ_{∗}(t)}(τ_{∗}(t) − s)^{n−2}( ∫_{s}^{+∞} ξ^{n−2}p(ξ)dξ)^{β_{1}}ds ≥

≥ \frac{α^{n−1−β_{1}}(β^{∗} − ε)^{β_{1}}t^{n−1−β_{1}}}{(1 − β_{1}) ⋯ (n − 1 − β_{1})} ( ∫_{t}^{+∞} s^{n−2}p(s)ds)^{−β_{1}} for t ∈ [t_{2}, +∞[,

where t_{2} ∈ [t_{1}, +∞[ is sufficiently large. Therefore

ψ_{n−1 3}(t, 0)t^{2−n} ∫_{t}^{+∞} s^{n−2}p(s)ds ≥ \frac{(β^{∗} − ε)α^{n−1−β_{1}}}{(1 − β_{1}) ⋯ (n − 1 − β_{1})} = β_{2}

for t ∈ [t_{2}, +∞[.

In a similar manner we show that

ψ_{n−1 i}(t, 0)t^{2−n}∫_{t}^{+∞}s^{n−2}p(s)ds≥ \frac{(β^{∗} − ε)α^{n−1−β_{i−1}}}{(1 − β_{i−1}) ⋯ (n − 1 − β_{i−1})} =

= β_{i} for t ∈ [t_{i−1}, +∞[ (i = 3, 4, …),  (5.56)

where t_{i} (i = 2, 3, …) is sufficiently large.

Putting

lim_{i→+∞} β_{i} = x_{0}^{∗},

from the equalities

β_{i} = \frac{(β^{∗} − ε)α^{n−1−β_{i−1}}}{(1 − β_{i−1}) ⋯ (n − 1 − β_{i−1})} (i = 2, 3, …)

we have

α^{x_{0}^{∗}−(n−1)}x_{0}^{∗}(1 − x_{0}^{∗}) ⋯ (n − 1 − x_{0}^{∗}) = β^{∗} − ε.

Thus x_{0}^{∗} is a solution of equation (5.54). Therefore by (5.56) there is k_{0} ∈ ℕ such that inequality (5.55) is fulfilled for t ∈ [t_{∗}, +∞[, where t_{∗} is sufficiently large. ■

Lemma 5.12′. Let for sufficiently large t there hold

τ(t) ≥ αt, p(t) ≥ \frac{β}{t^{n}}  (5.57)

and

\overline{lim}_{t→+∞}t ∫_{t}^{+∞} τ^{n−2}(s)p(s)ds > \frac{(n − 1)!}{α(1 + x_{0})},  (5.58)

where α ∈]0, 1], β ∈]0, M_{n}(α)] (M_{n}(α) is defined by (5.46)), and x_{0} is the smallest root of equation (5.51). Then (5.1) has no proper solution satisfying (2.14_{n−1}).

Proof. By (5.51) and (5.58) there exist β^{∗} ∈]0, β[ and ε ∈]0, β^{∗}[ such that

\overline{lim}_{t→+∞}t ∫_{t}^{+∞} τ^{n−2}(s)p(s)ds > \frac{(n − 1)!}{α(1 + x_{0}^{∗})},  (5.59)

where x_{0}^{∗} is the smallest root of (5.54). Due to (5.59) and Lemma 5.8 it suffices to show that there exists k_{0} ∈ ℕ such that

ψ_{n−1 k_{0}}(t, 0) ≥ \frac{t^{n−1}x_{0}^{∗}}{β} for t ∈ [t_{∗}, +∞[,  (5.60)

where t_{∗} ∈ R_{+} is some number.

By (5.12), (5.57) there exist β_{1} ∈]0, β] and t_{0} ∈ R_{+} such that

ψ_{n−1 2}(t, 0) ≥ β_{1}t^{n−1} for t ∈ [t_{0}, +∞[.

Since ββ_{1} < 1 (it is clear that (5.60) is fulfilled for ββ_{1} ≥ 1), by (5.57) we obtain

ψ_{n−1 3}(t, 0) ≥ \frac{1}{(n − 2)!} ∫_{t_{0}}^{τ_{∗}(t)}(τ_{∗}(t) − s)^{n−2} exp \{ ∫_{s}^{t} ββ_{1} \frac{dξ}{ξ} \}ds =

= \frac{t^{ββ_{1}}}{(n − 2)!} ∫_{t}^{τ_{∗}(t)} \frac{(τ_{∗}(t) − s)^{n−2}}{s^{ββ_{1}}} ds ≥ \frac{t^{n−1}α^{n−1−β^{∗}β_{1}}}{(1 − β^{∗}β_{1}) ⋯ (n − 1 − β^{∗}β_{1})}

for t ∈ [t_{1}, +∞[,

where t_{1} ∈ [t_{0}, +∞[ is sufficiently large. Therefore

t^{1−n}ψ_{n−1 3}(t, 0) ≥ \frac{α^{n−1−β^{∗}β_{1}}}{(1 − β^{∗}β_{1}) ⋯ (n − 1 − β^{∗}β_{1})} = β_{2} for t ∈ [t_{1}, +∞[.

Similarly we have

t^{1−n}ψ_{n−1 i}(t, 0) ≥ \frac{α^{n−1−β^{∗}β_{i−2}}}{(1 − β^{∗}β_{i−2}) ⋯ (n − 1 − β^{∗}β_{i−2})} =

= β_{i−1} (i = 3, 4, …) for t ∈ [t_{i−2}, +∞[,

where t_{i} (i = 1, 2, …) is sufficiently large.

Introducing the notation

lim_{i→∞} β^{∗}β_{i} = x_{0},

from the equalities

β_{i} = \frac{α^{n−1−β^{∗}β_{i−1}}}{(1 − β^{∗}β_{i−1}) ⋯ (n − 1 − β^{∗}β_{i−1})} (i = 2, 3, …)

we find that x_{0} is the root of the equation

x(1 − x) ⋯ (n − 1 − x)α^{x−(n−1)} = β^{∗}.

Hence we easily conclude that there exist numbers t_{∗} ∈ R_{+} and k_{0} ∈ ℕ for which (5.60) is fulfilled. ■

The validity of Lemmas 5.13 and 5.14 below can be proved similarly to that of Lemmas 5.11 and 5.12.

Lemma 5.13. Let l ∈ {1, …, n − 2}, l + n be odd (l + n be even),

\underline{lim}_{t→+∞}τ_{∗}(t) ∫_{t}^{+∞}(s − t)^{n−l−1}τ^{l−1}(s)p(s)ds > β > 0,

there exist ε > 0 such that

\underline{lim}_{t→+∞}τ_{∗}^{λ−l}(t) ∫_{η_{1}(1)}^{τ_{∗}(t)} \frac{(τ_{∗}(t) − s)^{l−1}}{τ_{∗}^{λ}(s)} ds >

> \frac{λ(l − 1)!(n − l − 1)!}{β} + ε for all λ ∈ [\frac{β}{l!(n − l − 1)!}, 1]

and

\overline{lim}_{t→+∞}(τ_{∗}(t) + \frac{1}{(n − l − 1)!} ∫_{0}^{τ_{∗}(t)} s^{n−l}τ_{∗}^{m}(s)p(s)ds) ×

× ∫_{t}^{+∞}(s − τ_{∗}(t))^{n−l−1}τ^{l−1}(s)p(s)ds > l!(n − l − 1)!  (5.61)

for some natural m. Then (5.1) ((5.2)) has no solution satisfying (2.14_{l}).

Corollary 5.8. Let l ∈ {1, …, n − 2}, l + n be odd (l + n be even), the inequality σ(t) ≥ αt with α ∈]0, 1] hold for sufficiently large t and

\underline{lim}_{t→+∞}t ∫_{t}^{+∞}(s − t)^{n−l−1}s^{l−1}p(s)ds > M_{l}(α)(n − l − 1)!,

where

M_{l}(α) = max{x(1 − x) ⋯ (l − x)α^{x−l} : x ∈ [0, 1]}.  (5.62)

Then (5.1) ((5.2)) has no solution satisfying (2.14_{l}).

Corollary 5.9. Let l ∈ {1, …, n − 2}, l + n be odd (l + n be even) and

lim_{t→+∞} \frac{t}{τ_{∗}(t)} = +∞, \underline{lim}_{t→+∞}τ_{∗}(t) ∫_{t}^{+∞}(s − t)^{n−l−1}τ^{l−1}(s)p(s)ds > 0.

Then for (5.1) ((5.2)) not to have a solution satisfying (2.14_{l}), it is sufficient that (5.61) be fulfilled for some natural m.

Corollary 5.10. Let l ∈ {1, …, n − 2}, l + n be odd (l + n be even) and

\underline{lim}_{t→+∞}t^{−α}τ(t) > 0, \underline{lim}_{t→+∞}t^{α} ∫_{t}^{+∞}(s − t)^{n−l−1}s^{α(l−1)}p(s)ds > 0,

where α ∈]0, 1[. Then (5.1) ((5.2)) has no solution satisfying (2.14_{l}).

Lemma 5.14. Let l ∈ {1, …, n − 2}, l + n be odd (l + n be even), for sufficiently large t inequality (5.57) be fulfilled which α ∈]0, 1], β ∈]0, M_{l}(α)(n − l)!] which M_{l}(α) is defined by (5.62) and

\overline{lim}_{t→+∞}t ∫_{t}^{+∞}(s − αt)^{n−l−1}τ^{l−1}(s)p(s)ds > \frac{(l − 1)!(n − l − 1)!}{α(1 + (n − l)x_{0})},

where x_{0} is the smallest root of the equation

x(1 − x) ⋯ (l − x)α^{x−l} = \frac{β}{(n − l)!}.

Then (5.1) ((5.2)) has no solution satisfying (2.14_{l}).

§ 6. Linear Differential Inequalities with a Deviating Argument and Property A (B)

The results obtained in §5 for the linear differential inequalities (5.1) and (5.2) will be used in this section to derive sufficient conditions for equation (0.1) to have property A (B).

6.1. Equations with Property A.

Theorem 6.1. Let F ∈ V (τ), condition (0.2) be fulfilled for some t_{0} ∈ ℝ_{+}, and

|F (u)(t)| ≥ ∫_{τ(t)}^{σ(t)} |u(s)|d_{s}r(s, t) for t ∈ [t_{0}, +∞[, u ∈ H_{t_{0},τ} , (6.1)

where

τ, σ ∈ C(ℝ_{+}; ℝ_{+}), τ(t) ≤ σ(t) for t ∈ ℝ_{+}, lim τ(t) = +∞,

r(s, ·) is a measurable function, r(·, t) is a nondecreasing function.

(6.2)

Let, in addition to the above, for some k_{0} ∈ ℕ the condition

\overline{lim}_{t→+∞}[ ∫_{τ_{∗}(t)}^{t} τ_{∗}^{n−1}(s)p(s)φ̃_{n−1 k_{0}}(s, t, 0)ds +

+ψ̃_{n−1 k_{0}}(t, 0) ∫_{t}^{+∞} τ_{∗}^{n−2}(s)p(s)ds] > (n − 1)!  (6.3)

hold, where

p(t) = r(σ(t), t) − r(τ(t), t),  (6.4)
τ_{∗}(t) = inf{τ_{0}(s) : s ≥ t}, τ_{0}(t) = min{t, τ(t)}  (6.5)

and the functions φ̃_{n−1 k_{0}} and φ_{n−1 k_{0}} (ψ̃_{n−1 k_{0}} and ψ_{n−1 k_{0}}) are defined by (5.19_{n−1}) and (5.9_{n−1}) ((5.20_{n−1}) and (5.12)). Then equation (0.1) has property A.

Proof. First of all note that condition (5.3) with the functions p and τ_{0} defined by (6.4) and (6.5), respectively, is fulfilled by virtue of (6.3).

Let us assume that equation (0.1) has no property A. Then by (0.2) and (6.1) the differential inequality

u^{(n)}(t) sign u(σ(t)) + ∫_{τ(t)}^{σ(t)} |u(s)|d_{s}r(s, t) ≤ 0  (6.6)

has no property A. Following Theorem 2.3, the equation

u^{(n)}(t) + p(t)u(τ_{∗}(t)) = 0  (6.7)

with p and τ_{∗} defined by (6.4) and (6.5), respectively, has no property A. Therefore (6.7) has a nonoscillatory proper solution u : [t_{1}, +∞[→ R satisfying (2.14_{l}) where l ∈ {0, …, n−1} (l +n is odd). Assuming now that n is odd and l = 0, by (5.3) we crearly see that (0.4) is fulfilled. Therefore l ∈ {1, …, n − 1} and thus by Lemma 5.1 equation (6.7) has a proper solution satisfying (2.14_{n−1}).

But by (6.3) and Lemma 5.8 equation (6.7) has no proper solution satisfying (2.14_{n−1}). The obtained contradiction proves the validity of the theorem. ■

Corollary 6.1. If F ∈ V (τ) and conditions (0.2), (6.1), (6.2) are fulfilled, then for equation (0.1) to have property A it is sufficient that

\overline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t} p(s)τ_{∗}^{n−1}(s) exp \{\frac{1}{(n − 1)!} ∫_{τ_{∗}(s)}^{s} p(ξ)τ_{∗}^{n−1}(ξ)dξ\}ds > (n − 1)!,

where p and τ_{∗} are defined by (6.4) and (6.5), respectively.

Corollary 6.2. If F ∈ V (τ) and conditions (0.2), (6.1), (6.2) are fulfilled, then for equation (0.1) to have property A it is sufficient that

\overline{lim}_{t→+∞}(τ_{∗}(t)+ \frac{1}{(n−1)!}∫_{0}^{τ_{∗}(t)}sτ_{∗}^{n−1}(s)p(s)ds)∫_{t}^{+∞}τ_{∗}^{n−2}(s)p(s)ds>(n − 1)!,

where p and τ_{∗} are defined by (6.4) and (6.5), respectively.

Corollary 6.2′. If p ∈ L_{loc}(ℝ_{+}; ℝ_{+}) and

\overline{lim}_{t→+∞}(t + \frac{1}{(n − 1)!}∫_{0}^{t} s^{n}p(s)ds)∫_{t}^{+∞}s^{n−2}p(s)ds>(n − 1)!,

then the equation

u^{(n)}(t) + p(t)u(t) = 0  (6.8)

has property A.

The particular case of Corollary 6.2′ is Theorem 2.3 in [11].

Theorem 6.2. If F ∈ V (τ), conditions (0.2), (6.1), (6.2) are fulfilled, and τ(t) < t for t ∈ R_{+}, then the condition

\underline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t} p(s)τ^{n−1}(s)ds > \frac{(n − 1)!}{e},  (6.9)

where p and τ_{∗} are defined by (6.4) and (6.5), respectively, is sufficient for equation (0.1) to have property A.

Proof. By Lemma 5.9 and (6.9) there exists k_{0} ∈ ℕ such that (6.3) holds. Therefore the conditions of Theorem 6.1 are fulfilled, which proves that Theorem 6.2 is valid. ■

Remark 6.1. Remark 5.1 clearly implies that (6.9) cannot be replaced by the condition

\underline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t} p(s)τ^{n−1}(s)ds ≥ \frac{(n − 1)!}{e} − ε,

where ε is an arbitrarily small positive number.

Theorem 6.3. Let F ∈ V (τ), conditions (0.2), (6.1), (6.2) be fulfilled,

\underline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t} p(s)τ_{∗}^{n−1}(s)ds ≥ c  (6.10)

and

\overline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t} τ_{∗}^{n−1}(s)p(s) exp \{\frac{x_{0}}{(n − 1)!} ×

× ∫_{τ_{∗}(s)}^{τ_{∗}(t)} p(ξ)τ_{∗}^{n−1}(ξ)dξ}ds > (n − 1)!,  (6.11)

where c ∈ ]0, \frac{(n−1)!}{e}], p and τ_{∗} are defined by (6.4) and (6.5)^{8} and x_{0} is the smallest root of the equation

exp\{\frac{c}{(n − 1)!}x\} = x.  (6.12)

Then equation (0.1) has property A.

Proof. By (6.10)–(6.12) there exists ε ∈]0, 1[ such that

\overline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t} τ_{∗}^{n−1}(s)p(s) ×

× exp \{\frac{x_{0} − ε}{(n − 1)!} ∫_{τ_{∗}(s)}^{τ_{∗}(t)} p(ξ)τ_{∗}^{n−1}(ξ)dξ\}ds > (n − 1)! .  (6.13)

8it is obvious that for c > \frac{(n−1)!}{e} the conditions of Theorem 6.2 are fulfilled and condition (6.10) becomes unnecessary.

On the other hand, as shown while proving Lemma 5.10, by (6.10)–(6.12) there exists k_{0} ∈ ℕ such that

\underline{lim}_{t→+∞}φ_{n−1 k_{0}}(t, 0) > x_{0} − ε,

where φ_{n−1 k_{0}} is defined by (5.9_{n−1}). On account of (6.13) it is obvious that condition (6.3) is fulfilled where φ̃_{n−1 k_{0}} is defined by (5.19_{n−1}). Therefore the conditions of theorem 6.1 are fulfilled, which proves that Theorem 6.3 is valid. ■

Theorem 6.4. Let F ∈ V (τ) and conditions (0.2), (6.1), (6.2), (5.36) − −(5.38) be fulfilled, where p and τ_{∗} are defined by (6.4) and (6.5), respectively. Then equation (0.1) has property A.

Proof. By analogy with the reasoning used while proving Lemma 5.11, there exists k_{0} ∈ ℕ such that (5.39) holds. Therefore by (5.38) it is clear that (6.3) is fulfilled. Thus the conditions of Theorem 6.1 are satisfied, which proves the validity of Theorem 6.4. ■

Corollary 6.3. Let F ∈ V (τ), conditions (0.2), (6.1) and (6.2) be fulfilled, and

\underline{lim}_{t→+∞}t ∫_{t}^{+∞} \frac{τ_{∗}^{n−1}(s)}{s}p(s)ds > M_{n},

where

M_{n} = max{x(1 − x) ⋯ (n − 1 − x) : x ∈ [0, 1]},  (6.14)

p and τ_{∗} are defined by equalities (6.4) and (6.5), respectively. Then equation (0.1) has property A.

Proof. The validity of the corollary follows from Corollary 5.4 and Lemma 5.1. ■

Corollary 6.4. Let F ∈ V (τ), conditions (0.2), (6.1) and (6.2) be fulfilled, and τ(t) ≥ αt, for t ∈ R_{+}, where α ∈]0, 1]. Then the condition

\underline{lim}_{t→+∞}t ∫_{t}^{+∞} s^{n−2}p(s)ds > M_{n}(α),

where M_{n}(α) and p are defined by (5.46) and (6.4), respectively, is sufficient for equation (0.1) to have property A.

Proof. The validity of the corollary follows from Corollary 5.5 and Lemma 5.1. ■

Corollary 6.4′ [12]. Let p ∈ L_{loc}(ℝ_{+}; ℝ_{+}) and

\underline{lim}_{t→+∞}t ∫_{t}^{+∞} s^{n−2}p(s)ds > M_{n},

where M_{n} is defined by (6.14). Then equation (6.7) has property A.

Corollary 6.5. Let F ∈ V (τ) and conditions (0.2), (6.1), (6.2) and (5.47) hold. If besides there exists m ∈ ℕ such that (5.38) is fulfilled with p and τ_{∗} defined by (6.4) and (6.5), then equation (0.1) has property A.

Proof. The validity of the corollary follows from Corollary 5.6 and Lemma 5.1. ■

Corollary 6.6. Let F ∈ V (τ), conditions (0.2), (6.1) and (6.2) hold and

\underline{lim}_{t→+∞}t^{−α}τ(t) > 0, \underline{lim}_{t→+∞}t^{α} ∫_{t}^{+∞} s^{α(n−2)}p(s)ds > 0,

where α ∈]0, 1[ and the function p is defined by (6.4). Then equation (0.1) has property A.

Proof. The validity of the corollary follows from Corollary 5.7 and Lemma 5.1. ■

Theorem 6.5. Let F ∈ V (τ), τ(t) ≥ αt for t ∈ R_{+}, conditions (0.2), (6.1), (6.2) and (5.49) be fulfilled and

\overline{lim}_{t→+∞}(αt + x_{0} ∫_{0}^{αt} s^{n−1}( ∫_{s}^{+∞} ξ^{n−2}p(ξ)dξ)^{−1}p(s)ds) ×

× ∫_{t}^{+∞} s^{n−2}p(s)ds > \frac{(n − 1)!}{α^{n−2}},  (6.15)

where α ∈]0, 1], β ∈]0, M_{n}(α)], M_{n}(α) is defined by (5.46), x_{0} is the smallest root of equation (5.51) and p is defined by (6.4). Then equation (0.1) has property A.

Proof. By (5.49), (6.15) and the same arguments as used in proving Lemma 5.12 there exist ε ∈]0, 1[ and k_{0} ∈ ℕ such that

\overline{lim}_{t→+∞}(αt + (x_{0} − ε) ∫_{0}^{αt} s^{n−1}( ∫_{s}^{+∞} ξ^{n−2}p(ξ)dξ)^{−1}p(s)ds) ×

× ∫_{t}^{+∞} s^{n−2}p(s)ds > \frac{(n − 1)!}{α^{n−2}},  (6.16)
ψ_{n−1 k_{0}}(t, 0) ≥ t^{n−2}( ∫_{t}^{+∞} ξ^{n−2}p(ξ)dξ)^{−1}(x_{0} − ε) for t ≥ t_{∗}, (6.17)

where t_{∗} is sufficiently large and ψ_{n−1 k_{0}} is defined by (5.12). Clearly condition (6.3), where τ_{∗}(t) = αt, holds by virtue of (6.16), (6.17) and (5.20_{n−1}). Therefore the conditions of Theorem 6.1 are fulfilled, which proves the validity of Theorem 6.5. ■

Using Lemmas 5.1 and 5.12, one can easily prove

Theorem 6.5′. If F ∈ V (τ), conditions (0.2), (6.1) and (5.57) are fulfilled and

\underline{lim}_{t→+∞}t ∫_{t}^{+∞} s^{n−2}p(s)ds > \frac{(n − 1)!}{α^{n−1}(1 + x_{0})},

where α ∈]0, 1], β ∈]0, M_{n}(α)], M_{n}(α) is defined by (5.46), x_{0} is the smallest root of equation (5.51) and p is given by (6.4), then equation (0.1) has property A.

Theorem 6.6. Let F ∈ V (τ) and conditions (0.2), (6.1)–(6.3) be fulfilled, where

p(t) = σ^{1−n}(t) ∫_{τ(t)}^{σ(t)} s^{n−1}d_{s}r(s, t),  (6.18)
τ_{∗}(t) = inf{τ_{0}(s) : s ≥ t}, τ_{0}(t) = min{t, σ(t)}  (6.19)

and φ̃_{n−1 k_{0}} and φ_{n−1 k_{0}} (ψ̃_{n−1 k_{0}} and ψ_{n−1 k_{0}}) are defined by (5.19_{n−1}) and (5.9_{n−1}) ((5.20_{n−1}) and (5.12)). Then equation (0.1) has property A.

Proof. Clearly condition (5.3), where p and τ_{∗} are defined by (6.18) and (6.19), respectively, is fulfilled by virtue of (6.3).

Assume that (0.1) has no property A. Then by (0.2) and (6.1) differential inequality (6.6) has no property A. Following Theorem 2.4′, equation (6.7) with the functions p and τ_{∗} defined by (6.18) and (6.19) has no property A.

Let u : [t_{0}, +∞[→ R be a nonoscillatory proper solution of equation (6.7). By Lemma 1.2 there exists l ∈ {0, …, n − 1} such that l + n is odd and condition (2.14_{l}) is fulfilled. Assuming that n is odd and l = 0, it is easy to show that (0.4) is fulfilled. Therefore l ∈ {1, …, n − 1}. Thus by Lemma 5.1 equation (6.7) has a proper solution satisfying (2.14_{n−1}).

But by Lemmas 6.3 and 5.8 (as has been said several times, the functions p and τ_{∗} are defined by (6.18) and (6.19), respectively) equation (6.7) has no proper solution satisfying (2.14_{n−1}). The obtained contradiction proves the validity of the theorem. ■

Corollary 6.7. Let F ∈ V (τ). Then for equation (0.1) to have property A it is sufficient that the conditions of Corollary 6.1 or 6.2 be fulfilled, where the functions p and τ_{∗} are defined by (6.18) and (6.19), respectively.

Using Theorems 6.2 – 6.5, 6.5′ and 6.6 one can easily prove

Theorem 6.7. Let F ∈ V (τ) and the conditions of anyone of Theorems 6.2–6.5, 6.5′ be fulfilled, where the functions p and τ_{∗} are defined by (6.18) and (6.19), respectively. Then equation (0.1) has property A.

6.2. Equations with Property B. Using Lemma 5.3, by a reasoning similar to that used in proving Theorem 6.1 one can prove

Theorem 6.8. Let F ∈ V (τ) and conditions (0.3), (6.1), (6.2) be fulfilled. Moreover, let there exist k_{0} ∈ ℕ such that

\overline{lim}_{t→+∞}[ ∫_{τ_{∗}(t)}^{t}(s − τ_{∗}(t))τ_{∗}^{n−2}(s)φ̃_{n−2 k_{0}}(s, t, 0)p(s)ds +

+ψ̃_{n−2 k_{0}}(t, 0) ∫_{t}^{+∞}(s − τ_{∗}(t))τ_{∗}^{n−3}(s)p(s)ds] > (n − 2)!, (6.20)

for even n, while for odd n (6.20) is fulfilled along with

\overline{lim}_{t→+∞}[ ∫_{τ_{∗}(t)}^{t}(s − τ_{∗}(t))^{n−2}τ_{∗}(s)φ̃_{1k_{0}}(s, t, 0)p(s)ds +

+ψ̃_{1k_{0}}(t, 0) ∫_{t}^{+∞}(s − τ_{∗}(t))^{n−2}p(s)ds] > (n − 2)!,  (6.21)

where p and τ_{∗} are defined by (6.4) and (6.5), respectively, while φ̃_{lk_{0}} and φ_{lk_{0}} (ψ̃_{lk_{0}} and ψ_{lk_{0}}) are defined by (5.19_{l}) and (5.9_{l}) ((5.20_{l}) and (5.13_{l})). Then equation (0.1) has property B.

Corollary 6.8. Let F ∈ V (τ) and conditions (0.3), (6.1), (6.2) be fulfilled. Then for equation (0.1) to have property B when n is even it is sufficient that

\overline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t}(s − τ_{∗}(t))τ_{∗}^{n−2}(s)p(s) exp \{\frac{1}{(n − 2)!} ×

× ∫_{τ_{∗}(s)}^{τ_{∗}(t)} p(ξ)(ξ − τ_{∗}(s))τ_{∗}^{n−2}(ξ)dξ}ds > (n − 2)!,  (6.22)

while for odd n it is sufficient that (6.22) be fulfilled and

\overline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t}(s − τ_{∗}(t))^{n−2}τ_{∗}(s)p(s) exp \{\frac{1}{(n − 2)!} ×

× ∫_{τ_{∗}(s)}^{τ_{∗}(t)} p(ξ)(ξ − τ_{∗}(s))^{n−2}τ_{∗}(ξ)dξ}ds > (n − 2)!,

where p and τ_{∗} are defined by (6.4) and (6.5).

Corollary 6.9. Let F ∈ V (τ) and conditions (0.3), (6.1), (6.2) be fulfilled. Then for equation (0.1) to have property B when n is even it is sufficient that

\overline{lim}_{t→+∞}(τ_{∗}(t) + \frac{1}{(n − 2)!} ∫_{0}^{τ_{∗}(t)} s^{2}τ_{∗}^{n−2}(s)p(s)ds) ×

× ∫_{t}^{+∞}(s − τ_{∗}(t))τ_{∗}^{n−3}(s)p(s)ds > (n − 2)!,  (6.23)

while for odd n it is sufficient that (6.23) be fulfilled and

\overline{lim}_{t→+∞}(τ_{∗}(t) + \frac{1}{(n − 2)!} ∫_{0}^{τ_{∗}(t)} s^{n−1}τ_{∗}(s)p(s)ds) ×

× ∫_{t}^{+∞}(s − τ_{∗}(t))^{n−2}p(s)ds > (n − 2)!,

where p and τ_{∗} are defined by (6.4) and (6.5).

Theorem 6.9. Let F ∈ V (τ) and conditions (0.3), (6.1), (6.2) be fulfilled. Then for equation (0.1) to have property B when n is even it is sufficient that

\underline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t} p(s)(s − τ_{∗}(t))τ_{∗}^{n−2}(s)ds > \frac{(n − 2)!}{e},  (6.24)

while for odd n it is sufficient that (6.24) be fulfilled and

\underline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t} p(s)(s − τ_{∗}(t))^{n−2}τ_{∗}(s)ds > \frac{(n − 2)!}{e},  (6.25)

where the functions p and τ_{∗} are defined by (6.4) and (6.5).

Proof. The validity of the theorem follows from Lemmas 5.3 and 5.9. ■

Remark 6.2. Condition (6.24) or (6.25) cannot be replaced by

\underline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t} p(s)(s − τ_{∗}(t))τ_{∗}^{n−2}(s)ds > \frac{(n − 2)!}{e} − ε,

\underline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t} p(s)(s − τ_{∗}(t))^{n−2}τ_{∗}(s)ds > \frac{(n − 2)!}{e} − ε,

where ε is an arbitrarily small positive number.

Theorem 6.10. Let F ∈ V (τ), conditions (0.3), (6.1), (6.2) be fulfilled, and

\underline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t} p(s)(s − τ_{∗}(t))τ_{∗}^{n−2}(s)ds = c_{1} > 0,

\underline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t} p(s)(s − τ_{∗}(t))^{n−2}τ_{∗}(s)ds = c_{2} > 0.

Then for equation (0.1) to have property B when n is even it is sufficient that

\overline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t}(s − τ_{∗}(t))τ_{∗}^{n−2}(s)p(s) exp \{\frac{x_{1}}{(n − 2)!} ×

× ∫_{τ_{∗}(s)}^{τ_{∗}(t)} p(ξ)(ξ − τ_{∗}(s))τ_{∗}^{n−2}(ξ)dξ}ds > (n − 2)!,  (6.26)

while for odd n it is sufficient that (6.26) be fulfilled and

\overline{lim}_{t→+∞} ∫_{τ_{∗}(t)}^{t}(s − τ_{∗}(t))^{n−2}τ_{∗}(s)p(s) exp \{\frac{x_{2}}{(n − 2)!} ×

× ∫_{τ_{∗}(s)}^{τ_{∗}(t)} p(ξ)(ξ − τ_{∗}(s))^{n−2}τ_{∗}(ξ)dξ}ds > (n − 2)!,

where x_{i} (i = 1, 2) is the smallest root of the equation

exp \{\frac{c_{i}}{(n − 2)!}x\} = x (i = 1, 2)

and p and τ_{∗} are defined by (6.4) and (6.5), respectively. Proof. The validity of the theorem follows from Lemmas 5.3 and 5.10. ■

Theorem 6.11. Let F ∈ V (τ), conditions (0.3), (6.1), (6.2) be fulfilled and

lim_{t→+∞} \frac{t}{τ_{∗}(t)} = +∞.

Moreover, let for even n

\underline{lim}_{t→+∞}τ_{∗}(t) ∫_{t}^{+∞}(s − t)τ^{n−3}(s)p(s)ds > 0  (6.27)

and for some m ∈ ℕ

\overline{lim}_{t→+∞}(τ_{∗}(t) + ∫_{0}^{τ_{∗}(t)} s^{2}τ_{∗}^{m}(s)p(s)ds) ×

× ∫_{t}^{+∞}(s − τ_{∗}(t))τ^{n−3}(s)p(s)ds > (n − 2)!,  (6.28)

while for odd n (6.27) and (6.28) be fulfilled,

\underline{lim}_{t→+∞}τ_{∗}(t) ∫_{t}^{+∞}(s − t)^{n−2}p(s)ds > 0

and for some m ∈ ℕ

\overline{lim}_{t→+∞}(τ_{∗}(t) + ∫_{0}^{τ_{∗}(t)} s^{n−1}τ_{∗}^{m}(s)p(s)ds) ×

× ∫_{t}^{+∞}(s − τ_{∗}(t))^{n−2}p(s)ds > (n − 2)!

with p and τ_{∗} defined by (6.4) and (6.5), respectively. Then equation (0.1) has property B.

Proof. The validity of the theorem follows from Lemma 5.3 and Corollary 5.9. ■

Corollary 6.10. Let F ∈ V (τ), conditions (0.3), (6.1), (6.2) be fulfilled and

\underline{lim}_{t→+∞} \frac{τ(t)}{t^{α}} > 0,

where α ∈]0, 1[. Besides, if for even n

\underline{lim}_{t→+∞}t^{α} ∫_{t}^{+∞}(s − t)s^{α(n−3)}p(s)ds > 0,  (6.29)

while for odd n (6.29) is fulfilled and

\underline{lim}_{t→+∞}t^{α} ∫_{t}^{+∞}(s − t)^{n−2}p(s)ds > 0,

where p is defined by (6.4), then equation (0.1) has property B.

Proof. The validity of the corollary readily follows from Lemma 5.3 and Corollary 5.10. ■

Theorem 6.12. Let F ∈ V (τ), conditions (0.3), (6.1), (6.2) be fulfilled and

\overline{lim}_{t→+∞}τ_{∗}(t) ∫_{t}^{+∞} τ_{∗}^{n−2}(s)p(s)ds > (n − 1)!,  (6.30)

where p and τ_{∗} are defined by (6.4) and (6.5), respectively. Then equation (0.1) has property B.

Proof. First of all we note that (5.3) is fulfilled by virtue of (6.30). Assume that equation (0.1) has no property B. Then by virtue of (0.3) and (6.1) the differential inequality

u^{(n)}(t) sign u(τ(t)) ≥ ∫_{τ(t)}^{σ(t)} |u(s)|d_{s}r(s, t)

has no property B. Therefore, following Theorem 2.4, the equation

u^{(n)}(t) − p(t)u(τ_{∗}(t)) = 0  (6.31)

with p and τ_{∗} defined by (6.4) and (6.5), respectively, has no property B.

Let u : [t_{0}, +∞[→ R be a nonoscillatory solution of equation (6.31). By Lemma 1.1 there is l ∈ {0, …, n}, l + n is even, such that (2.14_{l}) is fulfilled. Assuming that l = n (n is even and l = 0), we can easily to show that (0.5) ((0.4)) is fulfilled. Therefore l ∈ {1, …, n − 2}.

By Lemma 1.3 we have

|u(t)| ≥ \frac{t^{l}}{l!(n − l)!} ∫_{t}^{+∞} s^{n−l−1}p(s)|u(τ_{∗}(s))|ds for t ≥ t_{∗},

where t_{∗} is sufficiently large. Since t^{1−l}|u(t)| is a nondecreasing function, we now obtain

y(t) ≥ \frac{τ_{∗}(t)y(t)}{l!(n − l)!} ∫_{t}^{+∞} s^{n−l−1}τ_{∗}^{l−1}(s)p(s)ds for t ≥ t_{∗},

where y(t) = |u(τ_{∗}(t))|[τ_{∗}(t)]^{1−l}.

But this result contradicts (6.30). The obtained contradiction proves the theorem. ■

Using Theorem 2.4′ and the above reasoning one can easily prove

Theorem 6.13. Let F ∈ V (τ) and conditions (0.3), (6.1), (6.2) be fulfilled. If in addition to this it is assumed that the conditions of anyone of Theorems 6.8–6.12 is fulfilled, where

p(t) = σ^{2−n}(t) ∫_{τ(t)}^{σ(t)} s^{n−2}d_{s}r(s, t)

and τ_{∗} is defined by (6.19), then equation (0.1) has property B.

§ 7. Equations with a linear minorant having Properties A and B

7.1. Some Auxiliary Lemmas.

Lemma 7.1. Let t_{0} ∈ R_{+}, φ, ψ ∈ C([t_{0}, +∞[; ]0, +∞[), ψ be a nonincreasing function, and

lim_{t→+∞} φ(t) = +∞,  (7.1)
\underline{lim}_{t→+∞}ψ(t) · φ̃(t) = 0,  (7.2)

where φ̃(t) = inf{φ(s) : s ≥ t ≥ t_{0}}. Then there exists a sequence of {t_{k}} such that t_{k} ↑ +∞ as k ↑ +∞ and

φ̃(t_{k}) = φ(t_{k}), ψ(t)φ̃(t) ≥ ψ(t_{k})φ̃(t_{k})  (7.3)
for t_{0} ≤ t ≤ t_{k} (k = 1, 2, …).

Proof. Let t ∈]t_{0}, +∞[. Introduce the sets E_{i} (i = 1, 2) by:

t ∈ E_{1} ⇔ φ̃(t) = φ(t), t ∈ E_{2} ⇔ φ̃(s)ψ(s) ≥ φ̃(t)ψ(t) for s ∈ [t_{0}, t].

It is clear that by (7.1) and (7.2) sup E_{i} = +∞ (i = 1, 2). Show that

sup E_{1} ∩ E_{2} = +∞.  (7.4)

Indeed, if we assume that t_{∗} ∈ E_{2} and t_{∗} ∉ E_{1}, by (7.1) there exists t^{∗} > t_{∗} such that φ̃(t) = φ̃(t_{∗}) for t ∈ [t_{∗}, t^{∗}] and φ̃(t^{∗}) = φ(t^{∗}). On the other hand, since ψ is a nonincreasing function, we have ψ(t)φ̃(t) ≥ ψ(t^{∗})φ̃(t^{∗}) for t ∈ [t_{0}, t^{∗}]. Therefore t^{∗} ∈ E_{1} ∩ E_{2}. By the above reasoning we easily ascertain that (7.4) is fulfilled. Thus there exists a sequence of points {t_{k}} such that t_{k} ↑ +∞ for k ↑ +∞ and (7.3) holds. ■

Let σ ∈ C(ℝ_{+}; ℝ_{+}) and lim_{t→+∞} σ(t) = +∞. Denote by M^{+}(σ) the set of continuous mappings φ : C(ℝ_{+}; ℝ_{+}) → L_{loc}(ℝ_{+}; ℝ_{+}) which satisfy the conditions

φ(x)(t) ≥ φ(y)(t) for t ∈ ℝ_{+} and if x(s) ≥ y(s) ≥ 0 for s ≥ σ(t)
φ(xy)(t) ≥ x(σ(t))φ(y)(t) for t ∈ ℝ_{+}
if x(t) ↑ +∞ as t ↑ +∞ and y(s) > 0 for s ≥ σ(t).

(7.5)

Lemma 7.2. Let F ∈ V (τ) and condition (0.2) ((0.3)) be fulfilled, l ∈ {1, …, n − 1}, l + n be odd (l + n be even) and for some t_{0} ∈ ℝ_{+}

|F (u)(t)| ≥ φ(|u|)(t) for u ∈ H_{t_{0},τ} , t ∈ [t_{0}, +∞[,  (7.6)
\underline{lim}_{t→+∞} \frac{σ(t)}{t} > 0,  (7.7)

where φ ∈ M^{+}(σ). Besides, if it is assumed that for any λ ∈ [l − 1, l[ and there exists ε ∈]0, 1[ such that

\underline{lim}_{t→+∞}t^{l−λ} ∫_{t}^{+∞} s^{n−l−1}φ(θ_{λ})(s)ds ≥ ∏_{i=0;i≠l}^{n−1} |λ − i| + ε  (7.8_{l})

and

∫ ^{+∞} t^{n−1}φ(c)(t)dt = +∞ for all c ∈]0, +∞[,  (7.9)

where θ_{λ}(t) = t^{λ}, then equation (0.1) has no proper solution satisfying (2.14_{l}).

Proof. Assume the contrary, i.e. that (0.1) has a proper solution satisfying (2.14_{l}) where l ∈ {1, …, n − 1} and l + n is odd (l + n is even). By virtue of (7.5), (7.7) and (7.9) we have

∫ ^{+∞} t^{n−l}φ(cθ_{l−1})(t)dt = +∞ for all c ∈]0, +∞[,  (7.10_{l})

where θ_{l−1}(t) = t^{l−1}. By (7.6) and (7.10) we clearly obtain

∫ ^{+∞} t^{n−l}|u^{(n)}(t)|dt = +∞.

Therefore by Lemma 1.4 and (7.6)

lim_{t→+∞} t^{1−l}|u(t)| = +∞, lim_{t→+∞} t^{−l}|u(t)| < +∞,  (7.11)

|u(t)| ≥ \frac{1}{(l − 1)!(n − l − 1)!} ∫_{t_{1}}^{t}(t − s)^{l−1} ∫_{s}^{+∞}(ξ − s)^{n−l−1} ×

×φ(|u|)(s)dξ ds for t ≥ t_{1},  (7.12)

where t_{1} ∈ R_{+} is sufficiently large.

Denote by Λ the set of λ ∈ R_{+} satisfying

lim_{t→+∞} t^{−λ}|u(t)| = +∞.

Let λ_{0} = sup Λ. By (7.11) it is clear that λ_{0} ∈ [l − 1, l] and if it is assumed that λ_{0} = l − 1, then λ_{0} ∈ Λ. Therefore on account of (7.7) and (7.8_{l}) there exist t_{∗} ∈ [t_{1} + ∞[, ε_{0} ∈]0, ε[ and λ^{∗} ∈ [l − 1, λ_{0}] ∩ [l − 1, l[ such that

lim_{t→+∞} t^{−λ^{∗}}|u(t)| = +∞, lim_{t→+∞} t^{−λ^{∗}−ε_{0}}|u(t)| = 0,  (7.13)

t^{l−λ^{∗}} ∫_{t}^{+∞} s^{n−l−1}φ(θ_{λ^{∗}})(s)ds > ∏_{i=0}^{l−1}(λ^{∗} + ε_{0} − i) ×

× ∏_{i=l+1}^{n−1}(i − λ^{∗} + ε_{0})(\frac{α}{2})^{−ε_{0}} for t ≥ t_{∗},^{9}  (7.14)

where

α = \underline{lim}_{t→+∞}t^{−1}τ̃(t), τ̃(t) = inf \{ min\{s, σ(s)\} : s ≥ t\}, θ_{λ^{∗}}(t) = t^{λ^{∗}}.

Introducing the notation

ũ(t) = inf{s^{−λ^{∗}}|u(s)| : s ≥ t ≥ t_{∗}}.  (7.15)

by (7.13) we obtain

ũ(t) ↑ +∞ as t ↑ +∞,  (7.16)
\underline{lim}_{t→+∞}t^{−ε_{0}} ũ(t) = 0.  (7.17)

9For l = n − 1 we have ∏_{i=l+1}^{n−1}(i − λ^{∗} + ε_{0}) = 1.

By virtue of (7.15)–(7.17) and Lemma 7.1 there exists an increasing sequence of points {t_{k}} such that

lim_{k→+∞} t_{k} = +∞, (τ̃(t_{k}))^{−ε_{0}} ũ(τ̃(t_{k})) ≤

≤ (τ̃(t))^{−ε_{0}} ũ(τ̃(t)) for t_{∗} ≤ t ≤ t_{k},  (7.18)
ũ(τ̃(t_{k})) = (τ̃(t_{k}))^{−λ^{∗}}|u(τ̃(t_{k}))| (k = k_{0}, k_{0} + 1, …),  (7.19)

where k_{0} is sufficiently large. Using (7.5), from (7.12) we obtain

|u(τ̃(t_{k}))| ≥ \frac{1}{(l − 1)!(n − l − 1)!} ∫_{t_{∗}}^{_{τ}(t_{k})}(τ̃(t_{k}) − s)^{l−1}ũ(τ̃(s)) ×

× ∫_{s}^{+∞}(ξ − s)^{n−l−1}φ(θ_{λ^{∗}})(ξ)dξ ds (k = k_{0}, k_{0} + 1, …). (7.20)

It can be assumed without loss of generality that \frac{_{τ}(t)}{t} ≥ \frac{2}{3}α for t ≥ t_{∗}.

We shall first consider the case l = n − 1. By (7.14), (7.18) and (7.19) it follows from (7.20) that

|u(τ̃(t_{k}))| ≥ \frac{∏_{i=0}^{n−2}(λ^{∗} + ε_{0} − i)(\frac{α}{2})^{−ε_{0}}}{(n − 2)!} ∫_{t_{∗}}^{_{τ}(t_{k})}(τ̃(t_{k}) − s)^{n−2} ×

×s^{λ^{∗}+1−n}ũ(τ̃(s))ds ≥ \frac{∏_{i=0}^{n−2}(λ^{∗} + ε_{0} − i)(\frac{α}{2})^{−ε_{0}}}{(n − 2)!} \frac{ũ(τ̃(t_{k}))}{τ^{ε_{0}+λ^{∗}}(t_{k})} ×

×∫_{t_{∗}}^{_{τ}(t_{k})}(τ̃(t_{k}) − s)^{n−2}s^{λ^{∗}+1+ε_{0}−n}(\frac{τ̃(s)}{s})^{ε_{0}}ds≥ \frac{(\frac{4}{3})^{ε_{0}} ∏_{i=0}^{n−2}(λ^{∗} + ε_{0} − i)}{(n − 2)!} ×

×\frac{ũ(τ̃(t_{k}))}{(τ̃(t_{k}))^{ε_{0}+λ^{∗}}} ∫_{t_{∗}}^{_{τ}(t_{k})}(τ̃(t_{k}) − s)^{n−2}s^{λ^{∗}+1+ε_{0}−n}ds (k = k_{0}, k_{0} + 1, …),

which implies

|u(τ̃(t_{k}))| > \frac{∏_{i=0}^{n−2}(λ^{∗} + ε_{0} − i)(τ̃(t_{k}))^{ε_{0}+λ^{∗}}}{∏_{i=0}^{n−2}(λ^{∗} + ε_{0} − i)(τ̃(t_{k}))^{ε_{0}+λ^{∗}}} |u(τ̃(t_{k}))| =

= |u(τ̃(t_{k}))| (k = k_{1}, k_{1} + 1, …),

where k_{1} > k_{0} is a sufficiently large number. The obtained contradiction proves the validity of the lemma for l = n − 1.

Now consider the case l ∈ {1 …, n − 2}. From (7.20) we obtain

|u(τ̃(t_{k}))| ≥ −\frac{1}{(l − 1)!(n − l − 1)!} ∫_{t_{∗}}^{_{τ}(t_{k})}(τ̃(t_{k}) − s)^{l−1} ×

×ũ(τ̃(s)) ∫_{s}^{+∞}(ξ − s)^{n−l−1}ξ^{−n+l+1} ×

×d ∫_{ξ}^{+∞} ξ_{1}^{n−l−1}φ(θ_{λ^{∗}})(ξ_{1})dξ_{1} ds (k = k_{0}, k_{0} + 1, …).

Therefore

|u(τ̃(t_{k}))| ≥ \frac{1}{(l − 1)!(n − l − 1)!} ∫_{t_{∗}}^{_{τ}(t_{k})}(τ̃(t_{k}) − s)^{l−1}ũ(τ̃(s)) ×

× ∫_{s}^{+∞} ∫_{ξ}^{+∞} ξ_{1}^{n−l−1}φ(θ_{λ^{∗}})(ξ_{1})dξ_{1}((\frac{ξ − s}{ξ})^{n−l−1})′dξ ds.

Since ((\frac{ξ−s}{ξ})^{n−l−1})′ ≥ 0 for ξ ≥ s ≥ t_{∗}, by (7.14) the latter inequality yields

|u(τ̃(t_{k}))| ≥ \frac{(\frac{α}{2})^{−ε_{0}} ∏_{i=0}^{l−1} |λ^{∗} + ε_{0} − i| ∏_{i=l+1}^{n−1} |λ^{∗} + ε_{0} − i|}{(l − 1)!(n − l − 1)!} ×

× ∫_{t_{∗}}^{_{τ}(t_{k})}(τ̃(t_{k}) − s)^{l−1}ũ(τ̃(s)) ∫_{s}^{+∞} ξ^{λ^{∗}−l}((\frac{ξ − s}{ξ})^{n−l−1})′dξ ds =

= \frac{(l − λ^{∗})(\frac{α}{2})^{−ε_{0}} ∏_{i=0}^{l−1} |λ^{∗} + ε_{0} − i|}{(l − 1)!(n − l − 1)!} ∫_{t_{∗}}^{_{τ}(t_{k})}(τ̃(t_{k}) − s)^{l−1} ×

×ũ(τ̃(s)) ∫_{s}^{+∞}(ξ − s)^{n−l−1}ξ^{λ^{∗}−n}dξ ds (k = k_{0}, k_{0} + 1, …).

Hence due to (7.18), (7.19) we have

|u(τ̃(t_{k}))| > \frac{∏_{i=0}^{l−1} |λ^{∗} + ε_{0} − i| ∏_{i=l+1}^{n−1} |λ^{∗} + ε_{0} − i|(τ̃(t_{k}))^{ε_{0}+λ^{∗}}}{∏_{i=0}^{l−1} |λ^{∗} + ε_{0} − i| ∏_{i=l+1}^{n−1} |i − λ^{∗}|(τ̃(t_{k}))^{ε_{0}+λ^{∗}}} ×

×|u(τ̃(t_{k}))| ≥ |u(τ̃(t_{k}))| (k = k_{1}, k_{1} + 1, …),

where k_{1} > k_{0} is sufficiently large. The obtained contradiction proves the validity of the lemma. ■

Lemma 7.2′. Let F ∈ V (τ) and conditions (0.2), (7.6), (7.7), (7.9) ((0.3), (7.6), (7.7), (7.9)) be fulfilled, where φ ∈ M^{+}(τ). Besides, if it is assumed that l ∈ {1, …, n − 1} l + n is odd (l + n is even), and for any λ ∈ [l − 1, l[ there exists ε ∈]0, 1[ such that

\underline{lim}_{t→+∞}t ∫_{t}^{∞} s^{n−2−λ}φ(θ_{λ})(s)ds ≥ ∏_{i=0}^{n−1} |λ − i| + ε,  (7.21_{l})

where θ_{λ}(t) = t^{λ}, then equation (0.1) has no proper solution satisfying (2.14_{l}).

Proof. To prove the lemma it is sufficient to show that condition (7.21_{l}) implies the validity of (7.8_{l}).

By (7.21) there exist t_{0} ∈ R_{+} and ε_{0} ∈]0, ε[ such that for any λ ∈ [l −1, l[ we have

t ∫_{t}^{∞} s^{n−2−λ}φ(θ_{λ})(s)ds ≥ ∏_{i=0}^{n−1} |λ − i| + ε_{0} for t ∈ [t_{0}, +∞[,

so that

t^{l−λ} ∫_{t}^{∞} s^{n−1−λ}φ(θ_{λ})(s)ds = −t^{l−λ} ∫_{t}^{+∞} s^{λ−l+1} ×

×d ∫_{s}^{+∞} ξ^{n−2−λ}φ(θ_{λ})(ξ)dξ = t ∫_{t}^{+∞} s^{n−2−λ}φ(θ_{λ})(s)ds +

+(λ + 1 − l)t^{l−λ} ∫_{t}^{+∞} s^{λ−l} ∫_{s}^{∞} ξ^{n−2−λ}φ(θ_{λ})(ξ)dξ ds ≥

≥ ( ∏_{i=0}^{n−1} |λ − i| + ε_{0})(1 + (λ + 1 − l)t^{l−λ} ∫_{t}^{+∞} ξ^{λ−l−1}dξ) =

= ( ∏_{i=0}^{n−1} |λ − i| + ε_{0})(1 + \frac{λ + 1 − l}{l − λ} ) ≥ ∏_{i=0;i≠l}^{n−1} |λ − i| + ε_{0}

for t ∈ [t_{0}, +∞[, λ ∈ [l − 1, l[.

Therefore (7.8_{l}) is fulfilled. ■

7.2. Functional Differential Equations with a Linear Minorant Having Properties A and B.

Theorem 7.1. Let F ∈ V (τ) and conditions (0.2), (7.6), (7.7), (7.9) be fulfilled, where φ ∈ M^{+}(τ). Besides, if condition (7.8_{l}) holds for any l ∈ {1, …, n − 1} and λ ∈ [l − 1, l[, where l + n is odd, then equation (0.1) has property A.

Proof. Let u : [t_{0}, +∞[→ R be a nonoscillatory proper solution of equation (0.1). Then by Lemma 1.1 there exists l ∈ {0, …, n − 1} such that l + n is odd (2.14_{l}) is fulfilled. By Lemma 7.2 we have l ∉ {1, …, n − 1}. Assuming that n is odd and l = 0 and using (7.9), we can easily show that (0.4) holds. ■

Theorem 7.1′. Let F ∈ V (τ) and conditions (0.2), (7.6), (7.7), (7.9) be fulfilled, where φ ∈ M^{+}(τ). Besides, if condition (7.21_{l}), where l +n is odd, is fulfilled for any l ∈ {1, …, n − 1} and λ ∈ [l − 1, l[, then equation (0.1) has property A.

Proof. Since all the conditions of Theorem 7.1 are fulfilled on account of Lemma 7.2′, this proves the validity of Theorem 7.1′. ■

Theorem 7.2. Let F ∈ V (τ), and conditions (0.3), (7.6), (7.7), (7.9), where φ ∈ M^{+}(τ), be fulfilled. Besides, if condition (7.8_{l}) holds for any l ∈ {1, …, n − 1} and λ ∈ [l − 1, l[, where l + n is even, then equation (0.1) has property B.

Proof. Let u : [t_{0}, +∞[→ R be a nonoscillatory proper solution of equation (0.1). By Lemma 1.1 there exists l ∈ {0, …, n} such that l + n is even and (2.14_{l}) is fulfilled. By Lemma 7.2 we have l ∉ {1, …, n − 1}. If n is even and l = 0, then (0.4) is fulfilled.

Let l = n. By (7.5), (7.7) and (7.8_{l}) we obtain

∫ ^{+∞} φ(cθ_{n−1})(t)dt = +∞, for all c > 0,  (7.22)

where θ_{n−1}(t) = t^{n−1}.

On the other hand, by (2.14_{n}) there exist t_{0} ∈ R_{+} and c ∈]0, +∞[ such that |u(t)| ≥ ct^{n−1} for t ≥ t_{0}. Thus by (7.6) and (7.22) we find from (0.1) that

|u^{(n−1)}(t)| ≥ ∫_{t_{0}}^{t} φ(cθ_{n−1})(s)ds → +∞ for t → +∞.

By (2.14_{n}) it is now clear that (0.5) is fulfilled. ■

Theorem 7.2′. Let F ∈ V (τ) and conditions (0.3), (7.6), (7.7), (7.9) hold, where φ ∈ M^{+}(τ). Besides, if (7.21_{l}) is fulfilled for any l ∈ {1, …, n − 1} and λ ∈ [l − 1, l[ where l + n is even, then equation (0.1) has property B.

Proof. The Theorem is valid because all the conditions of Theorem 7.2 are fulfilled by Lemma 7.2′. ■

Theorem 7.3. Let F ∈ V (τ), condition (0.2) be fulfilled and let for any t_{0} ∈ R_{+}

|F (u)(t)| ≥ ∑_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} |u(s)|d_{s}r_{i}(s, t) for u ∈ H_{t_{0},τ} , t ≥ t_{0}, (7.23)

where

τ_{i}; σ_{i} ∈ C(ℝ_{+}; ℝ_{+}), τ_{i}(t) ≤ σ_{i}(t) for t ∈ R_{+},
r_{i}(s, t), r_{i} is measurable, r_{i}(·, t) ↑, as s ↑ (i = 1, …, m),

(7.24)

\underline{lim}_{t→+∞} \frac{τ_{i}(t)}{t} > 0 (i = 1, …, m).  (7.25)

Besides, if for any l ∈ {1, …, n − 1} and λ ∈ [l − 1, l[, where l + n is odd, there exists ε ∈]0, 1[ such that

\underline{lim}_{t→+∞}t^{l−λ}∫_{t}^{+∞}ξ^{n−l−1}∑_{i=1}^{m} ∫_{τ_{i}(ξ)}^{σ_{i}(ξ)} s^{λ}d_{s}r_{i}(s, ξ)dξ > ∏_{i=0;i≠l}^{n−1}|λ − i|+ε (7.26_{l})

and

∫ ^{+∞} t^{n−1} ∑_{i=1}^{m} (r_{i}(σ_{i}(t), t) − r_{i}(τ_{i}(t), t))dt = +∞,  (7.27)

then equation (0.1) has property A.

Proof. To prove the theorem it suffices to note that the conditions of Theorem 7.1 are fulfilled by virtue of (7.23) and (7.24)

\begin{matrix}φ(x)(t) = ∑_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} x(s)d_{s}r_{i}(s, t), \\ τ(t) = min\{τ_{i}(t) : i = 1, …, m\}. ■ \end{matrix}  (7.28)

Corollary 7.1. Let F ∈ V (τ), condition (0.2) be fulfilled and let for any t_{0} ∈ R_{+}

|F (u)(t)| ≥ ∑_{i=1}^{m} p_{i}(t)|u(τ_{i}(t))| for u ∈ H_{t_{0},τ} , t ≥ t_{0},  (7.29)

where

\begin{matrix}p_{i} ∈ L_{loc}(ℝ_{+}; ℝ_{+}), τ_{i} ∈ C(ℝ_{+}; ℝ_{+}), \\ \underline{lim}_{t→+∞} \frac{τ_{i}(t)}{t} > 0 (i = 1, …, m). \end{matrix}  (7.30)

Besides, if for any l ∈ {1, …, n − 1} and λ ∈ [l − 1, l[, where l + n is odd, there exists ε ∈]0, 1[ such that

\underline{lim}_{t→+∞}t^{l−λ} ∫_{t}^{+∞} s^{n−l−1} ∑_{i=1}^{m} p_{i}(s)τ_{i}^{λ}(s)ds ≥ ∏_{i=0;i≠l}^{n−1} |λ − i| + ε (7.31_{l})

and

∫ ^{+∞} t^{n−1} ∑_{i=1}^{m} p_{i}(t)dt = +∞,  (7.32)

Corollary 7.2. If F ∈ V (τ), condition (0.2) is fulfilled and for any t_{0} ∈ ℝ_{+} we have

|F (u)(t)| ≥ \frac{c}{t^{n+1}} ∫_{αt}^{\overline{α}t} |u(s)|ds for u ∈ H_{t_{0},τ} , t ≥ t_{0},  (7.33)

where 0 < α < \overline{α} and

c > max{−(λ + 1)λ(λ − 1) ⋯ (λ − n + 1) ×
×(\overline{α} ^{λ+1} − α^{λ+1})^{−1} : λ ∈ [0, n − 1]\},  (7.34)

then equation (0.1) has property A.

Corollary 7.3. Let c > 0, 0 < α < \overline{α}. Then for the equation

u^{(n)}(t) + \frac{c}{t^{n+1}} ∫_{αt}^{\overline{α}t} u(s)ds = 0  (7.35)

to have property A it is necessary and sufficient that (7.34) be fulfilled.

Theorem 7.3′. Let F ∈ V (τ) and conditions (0.2), (7.23)–(7.25), (7.27) be fulfilled. Besides, if for any l ∈ {1, …, n − 1} and λ ∈ [l − 1, l[, where l + n is odd, there exists ε ∈]0, 1[ such that the inequality

\underline{lim}_{t→+∞}t∫_{t}^{+∞}ξ^{n−2−λ}∑_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} s^{λ}d_{s}r_{i}(s, ξ)dξ ≥ ∏_{i=0}^{n−1} |λ − i| + ε, (7.36)

holds, then equation (0.1) has property A.

Proof. It suffices to note that the conditions of Theorem 7.1′ are fulfilled with φ and τ defined by (7.28). ■

Corollary 7.4. Let F ∈ V (τ) and conditions (0.2), (7.29), (7.30), (7.32) be fulfilled. Besides, if for any l ∈ {1, …, n − 1} and λ ∈ [l − 1, l[, where l + n is odd, there exists ε ∈]0, 1[ such that

\underline{lim}_{t→+∞}t ∫_{t}^{+∞} s^{n−2−λ} ∑_{i=1}^{m} p_{i}(s)τ_{i}^{λ}(s)ds ≥ ∏_{i=0}^{n−1} |λ − i| + ε, (7.37)

then equation (0.1) has property A.

Theorem 7.4. Let F ∈ V (τ) and conditions (0.3), (7.23)–(7.25), (7.27) be fulfilled. Besides, if (7.26_{l}) holds for any l ∈ {1, …, n − 1} and λ ∈ [l − 1, l[ where l + n is even, then equation (0.1) has property B.

Proof. It suffices to note that the conditions of Theorem 7.2 are fulfilled with φ and τ defined by (7.28). ■

Corollary 7.5. Let F ∈ V (τ) and conditions (0.3), (7.29), (7.30), (7.32) be fulfilled. Besides, if for any l ∈ {1, …, n − 1} and λ ∈ [l − 1, l[, where l + n is even, there exists ε ∈]0, 1[ such that (7.31_{l}) holds, then equation (0.1) has property B.

Corollary 7.6. Let F ∈ V (τ) and conditions (0.3), (7.33) hold for 0 < α < \overline{α}. Besides, if

c > max\{(λ + 1)λ(λ − 1) ⋯ (λ − n + 1)(\overline{α} ^{λ+1} − α^{λ+1})^{−1} : λ ∈ [0, n − 1]\},

then equation (0.1) has property B.

Corollary 7.7. Let c < 0 and 0 < α < \overline{α}. Then for equation (7.35) to have property B it is necessary and sufficient that

c < − max\{(λ + 1)λ(λ − 1) ⋯ (λ − n + 1)(\overline{α} ^{λ+1} − α^{λ+1})^{−1} : λ ∈ [0, n − 1]\}.

Theorem 7.4′. Let F ∈ V (τ) and conditions (0.3), (7.23)–(7.25), (7.27) be fulfilled. Besides, if for any l ∈ {1, …, n − 1} and λ ∈ [l − 1, l[, where l +n is even, there exists ε ∈]0, 1[ such that (7.36) holds, then equation (0.1) has property B.

Proof. It suffices to note that the conditions of Theorem 7.2′ are fulfilled with φ and τ defined by (7.28). ■

Corollary 7.8. Let F ∈ V (τ) and conditions (0.3), (7.29), (7.30), (7.32) be fulfilled. Besides, if for any l ∈ {1, …, n − 1} and λ ∈ [l − 1, l[, where l +n is even, there exists ε ∈]0, 1[ such that (7.37) holds, then equation (0.1) has property B.

Theorem 7.5. Let F ∈ V (τ), conditions (0.2), (7.23)–(7.25), (7.27) be fulfilled and

σ_{i}(t) ≤ t for t ∈ R_{+} (i = 1, …, m).  (7.38)

Besides, if for any λ ∈ [n−2, n−1] there exists ε ∈]0, 1] such that (7.26_{n−1}) holds, then equation (0.1) has property A.

Proof. Assume the contrary, that equation (0.1) has no property A. In that case by (0.2) and (7.23) the inequality

u^{(n)}(t) sign u(t) + ∑_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} |u(s)|d_{s}r_{i}(s, t) ≤ 0  (7.39)

has no property A.

Let u : [t_{0}, +∞[→ R be a nonoscillatory proper solution of (7.39). Then by Lemma 1.1 there exists l ∈ {0, …, n − 1} (l + n is odd) such that (2.14_{l}) is fulfilled. If it is assumed that n is odd and l = 0, then (0.4) will hold by virtue of (7.27) because following our assumption inequality (7.39) has no property A. Assume that l ∈ {1, …, n − 1}. By (7.25) and (7.27) it is clear that

∫ ^{+∞} t^{n−l}|u^{(n)}(t)|dt = +∞ for all l ∈ {1, …, n − 1}.

Therefore by Lemma 1.3

|u(t)|/t^{l} ↓ for t ↑ .

Hence on the interval [t_{1}, +∞[, where t_{1} is sufficiently large, the function u is a solution of the inequality

u^{(n)}(t) sign u(t) + ∑_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} s^{l}d_{s}r_{i}(s, t)\frac{|u(σ_{i}(t))|}{(σ_{i}(t))^{l}} ≤ 0.

Like in the case of Lemma 5.2 we can show that on the interval [t_{2}, +∞[ the last inequality has an oscillatory proper solution u_{1} satisfying (2.14_{n−1}) where t_{2} ≥ t_{1} is sufficiently large. On the other hand, by Lemma 1.3 we have

\frac{u_{1}(t)}{t^{l}} ↑ +∞ for t ↑ +∞.^{10}

Therefore on the interval [t_{2}, +∞[ the function u_{1} is a solution of inequality (7.39) satisfying (2.14_{n−1}). But Theorem 7.3 and condition (7.26_{n−1}) imply that (7.39) has no solution satisfying (2.14_{n−1}). The obtained contradiction proves the theorem. ■

Corollary 7.9. If F ∈ V (τ), conditions (0.2), (7.29), (7.30), (7.32) are fulfilled with λ ∈ [n − 2, n − 1] and there exists ε ∈]0, 1] such that condition (7.31_{n−1}) holds, then equation (0.1) has property A.

Corollary 7.10. If F ∈ V (τ) and conditions (0.2), (7.33) are fulfilled with 0 < α < \overline{α} ≤ 1, then for equation (0.1) to have property A it is sufficient that

c > max{−(λ + 1)λ(λ − 1) ⋯ (λ − n + 1) ×
×(\overline{α} ^{λ+1} − α^{λ+1})^{−1} : λ ∈ [n − 2, n − 1]\}.  (7.40)

Corollary 7.11. Let c > 0, 0 < α < \overline{α} ≤ 1. Then condition (7.40) is necessary and sufficient for equation (7.35) to have property A.

Corollary 7.12. Let F ∈ V (τ), condition (0.2) be fulfilled and for some t_{0} ∈ R_{+}

|F (u)(t)| ≥ \frac{1}{t^{n}} ∑_{i=1}^{m} c_{i}|u(α_{i}t)| for t ∈ [t_{0}, +∞[, u ∈ M_{t_{0}}, (7.41)

10It is assumed that l < n − 2 because otherwise l = n − 1, i.e. (7.39) has a solution satisfying (2.14_{n−1}).

where 0 < α_{i} ≤ 1 and c_{i} > 0 (i = 1, …, m). Then the condition

∑_{i=1}^{m} c_{i}α_{i}^{λ} > −λ(λ − 1) ⋯ (λ − n + 1) for λ ∈ [n − 2, n − 1]

is sufficient for equation (0.1) to have property A.

Corollary 7.13. [12]. Let p ∈ L_{loc}(ℝ_{+}; ℝ_{+}) and

\underline{lim}_{t→+∞}t ∫_{t}^{+∞} s^{n−2}p(s)ds > max\{−λ(λ−1) ⋯ (λ−n+1) : λ ∈ [n−2, n−1]\}.

Then equation (6.8) has property A.

Theorem 7.6. If F ∈ V (τ) and conditions (0.3), (7.23)–(7.25), (7.27), (7.38) are fulfilled, then for equation (0.1) to have property B it is sufficient in the case of an even n (of an odd n) that for any λ ∈ [n−1, n−2[ (for any λ ∈ [0, 1] ∪ [n − 3, n − 2]) there exist ε ∈]0, 1[ such that conditions (7.26_{n−2}) ((7.26_{1}) and (7.26_{n−2})) be fulfilled.

Proof. Let us assume the contrary, i.e. that equation (0.1) has no property B. Then by (0.3) and (7.23) the inequality

u^{(n)}(t) sign u(t) ≥ ∑_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} |u(s)|d_{s}r_{i}(s, t)  (7.42)

has no property B. Let u ∈ [t_{0}, +∞[→ R be a nonoscillatory proper solution of inequality (7.42). Following Lemma 1.1, there exists l ∈ {0, …, n} (l + n is even) such that (2.14_{l}) is fulfilled. If we assume that l = n (n is even and l = 0), then (0.5) ((0.4)) will be fulfilled by virtue of (7.25) and (7.27). Thus, since by our assumption inequality (7.42) has no property B, we conclude that l ∈ {1, …, n − 2}. If l ∈ {2, …, n − 2}, then by a reasoning similar to that used in considering Theorem 7.5 we prove that by Corollary 5.1 inequality (7.42) has an oscillatory proper solution satisfying (2.14_{n−2}). This means that for an even n (for an odd n) inequality (7.42) has a solution satisfying (2.14_{n−2}) ((2.14_{1}) or (2.14_{n−2})).

However by Theorem 7.4 and (7.26_{1}), (7.26_{n−2}) the differential inequality (7.42) has no proper solution satisfying (2.14_{1}), (2.14_{n−2}). The obtained contradiction proves the theorem. ■

Corollary 7.14. Let F ∈ V (τ) and conditions (0.3), (7.29), (7.30), (7.38) be fulfilled. Besides, if in the case of an even n (in the case of an odd n) for any λ ∈ [n − 2, n − 1] (for any λ ∈ [0, 1[∪[n − 2, n − 1[) there exists ε ∈]0, 1[ such that condition (7.31_{n−2}) ((7.31_{1}) and (7.31_{n−2})) holds, then equation (0.1) has property B.

Corollary 7.15. Let F ∈ V (τ) and conditions (0.3), (7.33) be fulfilled, where 0 < α < \overline{α} ≤ 1 and c ∈]0, +∞[. Moreover, if

c > max{(λ + 1)λ(λ − 1) ⋯ (λ − n + 1) ×
×(\overline{α} ^{λ+1} − α^{λ+1})^{−1} : λ ∈ [0, 1] ∪ [n − 3, n − 2]\},

then equation (0.1) has property B.

Corollary 7.16. Let c < 0 and 0 < α < \overline{α} ≤ 1. Then for equation (7.35) to have property B it is necessary and sufficient that

c < − max{(λ + 1)λ(λ − 1) ⋯ (λ − n + 1) ×
×(\overline{α} ^{λ+1} − α^{λ+1})^{−1} : λ ∈ [0, 1] ∪ [n − 3, n − 2]\}.

Theorem 7.6′. If F ∈ V (τ) and conditions (0.3), (7.23)–(7.25), (7.27), (7.38) are fulfilled, then for equation (0.1) to have property B it is sufficient in the case of an even n (in the case of an odd n) that for any λ ∈ [n−3, n−2] (λ ∈ [0, 1[∪[n − 3, n − 2[) there exist ε ∈]0, 1[ such that (7.36) be satisfied.

Proof. This theorem is proved like Theorem 7.6, if we replace Theorem 7.4 by Theorem 7.4′. ■

Corollary 7.17. If F ∈ V (τ), and conditions (0.3), (7.29), (7.30), (7.32), (7.38) are fulfilled, then for equation (0.1) to have property B it is sufficient in the case of an even n (in the case of an odd n) that for any λ ∈ [n−3, n−2[ (for any λ ∈ [0, 1[∪[n − 3, n − 2]) there exist ε ∈]0, 1[ such that condition (7.37) hold.

Corollary 7.18. Let p ∈ L_{loc}(ℝ_{+}; ℝ_{−}) and

\underline{lim}_{t→+∞}t∫_{t}^{+∞}s^{n−2}|p(s)|ds>max \{λ(λ−1) ⋯ (λ−n+1) : λ ∈[0, 1]∪[n−3, n−2]\}.

Then equation (6.8) has property B.

In the case of an even n Corollary 2 from [12] is a particular case of the above corollary.

Corollary 7.19. Let F ∈ V (τ) and conditions (0.3), (7.41) be fulfilled, where c_{i} > 0 and 0 < α_{i} ≤ 1 (i = 1, …, m). Besides, if

∑_{i=1}^{m} c_{i}α_{i}^{λ} > λ(λ − 1) ⋯ (λ − n + 1) for λ ∈ [0, 1] ∪ [n − 3, n − 2],

then equation (0.1) has property B.

Theorem 7.7. If F ∈ V (τ), conditions (0.2), (7.23), (7.24), (7.27) are fulfilled,

τ_{i}(t) ≥ t for t ∈ R_{+} (i = 1, …, m)  (7.43)

and in the case of an even n (in the case of an odd n) for any λ ∈ [0, 1[ (for any λ ∈ [1, 2[∪[n − 2, n − 1[) there exists ε ∈]0, 1[ such that condition (7.26_{1}) ((7.26_{2}) and (7.26_{n−1})) holds, then equation (0.1) has property A.

Proof. Assume the contrary, i.e. that (0.1) has no property A. Then by (0.2) and (7.23) inequality (7.39) has no property A.

Assuming u : [t_{0}, +∞[→ R to be an oscillatory proper solution of inequality (7.39), by Lemma 1.1 there exists l ∈ {0, …, n − 1} (l + n is odd) such that (2.14_{l}) is satisfied. If n is odd and l = 0, then (0.4) is fulfilled. Since (7.39) has no property A, we conclude that l ∈ {1, …, n − 1}. Similarly to Lemma 5.4 we show that for an even n (for an odd n) inequality (7.39) has a proper solution of form (2.14_{1}) ((2.14_{2}) or (2.14_{n−1})).

On the other hand, on account of Theorem 7.3 and (7.26_{1}) ((7.26_{2}) and (7.26_{n−1})) inequality (7.39) has no proper solution satisfying (7.14_{1}) ((7.14_{2}), (7.14_{n−1})). The obtained contradiction proves the theorem. ■

Corollary 7.20. Let F ∈ V (τ), conditions (0.2), (7.29) (7.32), (7.43) be fulfilled and in the case of an even n (in the case of an odd n) for any λ ∈ [0, 1[ (for any λ ∈ [1, 2[∪[n − 2, n − 1[) there exist ε ∈]0, 1[ such that condition (7.31_{1}) ((7.31_{2}) and (7.31_{n−1})) holds. Then equation (0.1) has property A.

Corollary 7.21. Let F ∈ V (τ) and conditions (0.2), (7.33) be fulfilled, where 1 ≤ α < \overline{α} and c > 0. Moreover, if for an even n (for an odd n)

c > max{−(λ + 1)λ(λ − 1) ⋯ (λ − n + 1) ×
×(\overline{α} ^{λ+1} − α^{λ+1})^{−1} : λ ∈ [0, 1]\}  (7.44)
(c > max{−(λ + 1)λ(λ − 1) ⋯ (λ − n + 1) ×
×(\overline{α} ^{λ+1} − α^{λ+1})^{−1} : λ ∈ [1, 2] ∪ [n − 2, n − 1]\}),  (7.45)

then equation (0.1) has property A.

Corollary 7.22. Let c > 0 and 1 ≤ α < \overline{α}. Then in the case of an even n (in the case of an odd n) condition (7.44) ((7.45)) is necessary and sufficient for equation (7.35) to have property A.

Theorem 7.7′. Let F ∈ V (τ), conditions (0.2), (7.23)–(7.25), (7.27), (7.43) be fulfilled and in the case of an even n (in the case of an odd n) for any λ ∈ [0, 1[ (for any λ ∈ [1, 2[∪[n − 2, n − 1[) there exist ε ∈]0, 1[ such that (7.36) holds. Then equation (0.1) has property A.

Proof. This theorem is proved like Theorem 7.7, if we replace Theorem 7.3 by Theorem 7.3′. ■

Corollary 7.23. Let F ∈ V (τ), conditions (0.2), (7.29), (7.32), (7.43) be fulfilled and in the case of an even n (in the case of an odd n) for any λ ∈ [0, 1[ (for any λ ∈ [1, 2[∪[n − 2, n − 1[) there exist ε ∈]0, 1[ such that (7.37) holds. Then equation (0.1) has property A.

Theorem 7.8. Let F ∈ V (τ), conditions (0.3), (7.23), (7.24), (7.27), (7.43) be fulfilled and in the case of an even n (in the case of an odd n) for any λ ∈ [1, 2[ (for any λ ∈ [0, 1[) there exist ε ∈]0, 1[ such that (7.26_{2}) ((7.26_{1})) holds. Then equation (0.1) has property B.

Proof. Assume the contrary, i.e. that (0.1) has no property B. Then by (0.3) and (7.23) inequality (7.42) has no property B.

Assume u : [t_{0}, +∞[→ R to be a nonoscillatory proper solution of inequality (7.42). Then by Lemma 1.1 there exists l ∈ {0, …, n} (l + n is even) such that (2.14_{l}) holds. If l = n (n is even and l = 0), then (0.5) ((0.4)) is satisfied. Therefore, since inequality (7.42) has no property B by virtue of our assumption, we conclude that l ∈ {1, …, n − 2}. Similarly to Lemma 5.5 it can be shown that for an even n (for an odd n) inequality (7.42) has a proper solution satisfying (2.14_{2}) ((2.14_{1})).

On the other hand, according to Theorem 7.4 and (7.26_{2}) ((7.26_{1})) for an even n (for an odd n) inequality (7.42) has no solution satisfying (7.14_{2}) ((7.14_{1})). The obtained contradiction proves the theorem. ■

Corollary 7.24. Let F ∈ V (τ), conditions (0.3), (7.29), (7.32), (7.43) be fulfilled and in the case of an even n (in the case of an odd n) for any λ ∈ [1, 2[ (for any λ ∈ [0, 1[) there exist ε ∈]0, 1[ such that (7.31_{2}) ((7.31_{1})) holds. Then equation (0.1) has property B.

Corollary 7.25. Let F ∈ V (τ) and conditions (0.3), (7.33) be fulfilled, where 1 ≤ α < \overline{α} and c > 0. Moreover, if

c > max\{(λ + 1)λ(λ − 1) ⋯ (λ − n + 1)(\overline{α}^{λ+1} − α^{λ+1})^{−1} : λ ∈ [0, 2]\},

then equation (0.1) has property B.

Corollary 7.26. Let c < 0 and 1 ≤ α < \overline{α}. Then for equation (7.35) to have property B it is necessary and sufficient that

c < − max\{(λ + 1)λ(λ − 1) ⋯ (λ − n + 1)(\overline{α} ^{λ+1} − α^{λ+1})^{−1} : λ ∈ [0, 2]\}.

Theorem 7.8′. Let F ∈ V (τ), conditions (0.3), (7.23), (7.24), (7.27), (7.43) be fulfilled and in the case of an even n (in the case of an odd n) for any λ ∈ [1, 2[ (for any λ ∈ [0, 1[) there exist ε ∈]0, 1[ such that (7.36) holds. Then equation (0.1) has property B.

Proof. This theorem is proved like Theorem 7.8, if we use Theorem 7.3′ instead of Theorem 7.3. ■

Corollary 7.27. Let F ∈ V (τ), conditions (0.3), (7.29), (7.32), (7.43) be fulfilled and in the case of an even n (in the case of an odd n) for any λ ∈ [1, 2[ (for any λ ∈ [0, 1[) there exist ε ∈]0, 1[ such that condition (7.37) holds. Then equation (0.1) has property B.

Corollary 7.28. Let F ∈ V (τ) and conditions (0.3), (7.41) be fulfilled, where α_{i} ≥ 1 and c_{i} > 0 (i = 1, …, m). Then the condition

∑_{i=1}^{m} c_{i}α_{i}^{λ} > λ(λ − 1) ⋯ (λ − n + 1) for λ ∈ [0, 2]

is sufficient for equation (0.1) to have property B.

7.3. Sufficient Conditions for the Existence of a Nonoscillatory Solution. The corollaries formulated in Subsection 7.2 are exact, which is testified to by the validity of the following

Theorem 7.9. Let F ∈ V (τ), condition (0.2) ((0.3)) be fulfilled and

|F (u)(t)| ≤ φ(u)(t) for t ∈ ℝ_{+}, u ∈ C(ℝ_{+}; ℝ_{+}),  (7.46)

where φ : C(ℝ_{+}; ℝ_{+}) → L_{loc}(ℝ_{+}; ℝ_{+}) is a continuous mapping,

φ(x)(t) ≥ φ(y)(t) if x, y ∈ C(ℝ_{+}; ℝ_{+}) and

x(s) ≥ y(s) for s ≥ τ(t). (7.47)

Moreover, if for some t_{0} ∈ ℝ_{+}, l ∈ {1, …, n − 1} and λ ∈ [l − 1, l[, ^{11} where l + n is odd (l + n is even) we have

t^{l−λ} ∫_{t}^{+∞} s^{n−l−1}φ(θ_{λ})(s)ds ≤ ∏_{i=0;i≠l}^{n−1} |λ − i| for t ≥ t_{0}, (7.48)

where θ_{λ}(t) = t^{λ}, then equation (0.1) has a proper solution satisfying (2.14_{l}).

Proof. Let U be the set of functions u ∈ C([t_{0}, +∞[; ℝ) satisfying the condition

u(t) = c_{l} for t ∈ [t_{0}, t_{∗}], c(t − t_{∗})^{l−1} ≤ u(t) ≤ t^{λ} for t > t_{∗}, (7.49)

where t_{∗} = max\{s : τ_{∗}(s) ≤ t_{0}\}, τ_{∗}(t) = min\{t, τ(t)\}, c ∈ [0, \frac{1}{l!} ∏_{i=0}^{l−2} |λ − i|] and c_{l} = c for l = 1, c_{l} = 0 for l > 1. Define the operator T : U → C([t_{0}, +∞[; ℝ) by

T (u)(t) = \begin{cases}c(t − t_{∗})^{l−1} + \frac{(−1)^{n+l+1}}{(l−1)!(n−l−1)!} ∫_{t_{∗}}^{t}(t − s)^{l−1} ∫_{s}^{+∞}(ξ − s)^{n−l−1}× \\ ×F (u)(ξ)dξ ds for t > t_{∗}, \\ c_{l} for t ∈ [t_{0}, t_{∗}]. \end{cases} (7.50)

11When λ = l − 1 we have φ(θ_{λ})(t) = 0 almost everywhere on the interval [t_{0}, +∞[.

Since in that case the validity of the theorem is obvious, it will be assumed below that λ ∈]l − 1, l[.

Show that T U ⊂ U. By (0.2) ((0.3)) and (7.49) and the assumption that l + n is odd (l + n is even) we find that if u ∈ U, then T (u)(t) ≥ c(t − t_{∗})^{l−1} for t ≥ t_{∗}. Show that if u ∈ U, then T (u)(t) ≤ t^{λ} for t ≥ t_{∗}.

Consider at first the case l = n−1. By virtue of (7.46)–(7.49) from (7.50) we obtain

T (u)(t) ≤ c(t − t_{∗})^{n−2} + \frac{∏_{i=0}^{n−2}(λ − i)}{(n − 2)!} ∫_{t_{∗}}^{t}(t − s)^{n−2}s^{λ−(n−1)}ds ≤

≤ c(t − t_{∗})^{n−2} − \frac{∏_{i=0}^{n−3}(λ − i)}{(n − 2)!}t_{∗}^{λ+2−n}(t − t_{∗})^{n−2} + t^{λ} ≤ t^{λ} for t > t_{∗}.^{12}

Let now l < n − 1. Then by (7.46)–(7.49) from (7.50) we have

T (u)(t) ≤ c(t − t_{∗})^{l−1} − \frac{1}{(l − 1)!(n − l − 1)!} ∫_{t_{∗}}^{t}(t − s)^{l−1} ×

× ∫_{s}^{+∞} (\frac{ξ − s}{ξ})^{n−l−1}d ∫_{ξ}^{+∞} ξ_{1}^{n−l−1}φ(θ_{λ})(ξ_{1})dξ_{1} ds = c(t − t_{∗})^{l−1} +

+\frac{1}{(l − 1)!(n − l − 1)!} ∫_{t_{∗}}^{t}(t − s)^{l−1} ∫_{s}^{+∞} ((\frac{ξ − s}{ξ})^{n−l−1})′ ×

× ∫_{ξ}^{+∞} ξ_{1}^{n−l−1}φ(θ_{λ})(ξ_{1})dξ_{1} dξ ds ≤ c(t − t_{∗})^{l−1} +

+\frac{∏_{i=0;i≠l}^{n−1} |λ − i|}{(l − 1)!(n − l − 1)!} ∫_{t_{∗}}^{t}(t − s)^{l−1} ∫_{s}^{+∞} ξ^{λ−l}((\frac{ξ − s}{ξ})^{n−l−1})′dξ ds =

= c(t − t_{∗})^{l−1} + \frac{∏_{i=0}^{n−1} |λ − i|}{(l − 1)!(n − l − 1)!} ∫_{t_{∗}}^{t}(t − s)^{l−1} ×

× ∫_{s}^{+∞}(ξ − s)^{n−l−1}ξ^{λ−n}dξ ds = c(t − t_{∗})^{l−1} + \frac{∏_{i=0}^{l−1} |λ − i|}{(l − 1)!} ×

× ∫_{t_{∗}}^{t}(t − s)^{l−1}s^{λ−l}ds ≤ c(t − t_{∗})^{l−1} − \frac{∏_{i=0}^{l−2} |λ − i|}{(l − 1)!}(t − t_{∗})^{l−1} ×

×t_{∗}^{λ−l+1} + t^{λ} ≤ t^{λ} for t > t_{∗}.

Therefore T U ⊂ U.

On the other hand, it is obvious that U is a closed bounded convex set. We easily ascertain that the operator T is continuous and T U is an equicontinuous set on every finite segment of the interval [t_{0}, +∞[. Therefore by Lemma 2.1 there exists u ∈ U such that T u = u. It is easy to see that the function u is a proper solution of equation (0.1) satisfying (2.14_{l}). ■

Theorem 7.9′. Let conditions (0.2), (7.46), (7.47) ((0.3), (7.46), (7.47)) be fulfilled and assume that for some t_{0} ∈ ℝ_{+}, l ∈ {1, …, n − 1} and λ ∈ [l − 1, l[, where l + n is odd (l + n is even) we have

12Without loss of generality it can be assumed here that t_{∗} ≥ 1.

t ∫_{t}^{+∞} ξ^{n−2−λ}φ(θ_{λ})(ξ)dξ ≤ ∏_{i=0}^{n−1} |λ − i| for t ≥ t_{0},  (7.51)

with θ_{λ}(t) = t^{λ}. Then equation (0.1) has a proper solution satisfying (2.14_{l}). Proof. To prove the theorem it suffices to show that (7.51) implies (7.48). By (7.51) we have

t^{l−λ} ∫_{t}^{+∞} ξ^{n−l−1}φ(θ_{λ})(ξ)dξ = −t^{l−λ} ∫_{t}^{+∞} ξ^{λ−l+1} ×

×d ∫_{ξ}^{+∞} s^{n−2−λ}φ(θ_{λ})(s)ds = t ∫_{t}^{+∞} ξ^{n−2−λ}φ(θ_{λ})(ξ)dξ +

+(λ + 1 − l)t^{l−λ} ∫_{t}^{+∞} ξ^{λ−l} ∫_{ξ}^{+∞} s^{n−2−λ}φ(θ_{λ})(s)ds dξ ≤

≤ ∏_{i=0}^{n−1} |λ − i|(1 + \frac{λ + 1 − l}{l − λ} ) = ∏_{i=0;i≠l}^{n−1} |λ − i|. ■

CHAPTER 3

ON KNESER-TYPE SOLUTIONS

§ 8. Some auxiliary statements

Definition 8.1. A proper solution u : [t_{0}, +∞[→ ℝ of equation (0.1) is called Kneser-type if there exists t_{1} ∈ [t_{0}, +∞[ such that

(−1)^{i}u^{(i)}(t)u(t) > 0 for t ≥ t_{1} (i = 0, …, n − 1).  (8.1)

Lemma 8.1. Let

p ∈ L_{loc}(ℝ_{+}; ℝ_{+}), δ ∈ C(ℝ_{+}; ℝ_{+}),  (8.2)
δ(t) ≤ t for t ∈ ℝ_{+}, lim_{t→+∞} δ(t) = +∞  (8.3)

and δ BE nondecreasing. Let, moreover, u : [t_{0}, +∞[→ ℝ be a solution of the equation

u^{(n)}(t) + (−1)^{n+1}p(t)u(δ(t)) = 0  (8.4)

satisfying (8.1). Then for any k ∈ {0, …, n − 1} we have

ρ_{k}(t)|u(δ(t))| ≤ (n − k − 1)!(n − 1)!u_{k}(t) for t ≥ t_{∗},  (8.5)

where

t_{∗} = η_{δ}(t_{1}), η_{δ}(t) = max{s : δ(s) ≤ t},  (8.6)
u_{k}(t) = ∑_{i=k}^{n−1} \frac{t^{i−k}|u^{(i)}(t)|}{(i − k)!},  (8.7)
ρ_{k}(t) = max{ψ_{k}(t, s, τ) : τ ∈ [t, η_{δ}(t)], s ∈ [δ(τ), t]}  (8.8_{k})

and

ψ_{k}(t, s, τ) = ∫_{s}^{t} ξ^{n−k−1}p(ξ)dξ ∫_{t}^{τ} ξ^{n−k−1}p(ξ)dξ ×

×(s^{k+1−n}(s − δ(τ))^{n−1} + \frac{1}{(n − 1)!} ∫_{δ(τ)}^{s}(ξ − δ(ξ))^{n−1} ×

×p(ξ)(ξ − δ(ξ))^{n−1}ξ^{k+1−n}dξ).  (8.9_{k})

Proof. By (8.4) and (8.1)

u_{k}(s) ≥ \frac{1}{(n − k − 1)!} ∫_{s}^{t} ξ^{n−k−1}p(ξ)|u(δ(ξ))|dξ for t_{∗} ≤ s ≤ t, (8.10)

where t_{∗} is defined by (8.6) and u_{k} – by (8.7).

Let t ∈ [t_{∗}, +∞[ and (s_{0}, τ_{0}) be the point of maximum of the function ψ_{k}(t, ·, ·) on [δ(τ_{0}), t] × [t, η_{δ}(t)]. Then by (8.10)

u_{k}(s_{0}) ≥ \frac{1}{(n − k − 1)!} ∫_{s_{0}}^{t} s^{n−k−1}p(s)|u(δ(s))|ds ≥

≥ \frac{1}{(n − k − 1)!} ∫_{s_{0}}^{t} s^{n−k−1}p(s)ds|u(δ(t))|,  (8.11)

u_{k}(t) ≥ \frac{1}{(n − k − 1)!} ∫_{t}^{τ_{0}} s^{n−k−1}p(s)|u(δ(s))|ds ≥

≥ \frac{1}{(n − k − 1)!} ∫_{t}^{τ_{0}} s^{n−k−1}p(s)ds|u(δ(τ_{0}))|.  (8.12)

On the other hand, since u_{k}(t) is nonincreasing, from (8.2) and (1.6_{0n}) we obtain

|u(δ(τ_{0}))| ≥ \frac{(n − k − 1)!}{(n − 1)!}[s_{0}^{k+1−n}(s_{0} − δ(τ_{0}))^{n−1} +

+\frac{1}{(n − 1)!} ∫_{δ(τ_{0})}^{s_{0}}(ξ − δ(τ_{0}))^{n−1} ×

×p(ξ)(ξ − δ(ξ))^{n−1}ξ^{k+1−n}dξ]u_{k}(s_{0})

whence by (8.11) and (8.12) it follows the validity of (8.5), where ρ_{k} is defined by (8.8_{k}) and (8.9_{k}). ■

Lemma 8.2. Let (8.2) and (8.3) be fulfilled and

\underline{lim}_{t→+∞} ∫_{δ(t)}^{t} p(s)ds > 0,  (8.13)
vrai sup{p(t) : t ∈ ℝ_{+}} < +∞,  (8.14)

where the function δ is nondecreasing. Let, moreover, u : [t_{0}, +∞[→ ℝ be a proper solution of (8.4) satisfying (8.1). Then

\overline{lim}_{t→+∞} \frac{|u(δ(t))|}{|u^{(n−1)}(t)|} < +∞.  (8.15)

Proof. According to Lemma 8.1, it suffices to show that

\underline{lim}_{t→+∞}ρ_{n−1}(t) > 0,  (8.16)

where the function ρ_{n−1} is defined by (8.8_{n−1}) and (8.9_{n−1}).

In view of (8.13) there exist c > 0 and t_{2} > t_{1} such that

∫_{δ(t)}^{t} p(s)ds ≥ c for t ≥ t_{2}.  (8.17)

Let t ∈ [t_{2}, +∞[. Then by (8.17) there exist t^{∗} ∈]t, η_{δ}(t)], \overline{t} ∈]t, t^{∗}[ and \underline{t} ∈]δ(t^{∗}), t[ such that ^{13}

∫ \underline{t} ; δ(t^{∗}) p(s)ds ≥ \frac{c}{4}, ∫ t ; \overline{t} p(s)ds ≥ \frac{c}{4}, ∫ \overline{t} ; t p(s)ds ≥ \frac{c}{4}.  (8.18)

According to (8.8_{n−1}) and (8.9_{n−1}) it is clear that

ρ_{n−1}(t) ≥ ∫ t ; \underline{t} p(s)ds ∫ \overline{t} ; t p(s)ds(\underline{t} − δ(\overline{t}))^{n−1}.  (8.19)

On the other hand, by (8.14) from (8.17) we have \underline{t} − δ(\overline{t}) ≥ \underline{t} − δ(t^{∗}) ≥ \frac{c}{4r}, where r = vrai sup{p(t) : t ∈ ℝ_{+}}. Therefore (8.18) and (8.19) imply

ρ_{n−1}(t) ≥ \frac{c^{3}}{64r} for t ≥ t_{2}

whence it follows the validity of (8.16). ■

Lemma 8.3. Let (8.2), (8.3) be fulfilled and for some k ∈ {0, …, n − 1}

\underline{lim}_{t→+∞} ∫_{δ(t)}^{t} s^{n−k−1}p(s)ds > 0,  (8.20_{k})
vrai sup{t^{n−k}p(t) : t ∈ ℝ_{+}} < +∞,  (8.21_{k})

where δ is nondecreasing. Let, moreover, u : [t_{0}, +∞[→ ℝ be a proper solution of (8.4) satisfying (8.1). Then

\overline{lim}_{t→+∞} \frac{δ^{k}(t)|u(δ(t))|}{u_{k}(t)} < +∞,  (8.22)

where u_{k} is defined by (8.7).

Proof. According to Lemma 8.1 it suffices to show that

\underline{lim}_{t→+∞}ρ_{k}(t)δ^{−k}(t) > 0.  (8.23)

By (8.20_{k}) there exist c > 0 and t_{2} > t_{1} such that

∫_{δ(t)}^{t} s^{n−k−1}p(s)ds ≥ c for t ≥ t_{2}.  (8.24)

Let t ∈ [t_{2}, +∞[. By (8.24) there exist t^{∗} ∈]t, η_{δ}(t)], \overline{t} ∈]t, t^{∗}[ and \underline{t} ∈ ]δ(t^{∗}), t[ such that

\begin{matrix}∫ \underline{t} ; δ(t^{∗}) s^{n−k−1}p(s)ds ≥ \frac{c}{4}, ∫ t ; \underline{t} s^{n−k−1}p(s)ds ≥ \frac{c}{4}, \\ ∫ \overline{t} ; t s^{n−k−1}p(s)ds ≥ \frac{c}{4}. \end{matrix}  (8.25)

13the function η_{δ} is defined by (8.6)

From (8.8_{k}) and (8.9_{k}) it follows

ρ_{k}(t) ≥ ∫ t ; \underline{t} s^{n−k−1}p(s)ds ∫ \overline{t} ; t s^{n−k−1}p(s)ds \underline{t}^{k+1−n}(\underline{t} − δ(\overline{t}))^{n−1}. (8.26)

On the other hand, (8.21_{k}) and (8.25) imply δ(t^{∗}) ≤ exp\{−\frac{c}{4r_{k}} \}\underline{t} with r_{k} = vrai sup\{t^{n−k}p(t) : t ∈ ℝ_{+}\}. Hence, since \underline{t} ≥ δ(t), by (8.26) and (8.25) we have

ρ_{k}(t) ≥ \frac{c^{2}}{16}δ^{k}(t)(1 − exp \{ − \frac{c}{4r_{k}} \})^{n−1} for t ≥ t_{2}.

Therefore (8.23) is valid. ■

Lemma 8.4. Let (8.2) and (8.3) be fulfilled and for some t_{0} ∈ ℝ_{+}

∫_{δ(t)}^{t} p(s)ds > 0 for t ≥ t_{0}.  (8.27)

Let, moreover, u : [t_{0}, +∞[→ ℝ be a solution of the inequality

(−1)^{n+1}u^{(n)}(t) sign u(t) + p(t)|u(δ(t))| ≤ 0  (8.28)

satisfying (8.1). Then there exists t_{∗} ≥ t_{1} such that (8.4) has a proper solution u_{∗} : [t_{∗}, +∞[→]0, +∞[ satisfying

(−1)^{i}u_{∗}^{(i)}(t) > 0 for t ≥ t_{∗} (i = 0, …, n − 1),  (8.29)
|u_{∗}^{(i)}(t)| ≤ |u^{(i)}(t)| for t ≥ t_{∗} (i = 0, …, n − 1).  (8.30)

Proof. According to (8.28)

|u^{(i)}(t)|≥ \frac{1}{(n − i − 1)!}∫_{t}^{+∞}(s − t)^{n−i−1}p(s)|u(δ(s))|ds  (8.31)
for t ≥ t_{∗} (i = 0, …, n − 1),

where t_{∗} = η_{δ}(t_{1}) (the function η_{δ} is defined by (8.6)).

Consider the sequence {u_{i}}_{i=1}^{+∞} of functions defined by

u_{1}(t) = |u(t)| for t ≥ t_{1},

u_{i}(t) = \begin{cases}\frac{1}{(n−1)!} ∫_{t}^{+∞}(s − t)^{n−1}p(s)u_{i−1}(δ(s))ds for t ≥ t_{∗}, \\ u_{i}(t_{∗}) + |u(t)| − |u(t_{∗})| for t ∈ [t_{1}, t_{∗}[ (i = 2, 3, …). \end{cases}

This sequence is obviously decreasing. Its limit u_{∗} is a solution of the integral equation

u_{∗}(t) = \begin{cases}\frac{1}{(n−1)!}∫_{t}^{+∞}(s − t)^{n−1}p(s)u_{∗}(δ(s))ds for t ≥ t_{∗}, \\ u_{∗}(t_{∗}) + |u(t)| − |u(t_{∗})| for t ∈ [t_{1}, t_{∗}[. \end{cases}  (8.32)

on [t_{∗}, +∞[.

Show that

u_{∗}(t) > 0 for t ≥ t_{∗}.  (8.33)

Suppose the contrary. Then there exists t^{∗} ∈ [t_{∗}, +∞[ such that

u_{∗}(t) ≡ 0 for t ≥ t^{∗}, u_{∗}(t) > 0 for t ∈ [t_{1}, t^{∗}[.  (8.34)

Denote by E the set of all t ∈ [t^{∗}, +∞[ satisfying δ(t) = t^{∗} and put t^{0} = inf E. By (8.27) and (8.34) there exists t_{∗}^{0} ∈]t^{∗}, t^{0}] such that

∫_{t^{∗}}^{t_{∗}^{0}}(s − t^{∗})^{n−1}p(s)u_{∗}(δ(s))ds > 0.

Therefore (8.32) implies

u_{∗}(t^{∗}) ≥ \frac{1}{(n − 1)} ∫_{t^{∗}}^{+∞}(s − t^{∗})^{n−1}p(s)u_{∗}(δ(s))ds ≥

≥ \frac{1}{(n − 1)!} ∫_{t^{∗}}^{t_{∗}^{0}}(s − t^{∗})^{n−1}p(s)u_{∗}(δ(s))ds > 0.

But this contradicts (8.34). The obtained contradiction proves that (8.33) is fulfilled. On the other hand, according to (8.31), (8.32) and (8.33) u_{∗} is a solution of (8.4) satisfying (8.29) and (8.30). ■

Lemma 8.5. Let (8.2) and (8.3) be fulfilled, the function δ be nondecreasing and for some k ∈ {0, …, n − 1} and t_{0} ∈ ℝ_{+} let

ρ_{k}(t) > 0 for t ≥ t_{0}.  (8.35_{k})

Let, moreover, u : [t_{0}, +∞[→ ℝ be a proper solution of (8.28) satisfying (8.1). Then

\underline{lim}_{t→+∞}u_{k}(t) exp \{(n − 1)! ∫_{t_{1}}^{t} p(s)s^{n−k−1}(ρ_{k}(s))^{−1}ds\} > 0, (8.36)

where u_{k} and ρ_{k} are defined respectively by (8.7) and (8.8_{k}), (8.9_{k}).

Proof. According to Lemma 8.4 equation (8.4) has a proper solution u_{∗} : [t_{∗}, +∞[→]0, +∞[ satisfying (8.29) and (8.30), where t_{∗} is sufficiently large. By (8.35_{k}) and Lemma 8.1

\frac{u_{∗}(δ(t))}{u_{∗k}(t)} ≤ (n − k − 1)!(n − 1)!(ρ_{k}(t))^{−1} for t ≥ t_{2},

where t_{2} ≥ t_{∗} is sufficiently large and

u_{∗k}(t) = ∑_{i=k}^{n−1} \frac{t^{i−k}|u_{∗}^{(i)}(t)|}{(i − k)!}.  (8.37)

Therefore from (8.4) we have

u_{∗k}(t) = u_{∗k}(t_{2}) exp \{ − \frac{1}{(n − k − 1)!} ∫_{t_{2}}^{t} s^{n−k−1}p(s)\frac{u_{∗}(δ(s))}{u_{∗k}(s)}ds\} ≥

≥ u_{∗k}(t_{2}) exp { − (n − 1)! ∫_{t_{2}}^{t} s^{n−k−1}p(s)(ρ_{k}(s))^{−1}ds} for t ≥ t_{2}

whence, taking into account (8.7), (8.30) and (8.37), we deduce (8.36). ■

Taking into account Lemmas 8.1–8.5, we can easily ascertain the validity of the following corollaries.

Corollary 8.1. Let (8.2) and (8.3) be fulfilled, δ be nondecreasing, for some k ∈ {0, …, n − 1} (8.35_{k}) hold and

\overline{lim}_{t→+∞} \frac{1}{t} ∫_{t_{0}}^{t} p(s)s^{n−k−1}(ρ_{k}(s))^{−1}ds < +∞,  (8.38)

where ρ_{k} is defined by (8.8_{k}) and (8.9_{k}). Let, moreover, u : [t_{0}, +∞[→ ℝ be a proper solution of (8.28) satisfying (8.1). Then there exists λ > 0 such that

|u(t)|e^{λt} → +∞ for t → +∞.  (8.39)

Corollary 8.1′. Let (8.2), (8.3), (8.13) and (8.14) be fulfilled with δ nondecreasing. Let, moreover, u : [t_{0}, +∞[→ ℝ be a proper solution of (8.28) satisfying (8.1). Then λ > 0 exists such that (8.39) holds.

Corollary 8.2. Let (8.2) and (8.3) be fulfilled, δ be nondecreasing, for some k ∈ {0, …, n − 1} (8.35_{k}) hold and

\overline{lim}_{t→+∞} \frac{1}{ln t} ∫_{t_{0}}^{t} p(s)s^{n−k−1}(ρ_{k}(s))^{−1}ds < +∞,  (8.40)

where ρ_{k} is defined by (8.8_{k}) and (8.9_{k}). Let, moreover, u : [t_{0}, +∞[→ ℝ be a proper solution of (8.28) satisfying (8.1). Then there exists λ > 0 such that

|u(t)|t^{λ} → +∞ for t → +∞.  (8.41)

Corollary 8.2′. Let (8.2) and (8.3) be fulfilled with δ nondecreasing and (8.20_{k}) and (8.21_{k}) hold for some k ∈ {0, …, n − 1}. Let, moreover, u : [t_{0}, +∞[→ ℝ be a proper solution of (8.28) satisfying (8.1). Then there exists λ > 0 such that (8.41) holds.

Corollary 8.3. Let (8.2) and (8.3) be fulfilled, δ be nondecreasing, for some k ∈ {0, …, n − 1} (8.35_{k}) hold and for some r ∈ {2, 3, … }

\overline{lim}_{t→+∞} \frac{1}{ln_{r} t} ∫_{t_{0}}^{t} s^{n−k−1}p(s)(ρ_{k}(s))^{−1}ds < +∞,  (8.42_{k})

where ln_{1} t = ln t, ln_{i} t = ln ln_{i−1} t (i = 2, …, r) and ρ_{k} is defined by (8.8_{k}) and (8.9_{k}). Let, moreover, u : [t_{0}, +∞[→ ℝ be a proper solution of (8.28) satisfying (8.1). Then there exists λ > 0 such that

|u(t)| ln_{r−1}^{λ} t → +∞ for t → +∞.  (8.43)

Corollary 8.3′. Let (8.2), (8.3), (8.20_{k}) and (8.21_{k}) hold for some k ∈ {0, …, n − 1} and for some r ∈ {2, 3, … }

\overline{lim}_{t→+∞} \frac{1}{ln_{r} t} ∫_{t_{0}}^{t} s^{n−k−1}p(s)δ^{−k}(s)ds < +∞.  (8.44_{k})

Let, moreover, u : [t_{0}, +∞[→ ℝ be a proper solution of (8.28) satisfying (8.1). Then there exists λ > 0 such that (8.43) holds.

§ 9. On the existence of Kneser-type solutions

9.1. Functional Differential Equations with a Linear Minorant. Let t_{0} ∈ ℝ_{+}. Denote by H_{t_{0},τ}^{−} the set of all functions u ∈ C̃_{loc}^{n−1}(ℝ_{+}; ℝ) satisfying

(−1)^{i}u^{(i)}(t)u(t) > 0 (i = 0, …, n − 1),
(−1)^{n}u^{(n)}(t)u(t) ≥ 0 for t ≥ t_{∗},

where t_{∗} = min{t_{0}, τ_{∗}(t_{0})}, τ_{∗}(t) = inf{τ(s) : s ≥ t}.

Lemma 9.1. Let F ∈ V (τ) and for some t_{0} ∈ ℝ_{+}

(−1)^{n+1}F (u)(t) sign u(t) ≥ φ(|u|)(t)  (9.1)
for u ∈ H_{t_{0},τ}^{−} , t ≥ t_{0},

where φ ∈ M^{+}(σ)^{14} and

σ ∈ C(ℝ_{+}; ℝ_{+}) is nondecreasing, σ(t) ≤ t

for t ∈ ℝ_{+}, lim_{t→+∞} σ(t) = +∞. (9.2)

Let, moreover, u : [t_{0}, +∞[→ ℝ be a proper solution of (0.1) satisfying (8.1) and there exist γ ∈ C̃_{loc}([t_{0}, +∞[; ℝ_{+}), r_{2} > 0 and r_{1} ∈ [0, r_{2}[ such that

γ(t) ↑ +∞ as t ↑ +∞, lim_{t→+∞}(γ(t))^{r_{2}}|u(t)| = +∞,

\underline{lim}_{t→+∞}(γ(t))^{r_{1}}|u(t)| = 0, \overline{lim}_{t→+∞} \frac{γ(t)}{γ(σ(t))} < +∞.

(9.3)

Then

\underline{lim}_{t→+∞}(γ(t))^{r_{2}} ∫_{t}^{+∞}(s − t)^{n−1}φ(θ)(s)ds ≤ (n − 1)!c^{r_{2}−r_{1}}, (9.4)

14The definition of the set see on the page 75

where

θ(t) = (γ(t))^{−r_{2}}, c = \overline{lim}_{t→+∞} \frac{γ(t)}{γ(σ(t))}.

Proof. Denote

ũ(t) = inf{(γ(s))^{r_{2}}|u(s)| : s ≥ t}.  (9.5)

By (9.2), (9.3) and (9.5) we have

ũ(σ(t)) ↑ +∞ for t ↑ +∞  (9.6)

and

\underline{lim}_{t→+∞}ũ(σ(t))(γ(t))^{r_{1}−r_{2}} = 0.  (9.7)

According to (9.5), (9.7) and Lemma 7.1, there exists a sequence of numbers {t_{k}}_{k=1}^{+∞} such that t_{k} ↑ +∞ as k ↑ +∞ and

ũ(σ(t_{k})) = (γ(σ(t_{k}))^{r_{2}}|u(σ(t_{k}))|, (γ(t_{k}))^{r_{1}−r_{2}} ũ(σ(t_{k})) ≤
≤ (γ(t))^{r_{1}−r_{2}} ũ(σ(t)) for t_{∗} ≤ t ≤ t_{k},  (9.8)

where t_{∗} > t_{1} is sufficiently large.

On the other hand, taking into account (8.1) and (9.1) from (0.1) we have

|u(σ(t))| ≥ \frac{1}{(n − 1)!} ∫_{σ(t)}^{t}(s − σ(t))^{n−1}φ(|u|)(s)ds +

+\frac{1}{(n − 1)!} ∫_{t}^{+∞}(s − σ(t))^{n−1}φ(|u|)(s)ds for t ≥ t_{∗}.

Hence by (9.5), (9.8) and the fact that φ ∈ M^{+}(σ) we obtain

|u(σ(t_{k}))| ≥ \frac{1}{(n − 1)!} ∫_{σ(t_{k})}^{t_{k}}(s − σ(t_{k}))^{n−1}ũ(σ(s))φ(θ)(s)ds +

+\frac{1}{(n − 1)!} ∫_{t_{k}}^{+∞}(s − σ(t_{k}))^{n−1}ũ(σ(s))φ(θ)(s)ds ≥

≥ \frac{(γ(t_{k}))^{r_{1}−r_{2}}(γ(σ(t_{k})))^{r_{2}}|u(σ(t_{k}))|}{(n − 1)!} ∫_{σ(t_{k})}^{t_{k}}(s − σ(t_{k}))^{n−1} ×

×(γ(s))^{r_{2}−r_{1}}φ(θ)(s)ds + \frac{(γ(σ(t_{k})))^{r_{2}}|u(σ(t_{k}))|}{(n − 1)!} ×

× ∫_{t_{k}}^{+∞}(s − σ(t_{k}))^{n−1}φ(θ)(s)ds = \frac{(γ(σ(t_{k})))^{r_{2}}|u(σ(t_{k}))|}{(n − 1)!} ×

×( − (γ(t_{k}))^{r_{1}−r_{2}}∫_{σ(t_{k})}^{t_{k}}(γ(s))^{r_{2}−r_{1}}d∫_{s}^{+∞}(ξ − σ(t_{k}))^{n−1} × ×φ(θ)(s)ds+ ∫_{t_{k}}^{+∞}(s − σ(t_{k}))^{n−1}φ(θ)(s)ds) ≥

≥ \frac{(γ(σ(t_{k})))^{r_{2}}|u(σ(t_{k}))|}{(n − 1)!}(\frac{γ(σ(t_{k}))}{γ(t_{k})})^{r_{2}−r_{1}} ×

× ∫_{σ(t_{k})}^{+∞}(s − σ(t_{k}))^{n−1}φ(θ)(s)ds k = 2, 3, …,  (9.9)

where θ(t) = (γ(t))^{−r_{2}}. Suppose that ε ∈]0, ε_{0}[. Then (9.9) implies

(γ(σ(t_{k})))^{r_{2}} ∫_{σ(t_{k})}^{+∞}(s − σ(t_{k}))^{n−1}φ(θ)(s)ds ≤ (n − 1)!(c + ε)^{r_{2}−r_{1}}

for k = k_{0}, k_{0} + 1, …,

where k_{0} ∈ ℕ is sufficiently large. Since ε is arbitrary, hence it follows (9.4). ■

Theorem 9.1. Let F ∈ V (τ) and (9.1) and (9.2) be fulfilled, where φ ∈ M^{+}(σ) and

\underline{lim}_{t→+∞}(σ(t) − t) > −∞.  (9.10)

Moreover, let

φ(|u|)(t) ≥ p(t)|u(δ(t))| for u ∈ H_{t_{0},τ}^{−} , t ≥ t_{0},  (9.11)

for some t_{0} ∈ ℝ_{+} (8.2), (8.3) be fulfilled with δ nondecreasing and for some k ∈ {0, …, n − 1} (8.35_{k}), (8.38) hold, where ρ_{k} is defined by (8.8_{k}), (8.9_{k}). Then the condition

inf \{ \underline{lim}_{t→+∞}e^{λt} ∫_{t}^{+∞}(s − t)^{n−1}φ(θ)(s)ds : λ ∈]0, +∞[\} > (n − 1)!, (9.12)

where θ(t) = e^{−λt}, is sufficient for (0.1) not to have a Kneser-type solution.

Proof. Suppose, on the contrary, that (0.1) has a proper solution u : [t_{0}, +∞[ → ℝ satisfying (8.1). According to (8.35_{k}), (8.38), (9.1), (9.11) and Corollary 8.1 there exists λ > 0 such that (8.39) is fulfilled.

Denote by Λ the set of all λ satisfying (8.39) and put λ_{0} = inf Λ. By (9.12) there exist t_{∗} > t_{1} and ε > 0 such that

e^{λt} ∫_{t}^{+∞}(s − t)^{n−1}φ(θ)(s)ds ≥ (n − 1)! + ε  (9.13)
for t ≥ t_{∗}, λ ∈]λ_{0}, λ_{0} + ε].

Choose ε_{2} ∈]0, ε[ and ε_{1} ∈ [0, ε_{2}[ such that

\begin{matrix}λ_{0} − ε_{1} ≥ 0, c^{ε_{2}+ε_{1}}(n − 1)! < (n − 1)! + ε, \\ lim_{t→+∞} e^{(λ_{0}+ε_{2})t}|u(t)| = +∞, \underline{lim}_{t→+∞}e^{(λ_{0}−ε_{1})t}|u(t)| = 0, \end{matrix} (9.14)

where c = \overline{lim}_{t→+∞}e^{t−σ(t)}. According to (9.1), (9.10) and (9.14) the conditions

of Lemma 9.1 are obviously satisfied with γ(t) = e^{t}, r_{2} = λ_{0} + ε_{2} and r_{1} = λ_{0} − ε_{1}. Therefore by (9.14) this lemma implies

\underline{lim}_{t→+∞}e^{(λ_{0}+ε_{2})t} ∫_{t}^{+∞}(s − t)^{n−1}φ(θ)(s)ds ≤ c^{ε_{1}+ε_{2}}(n − 1)! < (n − 1)! + ε,

where θ(t) = e^{−(λ_{0}+ε_{2})t}. But this contradicts (9.13). ■

Corollary 9.1. Let F ∈ V (τ) and (8.2), (8.3), (9.1), (9.2), (9.10) and (9.11) be fulfilled, where φ ∈ M^{+}(σ). Let, moreover, for some k ∈ {0, …, n− 1} (8.35_{k}) and (8.38) hold with a nondecreasing δ. Then the condition

inf {λ^{−n}(vrai inf_{t≥t_{0}} e^{λt}φ(θ)(t)) : λ ∈]0, +∞[} > 1,  (9.15)

where θ(t) = e^{λt} and t_{0} ∈ ℝ_{+}, is sufficient for (0.1) not to have a Knesertype solution.

Proof. It suffices to note that (9.15) implies (9.12). ■

Theorem 9.1′. Let F ∈ V (σ) and (8.2), (8.3), (8.13), (8.14), (9.1), (9.2), (9.10) and (9.11) be fulfilled, where φ ∈ M^{+}(σ) and δ is nondecreasing. Then the condition (9.12) ((9.15)) is sufficient for (0.1) not to have a Kneser-type solution.

Proof. The assertion of the theorem follows from Corollary 8.1′ and Theorem 9.1 (Corollary 9.1). ■

Theorem 9.2. Let F ∈ V (τ) and for some t_{0} ∈ ℝ_{+}

(−1)^{n+1}F (u)(t) sign u(t) ≥ ∑_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} |u(s)|d_{s}r_{i}(s, t)  (9.16)
for u ∈ H_{t_{0},τ}^{−} , t ≥ t_{0},

where

τ_{i}; σ_{i} ∈ C(ℝ_{+}; ℝ_{+}), τ_{i}(t) ≤ σ_{i}(t) for t ∈ ℝ_{+}, (i = 1, …, m), (9.17)
\underline{lim}_{t→+∞}(τ_{i}(t) − t) > −∞ (i = 1, …, m),  (9.18)
r_{i}(s, t) are measurable, r_{i}(·, t) are nondecreasing (i = 1, …, m). (9.19)

Let, moreover, there exist i_{0} ∈ {1, …, m} and a nondecreasing function δ ∈ C(ℝ_{+}; ℝ_{+}) such that

τ_{i_{0}}(t) ≤ δ(t) ≤ min{t, σ_{i_{0}}(t)}  (9.20)

and for some k ∈ {0, …, n − 1} and t_{∗} ∈ ℝ_{+} let

ρ_{k}(t) > 0 for t ≥ t_{∗},  (9.21_{k})
\overline{lim}_{t→+∞} \frac{1}{t} ∫_{t_{∗}}^{t} s^{n−k−1}p(s)(ρ_{k}(s))^{−1}ds < +∞,  (9.22_{k})

where ρ_{k} is defined by (8.8_{k}), (8.9_{k}) and

p(t) = r_{i_{0}}(δ(t), t) − r_{i_{0}}(τ_{i_{0}}(t), t).  (9.23)

Then the condition

inf \{ \underline{lim}_{t→+∞}e^{λt} ∫_{t}^{+∞}(s − t)^{n−1} ∑_{i=1}^{m} ∫_{τ_{i}(s)}^{σ_{i}(s)} e^{−λξ}d_{ξ} ×

×r_{i}(ξ, s)ds : λ ∈]0, +∞[} > (n − 1)!.  (9.24)

is sufficient for the (0.1) not to have a Kneser-type solution.

Proof. It suffices to show that the operator defined by

φ(u)(t) = ∑_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} u(s)d_{s}r_{i}(s, t)  (9.25)

satisfies the conditions of the Theorem 9.1. Indeed, taking into account (9.20) and (9.25) we see that (9.11) holds with p defined by (9.23). On the other hand, by (9.18) the conditions (9.10) and

φ(xy)(t) ≥ x(σ(t))φ(y)(t)
for all x, y ∈ C(ℝ_{+}; ℝ_{+}), x(t) ↑ +∞ as t ↑ +∞

are fulfilled with

σ(t) = inf{min(τ_{i}(s) : i = 1, …, m) : s ≥ t}.  (9.26)

Therefore φ ∈ M^{+}(σ), so according to (9.21_{k}), (9.22_{k}) and (9.24) the operator defined by (9.25) satisfies all the conditions of Theorem 9.1. ■

Theorem 9.2′. Let F ∈ V (τ), the conditions (9.16)–(9.19) be fulfilled and let there exist i_{0} ∈ {1, …, m} and a nondecreasing function δ ∈ C(ℝ_{+}; ℝ_{+}) such that (9.20), (8.13) and (8.14) hold, where p is defined by (9.23). Then the condition (9.24) is sufficient for (0.1) not to have a Kneser-type solution.

Proof. The assertion of the theorem follows from Corollary 8.1′ and Theorem 9.2. ■

Corollary 9.2. Let F ∈ V (τ), c_{i} > 0, Δ_{i}, \overline{Δ}_{i} ∈ ℝ, \overline{Δ}_{i} < Δ_{i} (i = 1, …, m), Δ_{i_{0}} > 0 for some i_{0} ∈ {1, …, m} and for some t_{0} ∈ ℝ_{+}

(−1)^{n+1}F (u)(t) sign u(t) ≥ ∑_{i=1}^{m} c_{i} ∫_{t−Δ_{i}}^{t−\overline{Δ}_{i}} |u(s)|ds  (9.27)
for u ∈ H_{t_{0},τ}^{−} , t ≥ t_{0}.

Then the condition

inf \{λ^{−n−1} ∑_{i=1}^{m} c_{i}(e^{λΔ_{i}} − e^{λ\overline{Δ}_{i}}) : λ ∈]0, +∞[\} > 1  (9.28)

is sufficient for (0.1) not to have a Kneser-type solution.

Proof. It suffices to note that in view of (9.27) and (9.28) the conditions of Theorem 9.2′ are fulfilled with τ_{i}(t) = t − Δ_{i}, σ_{i}(t) = t − \overline{Δ}_{i}, r_{i}(s, t) = c_{i}s (i = 1, …, m). ■

Corollary 9.2′. Let c_{i} > 0, Δ_{i}, \overline{Δ}_{i} ∈ ℝ, \overline{Δ}_{i} < Δ_{i} (i = 1, …, m) and there exist i_{0} ∈ {1, …, m} such that Δ_{i_{0}} > 0. Then the condition (9.28) is necessary and sufficient for the equation

u^{(n)}(t) + (−1)^{n+1} ∑_{i=1}^{m} c_{i} ∫_{t−Δ_{i}}^{t−\overline{Δ}_{i}} u(s)ds = 0  (9.29)

not to have a Kneser-type solution.

Proof. The sufficiency follows from the Corollary 9.2. If we assume that (9.29) is violated, then (9.29) obviously has the proper solution u(t) = e^{λt} with λ < 0. ■

Theorem 9.3. Let F ∈ V (τ) and for some t_{0} ∈ ℝ_{+}

(−1)^{n+1}F (u)(t) sign u(t) ≥ ∑_{i=1}^{m} p_{i}(t)|u(δ_{i}(t))|  (9.30)
for u ∈ H_{t_{0},τ}^{−} , t ≥ t_{0},

where

\begin{matrix}δ_{i} ∈ C(ℝ_{+}; ℝ_{+}), lim_{t→+∞} δ_{i}(t) = +∞, \\ p_{i} ∈ L_{loc}(ℝ_{+}; ℝ_{+}) (i = 1, …, m), \end{matrix}  (9.31)
\underline{lim}_{t→+∞}(δ_{i}(t) − t) > −∞ (i = 1, …, m).  (9.32)

Let, moreover, i_{0} ∈ {1, …, m} exist such that δ_{i_{0}}(t) ≤ t and for t ∈ ℝ_{+}, for some k ∈ {0, …, n − 1} the conditions (8.35_{k}), (8.38) be fulfilled with ρ_{k}(t) defined by (8.8_{k}), (8.9_{k}), p(t) ≡ p_{i_{0}}(t) and δ(t) ≡ δ_{i_{0}}(t). Then the condition

inf \{ \underline{lim}_{t→+∞}e^{λt} ∫_{t}^{∞}(s − t)^{n−1} ∑_{i=1}^{m} p_{i}(s)e^{−λδ_{i}(s)}ds :

: λ ∈]0, +∞[} > (n − 1)!  (9.33)

is sufficient for (0.1) not to have a Kneser-type solution.

Proof. According to (9.30) the condition (9.16) is fulfilled with

\begin{matrix}τ_{i}(t) = δ_{i}(t) − 1, σ_{i}(t) = δ_{i}(t), \\ r_{i}(s, t) = p_{i}(t)e(s − δ_{i}(t)) (i = 1, …, m).^{15} \end{matrix}  (9.34)

Therefore by (9.30)–(9.33) all the conditions of Theorem 9.2 are satisfied. This proves the validity of the theorem. ■

Corollary 9.3. Let F ∈ V (τ), the conditions (9.30)–(9.32) be fulfilled and i_{0} ∈ {1, …, m} exist such that δ_{i_{0}}(t) ≤ t for t ∈ ℝ_{+}. Let, moreover, for some k ∈ {0, …, n−1} the conditions (8.35_{k}), (8.38) hold with ρ_{k}(t) defined by (8.8_{k}), (8.9_{k}) and p(t) ≡ p_{i_{0}}(t). Then the condition

inf {λ^{−n} vrai inf_{t≥t_{∗}} ( ∑_{i=1}^{m} p_{i}(t)e^{λ(t−δ_{i}(t))}) : λ ∈]0, +∞[} > 1 (9.35)

with t_{∗} ∈ ℝ_{+} is sufficient for (0.1) not to have a Kneser-type solution.

Proof. It suffices to note that (9.35) implies (9.33). ■

Corollary 9.3′. Let F ∈ V (τ), the conditions (9.30)– (9.32) be fulfilled and δ_{i}(t) ≤ t for t ∈ ℝ_{+} (i = 1, …, m). Let, moreover, i_{0} ∈ {1, …, m} exist such that δ_{i_{0}}(t) is nondecreasing, for some k ∈ {0, …, n − 1} the conditions (8.35_{k}), (8.38) hold with ρ_{k} defined by (8.8._{k}), (8.9_{k}) and p(t) = p_{i_{0}}(t). Then the condition

vrai inf \{ ∑_{i=1}^{m} p_{i}(t)(t − δ_{i}(t))^{n} : t ∈]t_{∗}, +∞[\} > (\frac{n}{e})^{n}  (9.36)

with t_{∗} ∈ ℝ_{+} is sufficient for (0.1) not to have a Kneser-type solution.

Proof. Since e^{x} ≥ x^{n}(\frac{e}{n})^{n} for x ≥ 0, (9.36) obviously implies (9.35). ■

e(t) = \begin{cases}0 for t ∈] − ∞, 0[, \\ 1 for t ∈ [0, +∞[. \end{cases}

15everywhere below by e(t) we mean

In view of Corollary 8.1′ Theorem 9.3 (Corollary 9.3) easily implies

Theorem 9.3′. Let F ∈ V (τ), the conditions (9.30)–(9.32) be fulfilled and i_{0} ∈ {1, …, m} exist such that δ_{i_{0}}(t) ≤ t for t ∈ ℝ_{+}. Let, moreover, (8.13), (8.14) hold with δ_{i_{0}}(t) ≡ δ(t) and p_{i_{0}}(t) ≡ p(t). Then the condition (9.33) ((9.35)) is sufficient for (0.1) not to have a Kneser-type solution.

Corollary 9.4. Let F ∈ V (τ), c_{i} > 0, Δ_{i} ∈ ℝ (i = 1, …, m), i_{0} ∈ {1, …, m} exists such that Δ_{i_{0}} > 0 and for some t_{0} ∈ ℝ_{+} let

(−1)^{n+1}F (u)(t) sign u(t) ≥ ∑_{i=1}^{m} c_{i}|u(t − Δ_{i})|

for u ∈ H_{t_{0}τ}^{−} , t ≥ t_{0}.

Then the condition

inf {λ^{−n} ∑_{i=1}^{m} c_{i}e^{λΔ_{i}} : λ ∈]0, +∞[} > 1  (9.37)

is sufficient for (0.1) not to have a Kneser-type solution.

Corollary 9.4′. Let c_{i} > 0, Δ_{i} ∈ ℝ (i = 1, …, m) and Δ_{i_{0}} > 0 for some i_{0} ∈ {1, …, m}. Then (9.37) is necessary and sufficient for the equation

u^{(n)}(t) + (−1)^{n+1} ∑_{i=1}^{m} c_{i}u(t − Δ_{i}) = 0

not to have a Kneser-type solution.

Theorem 9.4. Let F ∈ V (τ), conditions (9.1), (9.2), (9.11), (8.2) and (8.3) be fulfilled with φ ∈ M^{+}(σ) and

\underline{lim}_{t→+∞} \frac{σ(t)}{t} > 0.  (9.38)

Let, moreover, for some k ∈ {0, …, n − 1} conditions (8.35_{k}) and (8.40) hold with ρ_{k}(t) defined by (9.8_{k}) and (9.9_{k}). Then the condition

inf \{ \underline{lim}_{t→+∞}t^{λ} ∫_{t}^{+∞}(s − t)^{n−1}φ(θ)(s)ds : λ ∈]0, +∞[\} > (n − 1)! (9.39)

with θ(t) = t^{−λ} is sufficient for (0.1) not to have a Kneser-type solution.

Proof. Suppose, on the contrary, that (0.1) has a proper solution u : [t_{0}, +∞[ → ℝ satisfying (8.1). According to (8.35_{k}), (8.40), (9.1), (9.11) and Corollary 8.2 there exists λ > 0 such that (8.41) holds. Denote by Λ the set of all λ satisfying (8.41) and put λ_{0} = inf Λ. By (9.39) there exist t_{∗} > t_{1} and ε > 0 such that

t^{λ} ∫_{t}^{+∞}(s − t)^{n−1}φ(θ)(s)ds ≥ (n − 1)! + ε  (9.40)
for t ≥ t_{∗}, λ ∈]λ_{0}, λ_{0} + ε].

Choose ε_{2} ∈]0, ε[ and ε_{1} ∈ [0, ε_{2}[ such that

\begin{matrix}λ_{0} − ε_{1} ≥ 0, c^{ε_{1}+ε_{2}}(n − 1)! < (n − 1)! + ε, \\ lim_{t→+∞} t^{λ_{0}+ε_{2}}|u(t)| = +∞, \underline{lim}_{t→+∞}t^{λ_{0}−ε_{1}}|u(t)| = 0, \end{matrix}  (9.41)

where c = \overline{lim}_{t→+∞}(\frac{t}{σ(t)}). By (9.1), (9.38) and (9.41) all the conditions of

Lemma 9.1 are fulfilled with γ(t) = t, r_{2} = λ_{0} + ε_{2} and r_{1} = λ_{0} − ε_{1}. Therefore, taking into account (9.41), this lemma implies

\underline{lim}_{t→+∞}t^{λ_{0}+ε_{2}} ∫_{t}^{+∞}(s − t)^{n−1}φ(θ)(s)ds ≤ (n − 1)!c^{ε_{1}+ε_{2}} < (n − 1)! + ε,

where θ(t) = t^{−(λ_{0}+ε_{2})}. But this inequality contradicts (9.40). The obtained contradiction proves the theorem. ■

Corollary 9.5. Let F ∈ V (τ) and conditions (9.1), (9.2), (9.11), (8.2), (8.3) and (9.38) be fulfilled with φ ∈ M^{+}(σ). Let, moreover, for some k ∈ {0, …, n − 1} (8.35_{k}) and (8.40) hold with ρ_{k}(t) defined by (8.8_{k}) and (8.9_{k}). Then the condition

inf \{\frac{1}{∏_{i=0}^{n−1}(i + λ)} vrai inf_{t≥t_{0}}(t^{n+λ}φ(θ)(t)) : λ ∈]0, +∞[)\} > 1 (9.42)

with θ(t) = t^{−λ} and t_{∗} ∈ ℝ_{+} is sufficient for (0.1) not to have a Kneser-type solution.

Proof. It suffices to note that (9.42) implies (9.39). ■

In view of Corollary 8.2′ Theorem 9.4 (Corollary 9.5) implies

Theorem 9.4′. Let F ∈ V (τ) and conditions (9.1), (9.2), (9.11), (8.2), (8.3) and (9.38) be fulfilled with φ ∈ M^{+}(σ). Let, moreover, for some k ∈ {0, …, n−1} (8.20_{k}) and (8.21_{k}) hold. Then (9.39) ((9.42)) is sufficient for (0.1) not to have a Kneser-type solution.

Theorem 9.5. Let F ∈ V (τ), conditions (9.16), (9.17) and (9.19) be fulfilled and

\underline{lim}_{t→+∞} \frac{τ_{i}(t)}{t} > 0 (i = 1, …, m).  (9.43)

Let, moreover, there exist i_{0} ∈ {1, …, m} and a nondecreasing function δ ∈ C(ℝ_{+}; ℝ_{+}) such that (9.20)–(9.22_{k}) hold with function ρ_{k} ( function p) defined by (8.8_{k}) and (8.9_{k}) ((9.23)). Then the condition

inf \{ \underline{lim}_{t→+∞}t^{λ} ∫_{t}^{+∞}(s − t)^{n−1} ∑_{i=1}^{m} ∫_{τ_{i}(s)}^{σ_{i}(s)} ξ^{−λ}d_{ξ}r_{i}(ξ, s)ds :

: λ ∈]0, +∞[} > (n − 1)!  (9.44)

is sufficient for (0.1) not to have a Kneser-type solution.

Proof. It suffices to note that the operator defined by (9.25) satisfies all the conditions of Theorem 9.4. ■

In view of Corollary 8.2′ Theorem 9.5 implies

Theorem 9.5′. Let F ∈ V (τ) and conditions (9.16), (9.17), (9.19) and (9.43) be fulfilled. Let, moreover, there exist i_{0} ∈ {1, …, m} and a nondecreasing function δ ∈ C(ℝ_{+}; ℝ_{+}) such that (8.20_{k}) and (8.27_{k}) hold with p defined by (9.23). Then (9.44) is sufficient for (0.1) not to have a Knesertype solution.

Corollary 9.6. Let F ∈ V (τ), c_{i}, α_{i}, \overline{α}_{i} ∈]0, +∞[, α_{i} < \overline{α}_{i} (i = 1, …, m), α_{i_{0}} < 1 for some i_{0} ∈ {1, …, m} and for some t_{0} ∈ ℝ_{+} let

(−1)^{n+1}F (u)(t) sign u(t) ≥ ∑_{i=1}^{m} c_{i} ∫_{α_{i}t}^{\overline{α}_{i}t} s^{−n−1}|u(s)|ds  (9.45)
for u ∈ H_{t_{0},τ}^{−} , t ≥ t_{0}.

Then the condition

inf \{\frac{1}{∏_{i=0}^{n}(i + λ)} ∑_{i=1}^{m} c_{i}(α_{i}^{−n−λ} − \overline{α}_{i}^{−n−λ}) : λ ∈]0, +∞[\} > 1 (9.46)

is sufficient for (0.1) not to have a Kneser-type solution.

Proof. It suffices to note that by (9.45) and (9.46) all the conditions of Theorem 9.5′ are fulfilled with τ_{i}(t) = α_{i}t, σ_{i}(t) = \overline{α}_{i}t and r_{i}(s, t) = −\frac{c_{i}s^{−n}}{n} (i = 1, …, m). ■

Corollary 9.6′. Let c_{i}, α_{i}, \overline{α}_{i} ∈]0, +∞[, α_{i} < \overline{α}_{i}, (i = 1, …, m) and for some i_{0} ∈ {1, …, m} let α_{i_{0}} < 1. Then the condition (9.46) is necessary and sufficient for the equation

u^{(n)}(t) + (−1)^{n+1} ∑_{i=1}^{m} c_{i} ∫_{α_{i}t}^{\overline{α}_{i}t} s^{−n−1}u(s)ds = 0  (9.47)

not to have a Kneser-type solution.

Proof. Sufficiency follows from Corollary 9.6. If we assume that (9.46) is violated, then (9.47) has the solution u(t) = t^{−λ} with λ > 0. ■

Corollary 9.7. Let F ∈ V (τ), c_{i}, α_{i}, \overline{α}_{i} ∈]0, +∞[, α_{i} < \overline{α}_{i}, (i = 1, …, m), there exist i_{0} ∈ {1, …, m} such that α_{i0} < 1 and for some t_{0} ∈ ℝ_{+} let

(−1)^{n+1}F (u)(t) sign u(t) ≥ t^{−n−1} ∑_{i=1}^{m} c_{i} ∫_{α_{i}t}^{\overline{α}_{i}t} |u(s)|ds,  (9.48)
for u ∈ H_{t_{0},τ}^{−} , t ≥ t_{0}.

Then the condition

inf \{\frac{1}{∏_{i=−1}^{n−1}(i + λ)} ∑_{i=1}^{m} c_{i}(\overline{α}_{i}^{1−λ} − α_{i}^{1−λ}) :

: λ ∈]0, 1[∪]1, +∞[} > 1  (9.49)

is sufficient for (0.1) not to have a Kneser-type solution.

Proof. By (9.48) and (9.49) all the conditions of Theorem 9.5′ are fulfilled with τ_{i}(t) = α_{i}t, σ_{i}(t) = \overline{α}_{i}t and r_{i}(s, t) = c_{i}t^{−n−1}s (i = 1, …, m). ■

Corollary 9.7′. Let c_{i}, α_{i}, \overline{α}_{i} ∈]0, +∞[, α_{i} < \overline{α}_{i}, (i = 1, …, m) and for some i_{0} ∈ {1, …, m} let α_{i_{0}} < 1. Then (9.49) is necessary and sufficient for the equation

u^{(n)}(t) + (−1)^{n+1} ∑_{i=1}^{m} \frac{c_{i}}{t^{n+1}} ∫_{α_{i}t}^{\overline{α}_{i}t} u(s)ds = 0

not to have a Kneser-type solution.

Theorem 9.6. Let F ∈ V (τ), (9.30) and (9.31) be fulfilled and

\underline{lim}_{t→+∞} \frac{δ_{i}(t)}{t} > 0 (i = 1, …, m).  (9.50)

Let, moreover, there exist i_{0} ∈ {1, …, m} such that δ_{i_{0}} is nondecreasing, δ_{i_{0}}(t) ≤ t for t ∈ ℝ_{+} and for some k ∈ {0, …, n − 1} (8.35_{k}), (8.40) hold with ρ_{k} defined by (8.8_{k}), (8.9_{k}), p(t) ≡ p_{i_{0}}(t) and δ(t) ≡ δ_{i_{0}}(t). Then the condition

inf \{ \underline{lim}_{t→+∞}t^{λ} ∫_{t}^{+∞}(s − t)^{n−1} ∑_{i=1}^{m} p_{i}(s)δ_{i}^{−λ}(s)ds :

: λ ∈]0, +∞[} > (n − 1)!  (9.51)

is sufficient for (0.1) not to have a Kneser-type solution.

Proof. By (9.30) the inequality (9.16) holds with τ_{i}(t) = δ_{i}(t) − 1, σ_{i}(t) = δ_{i}(t) and r_{i}(s, t) = p_{i}(t)e(s − δ_{i}(t)) (i = 1, …, m) (the definition of the function e see on p.104). Therefore according to (8.35_{k}), (8.40), (9.50) and (9.51) all the conditions of Theorem 9.5 are satisfied. This proves the theorem. ■

According to Corollary 8.2′ Theorem 9.6 easily implies

Theorem 9.6′. Let F ∈ V (τ) and (9.30), (9.31) and (9.50) be fulfilled. Let, moreover, there exist i_{0} ∈ {1, …, m} such that δ_{i_{0}} is nondecreasing, δ_{i_{0}}(t) ≤ t for t ∈ ℝ_{+} and for some k ∈ {0, …, n − 1} (8.20_{k}), (8.21_{k}) hold with p(t) ≡ p_{i_{0}}(t) and δ(t) ≡ δ_{i_{0}}(t). Then (9.51) is sufficient for (0.1) not to have a Kneser-type solution.

Corollary 9.8. Let F ∈ V (τ) and (9.30), (9.31) and (9.50) be fulfilled. Let, moreover, there exist i_{0} ∈ {1, …, m} such that δ_{i_{0}} is nondecreasing, δ_{i_{0}}(t) ≤ t for t ∈ ℝ_{+} and for some k ∈ {0, …, n − 1} (8.20_{k}), (8.21_{k}) hold with p(t) ≡ p_{i_{0}}(t) and δ(t) ≡ δ_{i_{0}}(t). Then the condition

inf \{\frac{1}{∏_{i=0}^{n−1}(i + λ)} vrai inf_{t≥t_{∗}} (t^{n+λ} ∑_{i=1}^{m} p_{i}(t)δ_{i}^{−λ}(t)) :

: λ ∈]0, +∞[} > 1,  (9.52)

where t_{∗} ∈ ℝ_{+}, is sufficient for (0.1) not to have a Kneser-type solution.

Proof. It suffices to note that (9.52) implies (9.51). ■

Corollary 9.9. Let F ∈ V (τ), c_{i}, α_{i} ∈]0, +∞[ (i = 1, …, m), there exist i_{0} ∈ {1, …, m} such that α_{i_{0}} < 1 and for any t_{0} ∈ ℝ_{+} let

(−1)^{n+1}F (u)(t) sign u(t) ≥ ∑_{i=1}^{m} \frac{c_{i}}{t^{n}} |u(α_{i}t)|

for u ∈ H_{t_{0},τ}^{−} , t ≥ t_{0}.

Then the condition

inf \{\frac{1}{∏_{i=0}^{n−1}(i + λ)} ∑_{i=1}^{m} c_{i}α_{i}^{−λ} : λ ∈]0, +∞[\} > 1  (9.53)

is sufficient for (0.1) not to have a Kneser-type solution.

Corollary 9.9′. Let c_{i}, α_{i} ∈]0, +∞[ (i = 1, …, m) and for some i_{0} ∈ {1, …, m} let α_{i_{0}} < 1. Then (9.53) is necessary and sufficient for the equation

u^{(n)}(t) + (−1)^{n+1} ∑_{i=1}^{m} \frac{c_{i}}{t^{n}} u(α_{i}t) = 0

not to have a Kneser-type solution.

Theorem 9.7. Let n > 1, F ∈ V (τ), (8.2) and (8.3) be fulfilled and for some t_{0} ∈ ℝ_{+} let

(−1)^{n+1}F (u)(t) sign u(t) ≥ p(t)|u(δ(t))|  (9.54)
for u ∈ H_{t_{0},τ}^{−} , t ≥ t_{0},

where δ is nondecreasing. Let, moreover, for some k ∈ {1, …, n − 1} and r ∈ {2, 3, … } (8.35_{k}) and (8.42_{k}) hold with ρ_{k} defined by (9.8_{k}) and (8.9_{k}). Then (0.1) has no Knezer-type solution.

Proof. Suppose, on the contrary, that (0.1) has a proper solution u : [t_{0}, +∞[ → ℝ satisfying (8.1). Then according to Lemma 8.5 there exist λ > 0 and t_{∗} > t_{0} such that u_{k}(t) ≥ ln^{−λ} t for t ≥ t_{∗}, where u_{k} is defined by (8.7). The last inequality easily implies the existence of the numbers t^{∗} > t_{∗} and c > 0 such that |u^{(k)}(t)| ≥ c ln^{−λ} t for t ≥ t^{∗}. Therefore, since k ∈ {1, …, n − 1}, by (8.1) we have

+∞ > ∫_{t^{∗}}^{+∞} |u^{(k)}(t)|dt ≥ c ∫_{t^{∗}}^{+∞} ln^{−λ} t dt = +∞.

The obtained contradiction proves the theorem. ■

Corollary 9.10. Let F ∈ V (τ) and (8.2), (8.3) and (9.54) be fulfilled. Let, moreover, for some k ∈ {1, …, n − 1} and r ∈ {2, 3, … } (8.20_{k}), (8.21_{k}) and (8.44_{k}) hold. Then (0.1) has no Kneser-type solution.

Theorem 9.8. Let F ∈ V (τ), conditions (8.2), (8.3), (9.1), (9.2) and (9.11) be fulfilled with φ ∈ M^{+}(σ) and for some r ∈ {2, 3, … } let

\overline{lim}_{t→+∞} \frac{ln_{r−1} t}{ln_{r−1} σ(t)} < +∞.  (9.55)

Let, moreover (8.35_{0}) and (8.42_{0}) hold with ρ_{0} defined by (8.8_{0}) and (8.9_{0}). Then the condition

inf\{ \underline{lim}_{t→+∞}ln_{r−1}^{λ}t∫_{t}^{+∞}(s − t)^{n−1}φ(θ)(s)ds:λ ∈]0, k]\}>(n − 1)! (9.56)
for all k ∈ ℕ

with θ(t) = (ln_{r−1} t)^{−λ} is sufficient for (0.1) not to have a Kneser-type solution.

Proof. Suppose, on the contrary, that (0.1) has a proper solution u : [t_{0}, +∞[ → ℝ satisfying (8.1). According to (8.35_{0}), (8.42_{0}), (9.1), (9.11) and Corollary 8.3, there exists λ > 0 such that (8.43) holds. Denote by Λ the set of all such λ and put λ_{0} = inf Λ. By (9.56) there exist ε > 0 and t_{∗} > t_{1} satisfying

ln_{r−1}^{λ} t ∫_{t}^{+∞}(s − t)^{n−1}φ(θ)(s)ds ≥ (n − 1)! + ε  (9.57)
for t ≥ t_{∗}, λ ∈]λ_{0}, λ_{0} + ε].

Choose ε_{2} ∈]0, ε[ and ε_{1} ∈ [0, ε_{2}[ such that

c^{ε_{1}+ε_{2}}(n − 1)! < (n − 1)! + ε,

lim_{t→+∞}(ln_{r−1} t)^{λ_{0}+ε_{2}}|u(t)|=+∞, \underline{lim}_{t→+∞}(ln_{r−1} t)^{λ−ε_{1}}|u(t)|=0, (9.58)

where c = \overline{lim}_{t→+∞} \frac{ln_{r−1} t}{ln_{r−1} σ(t)}. By (9.1), (9.55) and (9.58) all the conditions of

Lemma 9.1 are obviously fulfilled with γ(t) = ln_{r−1} t, r_{2} = λ_{0} + ε_{2} and r_{1} = λ_{0} − ε_{1}. Therefore, using this lemma and taking into account the first inequality of (9.58), we obtain

\underline{lim}_{t→+∞}(ln_{r−1} t)^{λ_{0}+ε_{2}} ∫_{t}^{+∞}(s−t)^{n−1}φ(θ)(s)ds ≤ c^{ε_{1}+ε_{2}}(n−1)! < (n−1)!+ε,

where θ(t) = (ln_{r−1} t)^{−(λ_{0}+ε_{2})}. But this contradicts (9.57). ■

Corollary 9.11. Let F ∈ V (τ), conditions (8.2), (8.3), (9.1), (9.2) and (9.11) be fulfilled and for some r ∈ {2, 3, … } (9.55) hold with φ ∈ M^{+}(σ). Let, moreover, (8.35_{0}) and (8.42_{0}) hold with ρ_{0} defined by (8.8_{0}) and (8.9_{0}). Then the condition

inf \{\frac{1}{λ} vrai inf_{t≥t_{k}}(t^{n} ln_{1} t ⋯ (ln_{r−1} t)^{λ+1}φ(θ)(t)) :

: λ ∈]0, k]} > (n − 1)! for all k ∈ ℕ  (9.59)

with θ(t) = (ln_{r−1} t)^{−λ} is sufficient for (0.1) not to have a Kneser-type solution.

Proof. It suffices to show that (9.59) implies (9.56). Let k ∈ ℕ. By (9.59) there exist t_{k} ∈ ℝ_{+} and ε ∈]0, 1] such that

φ(θ)(t) ≥ \frac{((n − 1)! + ε)λ}{t^{n} ln_{1} t ⋯ (ln_{r−1} t)^{λ+1}} for t ≥ t_{k}, λ ∈]0, k].

Therefore

∫_{t}^{+∞}(s−t)^{n−1}φ(θ)(s)ds≥λ((n−1)!+ε)∫_{t}^{+∞} \frac{(s − t)^{n−1}ds}{s^{n} ln_{1} s⋯(ln_{r−1}s)^{λ+1}} (9.60)
for t≥t_{k}, λ∈]0, k].

Choose x ∈]0, +∞[ such that

(\frac{x}{1 + x})^{n−1}((n − 1)! + ε) > (n − 1)! + \frac{ε}{2}.

Then in view of (9.60) we obtain

(ln_{r−1} t)^{λ} ∫_{t}^{+∞}(s − t)^{n−1}φ(θ)(s)ds ≥ −((n − 1)! + ε)(ln_{r−1} t)^{λ} ×

× ∫_{(1+x)t}^{+∞} (1 − \frac{t}{s})^{n−1}d(ln_{r−1} s)^{−λ} ≥ ((n − 1)! + ε)(\frac{x}{1 + x})^{n−1} ×

×(\frac{ln_{r−1} t}{ln_{r−1}(1 + x)t})^{λ} for t ≥ t_{k}, λ ∈]0, k].

So

(ln_{r−1} t)^{λ} ∫_{t}^{+∞}(s − t)^{n−1}φ(θ)(s)ds ≥ (n − 1)! + \frac{ε}{3} for t ≥ t′_{k}, λ ∈]0, k],

where t′_{k} > t_{k} is sufficiently large. But this means that (9.56) is true. ■

According to Corollary 8.3′ from Theorem 9.8 (Corollary 9.11) immediately follows

Theorem 9.8′ Let F ∈ V (τ), conditions (8.2), (8.3), (9.1), (9.2) and (9.11) be fulfilled and for some r ∈ {2, 3, … } (9.55) hold with φ ∈ M^{+}(σ). Let, moreover, (8.20_{0}), (8.21_{0}) and (8.42_{0}) hold. Then (9.56), ((9.59)) is sufficient for (0.1) not to have a Kneser-type solution.

Theorem 9.9. Let F ∈ V (τ), conditions (9.16), (9.17) and (9.19) be fulfilled and for some r ∈ {2, 3, … } let

\overline{lim}_{t→+∞} \frac{ln_{r−1} t}{ln_{r−1} τ_{i}(t)} < +∞ (i = 1, …, m).  (9.61)

Let, moreover, there exist i_{0} ∈ {1, …, m} and a nondecreasing function δ ∈ C(ℝ_{+}; ℝ_{+}) such that (9.20), (8.35_{0}) and (8.42_{0}) hold with ρ_{0} (p) defined by (8.8_{0}), (8.9_{0}) ((9.23)). Then the condition

inf \{ \underline{lim}_{t→+∞}(ln_{r−1} t)^{λ} ∫_{t}^{+∞}(s − t)^{n−1} ∑_{i=1}^{m} ∫_{τ_{i}(s)}^{σ_{i}(s)}(ln_{r−1} ξ)^{−λ}d_{ξ}r_{i}(ξ, s)ds :

: λ ∈]0, k]} > (n − 1)! for all k ∈ ℕ  (9.62)

is sufficient for (0.1) not to have a Kneser-type solution.

Proof. It suffices to note that the operator defined by (9.25) satisfies all the conditions of Theorem 9.8. ■

Theorem 9.8 and Corollary 8.3′ imply

Theorem 9.9′. Let F ∈ V (τ), conditions (9.16), (9.17) and (9.19) be fulfilled and for some r ∈ {2, 3, … } (9.61) hold. Let, moreover, there exist i_{0} ∈ {1, …, m} and a nondecreasing function δ ∈ C(ℝ_{+}; ℝ_{+}) such that (9.20), (8.20_{0}), (8.21_{0}) and (8.44_{0}) hold with p defined by (9.23). Then the condition (9.62) is sufficient for (0.1) not to have a Kneser-type solution.

Corollary 9.12. Let F ∈ V (τ), c_{i}, α_{i}, \overline{α}_{i} ∈]0, +∞[, α_{i} < \overline{α}_{i}, (i = 1, …, m), α_{i_{0}} < 1 for some i_{0} ∈ {1, …, m} and for some t_{0} ∈ ℝ_{+} let

(−1)^{n+1}F (u)(t) sign u(t) ≥ ∑_{i=1}^{m} \frac{c_{i}}{t^{n}} ∫_{t^{α_{i}}}^{t^{\overline{α}^{i}}} \frac{|u(s)|}{s ln^{2} s}ds

for u ∈ H_{t_{0},τ}^{−} , t ≥ t_{0}.

Then the condition

inf\{\frac{1}{λ(λ + 1)}∑_{i=1}^{m} c_{i}(α_{i}^{−λ−1} − \overline{α}_{i}^{−λ−1}) : λ ∈]0, +∞[\}>(n − 1)! (9.63)

is sufficient for (0.1) not to have a Kneser-type solution.

Remark 9.1. Condition (9.66) cannot be replaced by

inf \{\frac{1}{λ(λ + 1)} ∑_{i=1}^{m} c_{i}(α_{i}^{−λ−1} − \overline{α}_{i}^{−λ−1}) : λ ∈]0, +∞[\} ≥ (n − 1)! − ε,

however small ε would be.

Theorem 9.10. Let F ∈ V (τ), conditions (9.30) and (9.31) be fulfilled and for some r ∈ {2, 3, … } let

\overline{lim}_{t→+∞} \frac{ln_{r−1} t}{ln_{r−1} δ_{i}(t)} < +∞ (i = 1, …, m).  (9.64)

Let, moreover, there exist i_{0} ∈ {1, …, m} such that δ_{i_{0}} is nondecreasing, δ_{i_{0}}(t) ≤ t for t ∈ ℝ_{+} and (8.20_{0}), (8.21_{0}) hold with δ = δ_{i_{0}}. Then the condition

inf \{ \underline{lim}_{t→+∞}(ln_{r−1} t)^{λ} ∫_{t}^{+∞}(s − t)^{n−1} ∑_{i=1}^{m} p_{i}(s)(ln_{r−1} δ_{i}(s))^{−λ}ds :

: λ ∈]0, k]} > (n − 1)! for all k ∈ ℕ  (9.65)

is sufficient for (0.1) not to have a Kneser-type solution.

Proof. According to (9.30) inequality (9.16) is valid with σ_{i}(t) = δ_{i}(t), τ_{i}(t) = δ_{i}(t) − 1 and r_{i}(s, t) = p_{i}(t)e(s − δ_{i}(t)) (i = 1, …, m). Therefore, using (9.64) and (9.65), we can easily show that all the conditions of Theorem 9.9′ are satisfied. This proves the theorem. ■

Corollary 9.13. Let F ∈ V (τ), conditions (9.30) and (9.31) be fulfilled and for some r ∈ {2, 3, … } (9.64) hold. Let, moreover, there exist i_{0} ∈ {1, …, m} such that δ_{i_{0}} is nondecreasing, δ_{i_{0}}(t) ≤ 0 for t ∈ ℝ_{+} and (8.20_{0}), (8.21_{0}), (8.44_{0}) hold with δ(t) ≡ δ_{i_{0}}(t). Then the condition: for any k ∈ ℕ there exists t_{k} ∈ ℝ_{+} such that

inf \{\frac{1}{λ} vrai inf_{t≥t_{k}} (t^{n} ln_{1} t ⋯ (ln_{r−1} t)^{λ+1} ∑_{i=1}^{m} p_{i}(t)(ln_{r−1} δ_{i}(t))^{−λ}) :

: λ ∈]0, k]} > (n − 1)!  (9.66)

is sufficient for (0.1) not to have a Kneser-type solution.

Proof. It suffices to note that (9.66) implies (9.65). ■

Corollary 9.14. Let F ∈ V (τ), c_{i}, α_{i} ∈]0, +∞[, α_{i_{0}} < 1 for some i_{0} ∈ {1, …, m} and for some t_{0} t_{0} ∈ ℝ_{+} let

(−1)^{n+1}F (u)(t) sign u(t) ≥ \frac{1}{t^{n} ln t} ∑_{i=1}^{m} c_{i}|u(t^{α_{i}})|

for u ∈ H_{t_{0},τ}^{−} , t ≥ t_{0}.

Then the condition

inf \{\frac{1}{λ} ∑_{i=1}^{m} c_{i}α_{i}^{−λ} : λ ∈]0, +∞[\} > (n − 1)!

is sufficient for (0.1) not to have a Kneser-type solution.

9.2. Linear inequalities with deviated arguments.

Theorem 9.11. Let F ∈ V (τ), conditions (8.2), (8.3) and (9.54) be fulfilled and for some i ∈ {0, …, n − 1} and k ∈ ℕ

\overline{lim}_{t→+∞} ∫_{δ(t)}^{t} p(s)(s − δ(t))^{n−i−1}(δ(t) − δ(s))^{i} ×

×g_{k}(δ(s), δ(t))ds > i!(n − i − 1)!,  (9.67)

where the function δ is nondecreasing and

g_{k}(t, s) = exp \{\frac{1}{(n − 1)!} ∫_{t}^{s}(ξ − t)^{n−1}ψ_{k}(ξ)p(ξ)dξ\},  (9.68)

ψ_{1}(t) = 1, ψ_{j}(t) = exp \{\frac{1}{(n − 1)!} ∫_{δ(t)}^{t}(ξ − δ(t))^{n−1} ×

×p(ξ)ψ_{j−1}(ξ)dξ} (j = 2, …, k).  (9.69)

Then equation (0.1) has no Kneser-type solution.

Proof. Suppose, on the contrary, that (0.1) has a proper solution u : [t_{0}, +∞[ → ℝ satisfying (8.1). Then by (0.1) and (9.30) we have

−\frac{1}{(n − 1)} ∫_{δ(t)}^{s} \frac{(ξ − δ(t))^{n−1}|u^{(n)}(ξ)|}{u_{0}(ξ, δ(t))}dξ + \frac{1}{(n − 1)} ×

× ∫_{δ(t)}^{s}(ξ − δ(t))^{n−1}p(ξ)\frac{|u(δ(ξ))|}{u_{0}(ξ, δ(t))}dξ ≤ 0 for t_{∗} ≤ t ≤ s,

where t_{∗} > t_{1} is sufficiently large and

u_{0}(t, s) = ∑_{j=0}^{n−1} \frac{(t − s)^{j}}{j!} |u^{(j)}(t)|.

Hence we obtain

ln \frac{|u(δ(t))|}{u_{0}(s, δ(t))} ≥ \frac{1}{(n − 1)!} ∫_{δ(t)}^{s}(ξ − δ(t))^{n−1}p(ξ)\frac{|u(δ(ξ))|}{u_{0}(ξ, δ(t))}dξ (9.70)
for t_{∗} ≤ δ(t) ≤ s ≤ t.

Since u_{0}(t, s_{1}) ≤ u_{0}(t, s_{2}) for t_{∗} ≤ s_{2} ≤ s_{1} ≤ t, (9.70) implies

\frac{|u(δ(t))|}{u_{0}(s, δ(t))} ≥ exp \{\frac{1}{(n − 1)!} ∫_{δ(t)}^{s}(ξ − δ(t))^{n−1}p(ξ)\frac{|u(δ(ξ))|}{u_{0}(ξ, δ(t))}dξ\}

whence we easily conclude that

|u(δ(t))| ≥ exp \{\frac{1}{(n − 1)!} ∫_{δ(t)}^{s}(ξ − δ(t))^{n−1}p(ξ) ×

×ψ_{k}(ξ)dξ}u_{0}(s, δ(t)) for t ∈ [η_{δk}(t_{∗}), +∞[,  (9.71)

where η_{δ1} = sup{s : δ(s) < t}, η_{δj} = η_{δ1}(η_{δ j−1}(t)) (j = 2, …, k) and ψ_{k} is defined by (9.69).

On the other hand, by (81) and (9.54) from (1.6_{in}) we have

|u^{(i)}(δ(t))| ≥ \frac{1}{(n − i − 1)!} ∫_{δ(t)}^{s}(s − δ(t))^{n−i−1}p(s)|u(δ(s))|ds (9.72)
for t ≥ η_{δk}(t_{∗}).

Since

u_{0}(δ(t), δ(s)) ≥ \frac{(δ(t) − δ(s))^{i}}{i!} |u^{(i)}(δ(t))| for t ≥ η_{δ1}(t_{∗}),

from (9.71) and (9.72) it follows

∫_{δ(t)}^{t}(s − δ(t))^{n−i−1}p(s)(δ(t) − δ(s))^{i}g_{k}(δ(s), δ(t))ds ≤
≤ i!(n − i − 1)! for t ≥ η_{δk}(t_{∗}),

where the function g_{k} is defined by (9.68). But this inequality contradicts (9.67). The obtained contradiction proves the theorem. ■

Corollary 9.15. Let F ∈ V (τ) and (9.54), (8.2) and (8.3) be fulfilled, where δ is nondecreasing. Then the condition

\overline{lim}_{t→+∞} ∫_{δ(t)}^{t} p(s)(s − δ(t))^{n−1}ds > (n − 1)!  (9.73)

is sufficient for (0.1) not to have a Kneser-type solution.

Remark 9.2. Condition (9.73) cannot be replaced by

\overline{lim}_{t→+∞} ∫_{δ(t)}^{t} p(s)(s − δ(t))^{n−1}ds > (n − 1)!(1 − ε)^{n},  (9.74)

however small ε > 0 would be.

Indeed, let ε ∈]0, 1[. By the Stirling formula n_{0} ∈ ℕ can be found such that

\sqrt[n]{n!} < \frac{(1 + ε)n}{e} for n ≥ n_{0}.

Choose Δ > 0 and c > 0 such that

cΔ^{n} = (1 − ε)^{n}n!.

Then since

eΔ \sqrt[n]{p} ≤ n for n ≥ n_{0},

the equation

u^{(n)}(t) + (−1)^{n+1}cu(t − Δ) = 0

has a Kneser-type solution. On the other hand, (9.74) holds with p(t) ≡ c and δ(t) = t − Δ.

Theorem 9.12. Let F ∈ V (τ), conditions (9.54), (8.2) and (8.3) be fulfilled and

\underline{lim}_{t→+∞} ∫_{δ(t)}^{t}(s − δ(t))^{n−1}p(s)ds > \frac{(n − 1)!}{e}.  (9.75)

then equation (0.1) has no Kneser-type solution.

The proof of this theorem is analogous to that of Theorem 6.2.

Remark 9.3. Suppose that n ≥ 2. Then (9.75) cannot be replaced by

\underline{lim}_{t→+∞} ∫_{δ(t)}^{t}(s − δ(t))^{n−1}p(s)ds ≥ \frac{(n − 1)!}{e} − ε,  (9.76)

however small ε > 0 would be.

Indeed, let ε ∈ ]0, \frac{(n−1)!}{e}[. Choose λ ∈]0, 1[ such that

(λ + 1)(λ + 2) ⋯ (λ + n − 1)(1 − λ ∑_{i=0}^{n−1} \frac{1}{n − i}) > (n − 1)! − eε. (9.77)

Put

α = e^{− \frac{1}{λ}} , p(t) = \frac{λ(λ + 1) ⋯ (λ + n − 1)}{et^{n}}.  (9.78)

Then the equation

u^{(n)}(t) + (−1)^{n+1}p(t)u(αt) = 0

has the solution u(t) = t^{−λ}. On the other hand, according to (9.77) and (9.78) condition (9.76) is fulfilled.

In the case n = 1 it holds the following

Theorem 9.13. Let n = 1, F ∈ V (τ) and for some t_{0} ∈ ℝ_{+}

\begin{matrix}0 ≤ F (u)(t) sign u(t) ≤ p(t)|u(δ(t))| \\ for u ∈ H_{t_{0},τ} , t ≥ t_{0}, \end{matrix}  (9.79)

where p ∈ L_{loc}(ℝ_{+}; ℝ_{+}), δ ∈ C(ℝ_{+}; ℝ_{+}), lim_{t→+∞} δ(t) = +∞ and

∫_{δ_{∗}(t)}^{t} p(s)ds ≤ \frac{1}{e} for t ≥ t_{0}  (9.80)

with δ_{∗}(t) ≤ min{t, δ(t)}. Then there exists t_{1} ≥ t_{0} such that (0.1) has a proper solution u : [t_{1}, +∞[→]0, +∞[ satisfying

exp { − e ∫_{δ(t_{1})}^{t} p(s)ds} ≤ u(t) ≤ 1 for t ≥ t_{1}.^{16}  (9.81)

Proof. Let t_{1} ≥ max{s : δ_{∗}(s) ≤ t_{0}} and U ∈ C_{loc}([t_{0}, +∞[; ℝ) be the set of all functions u : [t_{1}, +∞[→ ℝ satisfying

exp { − e ∫_{t_{1}}^{t} p(s)ds} ≤ u(t) ≤ 1 for t ≥ t_{1},
u(t) = 1 for t ∈ [t_{0}, t_{1}[, u(δ(t)) ≤ eu(t) for t ≥ t_{1}.

(9.82)

Define the operator T : U → C_{loc}([t_{0}, +∞[: ℝ) by

T (u)(t) = \begin{cases}exp \{ − ∫_{t_{1}}^{t} & \frac{F (u)(s)}{u(s)}ds\} for t ≥ t_{1}, \\ 1 & for t ∈ [t_{0}, t_{1}[. \end{cases}  (9.83)

16For the existence of Kneser-type solutions of higher order differential equations with deviating arguments see [77,88]

By (9.80), (9.82) and (9.83) T satisfies all the conditions of Lemma 2.1. Therefore it has a fixed point which, as it can be easily checked up, is a solution of (0.1) satisfying (9.81). ■

Analogously to Theorem 6.3 can be proved

Theorem 9.14. Let F ∈ V (τ), conditions (8.2), (8.3) and (9.54) be fulfilled and

\underline{lim}_{t→+∞} ∫_{δ(t)}^{t}(s − δ(t))^{n−1}p(s)ds = c,

where the function δ is nondecreasing and c ∈ ]0, \frac{(n−1)!}{e}].^{17} Then the fulfillement for some i ∈ {0, …, n − 1} of the condition

\overline{lim}_{t→+∞} ∫_{δ(t)}^{t}(s − δ(t))^{n−i−1}(δ(t) − δ(s))^{i}p(s) exp \{\frac{x_{0}}{(n − 1)!} ×

× ∫_{δ(s)}^{δ(t)}(ξ − δ(s))^{n−1}p(ξ)dξ}ds > i!(n − i − 1)!  (9.84)

is sufficient for (0.1) not to have a Kneser-type solution, where x_{0} is the least root of the equation x = exp \{\frac{cx}{(n−1)!}\}.

9.3. Nonlinear equations.

Theorem 9.15. Let F ∈ V (τ) and for some t_{0} ∈ ℝ_{+}

(−1)^{n+1}F (u)(t) sign u(t) ≥ φ(u)(t) sign u(t) ≥ 0  (9.85)
for u ∈ H_{t_{0},τ}^{−} , t ≥ t_{0},

where φ ∈ M(τ, σ) and σ(t) ≤ t for t ∈ ℝ_{+}. Then the condition

φ̃_{0} ∈ M_{1}(τ, σ)  (9.86)

with

φ̃_{0}(u)(t) = \frac{t^{n−1}}{(n − 1)!}φ((\frac{t − σ(t)}{t})^{n−1}u(t))(t)

is sufficient for (0.1) not to have a Kneser-type solution.

Proof. Suppose, on the contrary, that (0.1) has a proper solution u : [t_{0}, +∞[ → ℝ satisfying (8.1). Without loss of the generality we can assume that u(t) > 0 for t ≥ t_{1}. Then by (8.1) from (1.6_{0n}) and (8.1) we have

u(σ(t)) ≥ ∑_{i=0}^{n−1} \frac{|u^{(i)}(t)|}{i!}(t − σ(t))^{i} ≥

≥ (\frac{t − σ(t)}{t})^{n−1} ∑_{i=0}^{n−1} \frac{|u^{(i)}(t)|}{i!}t^{i} for t ≥ η_{τ} (t_{1}),

17If c > \frac{(n−1)!}{e} then (9.75) holds and (9.84) is unnecessary

and

∑_{i=0}^{n−1} \frac{t^{i}|u^{(i)}(t)|}{i!} ≥ \frac{1}{(n − 1)!} ∫_{t}^{+∞} s^{n−1}|u^{(n)}(s)|ds

for t ≥ η_{τ} (t_{1}),

where η_{τ} (t) = sup{s : τ(s) < t}. Hence taking onto account (0.1), (8.1) and (9.85) we obtain

y(t) ≥ ∫_{t}^{+∞} φ̃(y)(s)ds for t ≥ η_{τ} (t_{1})

with y(t) = ∑_{i=0}^{n−1} \frac{|u^{(i)}(t)|}{i!}t^{i} > 0 for t ≥ η_{τ} (t_{1}). But this contradicts (9.86). The obtained contradiction proves the theorem. ■

Corollary 9.16. Let F ∈ V (τ) and for some t_{0} ∈ ℝ_{+}

(−1)^{n+1}F (u)(t) sign u(t) ≥ ∏_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} |u(s)|^{λ_{i}}d_{s}r_{i}(s, t)

for u ∈ H_{t_{0},τ}^{−} , t ≥ t_{0},

where τ_{i}, σ_{i} ∈ C(ℝ_{+}; ℝ_{+}), τ_{i}(t) ≤ σ_{i}(t) ≤ t, lim τ_{i}(t) = +∞, the functions r_{i}(s, t) are measurable, r_{i}(·, t) is nondecreasing, λ_{i} ∈]0, 1[ (i = 1, …, m) and ∑_{i=1}^{m} λ_{i} = λ < 1. Then the condition

∫ ^{+∞} t^{(n−1)(1−λ)} ∏_{i=1}^{m}(t − σ_{i}(t))^{λ_{i}(n−1)}(r_{i}(σ_{i}(t), t) − r_{i}(τ_{i}(t), t))dt = +∞

is sufficient for (0.1) not to have a Kneser-type solution.

Analogously to Theorem 9.15 can be proved

Theorem 9.15′. Let F ∈ V (τ) and for some t_{0} ∈ ℝ_{+}

(−1)^{n+1}F (u)(t) sign u(t) ≥ φ(t, u(δ_{1}(t)), …, u(δ_{m}(t))) ×
× sign u(t) ≥ 0 for u ∈ H_{t_{0},τ}^{−} , t ≥ t_{0},

where φ ∈ K_{loc}(ℝ_{+} × ℝ^{m}; ℝ), δ_{i} ∈ C(ℝ_{+}; ℝ_{+}), lim_{t→+∞} δ_{i}(t) = +∞, δ_{i}(t) ≤ t for t ∈ ℝ_{+} (i = 1, …, m) and

φ(t, x_{1}, …, x_{m}) sign x_{1} ≥ φ(t, y_{1}, …, y_{m}) sign y_{1} ≥ 0
for t ∈ ℝ_{+}, |x_{i}| ≥ |y_{i}|, x_{i}y_{i} > 0 (i = 1, …, m).

Then condition (9.86) with

φ̃(u)(t) = \frac{t^{n−1}}{(n − 1)!}φ(t, (\frac{t − δ_{1}(t)}{t})^{n−1}u(t), …, (\frac{t − δ_{m}(t)}{t})^{n−1}u(t))

is sufficient for (0.1) not to have a Kneser-type solution.

Corollary 9.17. Let F ∈ V (τ) and for some t ∈ ℝ_{+}

(−1)^{n+1}F (u)(t) sign u(t) ≥ ∏_{i=1}^{m} p_{i}(t)|u(δ_{i}(t))|^{λ_{i}}

for u ∈ H_{t_{0},τ}^{−} , t ≥ t_{0},

where p_{i} ∈ L_{loc}(ℝ_{+}; ℝ_{+}), δ_{i}(t) ≤ t for t ∈ ℝ_{+}, lim_{t→+∞} δ_{i}(t) = +∞, λ_{i} ∈]0, 1[ (i = 1, …, m) and ∑_{i=1}^{m} λ_{i} = λ < 1. Then the condition

∫ ^{+∞} t^{(n−1)(1−λ)} ∏_{i=1}^{m} p_{i}(t)(t − δ_{i}(t))^{n−1}dt = +∞

is sufficient for (0.1) not to have a Kneser-type solution.

CHAPTER 4

MONOTONICALLY INCREASING SOLUTIONS

§ 10. Auxiliary statements

In chapter 4 sufficient conditions are given for equation (0.1) not to have a solution satisfying

u^{(i)}(t)u(t) > 0 for t ≥ t_{0} (i = 0, …, n − 1),  (10.1)

as well as necessary and sufficient ones.

Lemma 10.1. Suppose that

p ∈ L_{loc}(ℝ_{+}; ℝ_{+}), δ ∈ C(ℝ_{+}; ℝ_{+}), δ ↑, δ(t) ≥ t for t ∈ ℝ_{+} (10.2)

and let u : [t_{0}, +∞[→ ℝ be a solution of the differential inequality

u^{(n)}(t) sign u(t) ≥ p(t)|u(δ(t))|  (10.3)

satisfying (10.1). Then for any k ∈ {0, …, n − 1} we have

ρ_{k}^{∗}(t)|u(δ_{i}(t))| ≤ (n − 1)!t^{n−k−1}|u^{(n−1)}(t)| for t ≥ δ(t_{0}), (10.4)

where

ρ_{k}^{∗}(t)=max{ψ_{k}^{∗}(t, s, τ) : s∈[η_{δ}^{∗}(t), t], τ ∈[t, δ(s)]},  (10.5_{k})

ψ_{k}^{∗}(t, s, τ)=∫_{s}^{t} ξ^{n−k−1}p(ξ)dξ∫_{t}^{τ} ξ^{n−k−1}p(ξ)dξ τ^{k+1−n}[(δ(s) − τ)^{n−1}+

+\frac{1}{(n − 1)!} ∫_{τ}^{δ(s)}(δ(s) − ξ)^{n−1}(δ(ξ) − ξ)^{n−1}p(ξ)dξ]  (10.6_{k})
η_{δ}^{∗}(t) = min{s : δ(s) ≥ t}.  (10.7)

Proof. By (10.1) from (10.3) we have

t^{n−k−1}|u^{(n−1)}(t)| ≥ ∫_{t_{0}}^{t} s^{n−k−1}p(s)|u(δ(s))|ds for t ≥ t_{0}. (10.8)

Let t ≥ δ(t_{0}) and (s_{0}, τ_{0}) be a point of maximum of the function ψ_{k}^{∗}(t, ·, ·) on [η_{δ}^{∗}(t), t] × [t, δ(s_{0})]. Then according to (10.8)

t^{n−k−1}|u^{(n−1)}(t)| ≥ ∫_{s_{0}}^{t} ξ^{n−k−1}p(ξ)|u(δ(ξ))|dξ ≥
≥ ∫_{s_{0}}^{t} ξ^{n−k−1}p(ξ)dξ|u(δ(s_{0}))|,  (10.9)
τ_{0}^{n−k−1}|u^{(n−1)}(τ_{0})| ≥ ∫_{t}^{τ_{0}} ξ^{n−k−1}p(ξ)|u(δ(ξ))|dξ ≥
≥ ∫_{t}^{τ_{0}} ξ^{n−k−1}p(ξ)dξ|u(δ(t))|.  (10.10)

On the other hand, (1.6_{0n}) in view of (10.1) implies

|u(δ(s_{0}))| ≥ |u^{(n−1)}(τ_{0})|\frac{1}{(n − 1)!}[(δ(s_{0}) − τ_{0})^{n−1} +

+\frac{1}{(n − 1)!} ∫_{τ_{0}}^{δ(s_{0})}(δ(s_{0}) − ξ)^{n−1}p(ξ)(δ(ξ) − ξ)^{n−1}dξ].

Therefore by (10.9) and (10.10) we obtain

t^{n−k−1}|u^{(n−1)}(t)| ≥ \frac{1}{(n − 1)!} ∫_{s_{0}}^{t} ξ^{n−k−1}p(ξ)dξ ∫_{t}^{τ_{0}} ξ^{n−k−1}p(ξ)dξ ×

×[(δ(s_{0})−τ_{0})^{n−1}+ \frac{1}{(n−1)!}∫_{τ_{0}}^{δ(s_{0})}(δ(s_{0})−ξ)^{n−1}p(ξ)(δ(ξ)−ξ)^{n−1}dξ]|u(δ(t))|

whence it follows the validity of (10.4). ■

Lemma 10.2. Let (10.2) be fulfilled and

\underline{lim}_{t→+∞} ∫_{t}^{δ(t)} p(s)ds > 0,  (10.11)
vrai sup{p(t) : t ∈ ℝ_{+}} < +∞.  (10.12)

Let, moreover, u : [t_{0}, +∞[→ ℝ be a solution of (10.3) satisfying (10.1). Then

\overline{lim}_{t→+∞} \frac{|u(δ(t))|}{|u^{(n−1)|}(t)|} < +∞.  (10.13)

Proof. According to Lemma 10.1 it suffices to prove that

\underline{lim}_{t→+∞}ρ_{n−1}^{∗}(t) > 0  (10.14)

with ρ_{n−1}^{∗}(t) defined by (10.5_{n−1}) and (10.6_{n−1}).

By (10.11) there exist c > 0 and t_{1} ≥ t_{0} such that

∫_{t}^{δ(t)} p(s)ds ≥ c for t ≥ t_{1}.

Therefore for any t ≥ δ(t_{1}) there exist ξ_{1} ∈ [η_{δ}^{∗}(t), t] and ξ_{2} ∈ [t, δ(ξ_{1})] such that

∫_{ξ_{1}}^{t} p(s)ds ≥ \frac{c}{4}, ∫_{t}^{ξ_{2}} p(s)ds ≥ \frac{c}{4}, ∫_{ξ_{2}}^{δ(ξ_{1})} p(s)ds ≥ \frac{c}{4}. (10.15)

According to (10.5_{n−1}) and (10.6_{n−1}) we have

ρ_{n−1}^{∗}(t) ≥ ∫_{ξ_{1}}^{t} p(s)ds ∫_{t}^{ξ_{2}} p(s)ds(δ(ξ_{1}) − ξ_{2})^{n−1}.  (10.16)

On the other hand, in view of (10.12) and (10.15) δ(ξ_{1}) − ξ_{2} ≥ \frac{c}{4r} with r = vrai sup{p(t) : t ∈ ℝ_{+}}. Thus the validity of (10.14) obviously follows from (10.15) and (10.16). ■

Lemma 10.3. Let (10.2) be fulfilled and for some k ∈ {0, …, n − 1}

\underline{lim}_{t→+∞} ∫_{t}^{δ(t)} s^{n−k−1}p(s)ds > 0,  (10.17_{k})
vrai sup{t^{n−k}p(t) : t ∈ ℝ_{+}} < +∞.  (10.18_{k})

Let, moreover, u : [t_{0}, +∞[→ ℝ be a solution of (10.3) satisfying (10.1). Then

\overline{lim}_{t→+∞} \frac{t^{2k+1−n}|u(δ(t))|}{|u^{(n−1)}(t)|} < +∞.  (10.19)

Proof. According to Lemma 10.1 it suffices to show that

\underline{lim}_{t→+∞}ρ_{k}^{∗}(t)t^{−k} > 0  (10.20)

with ρ_{k}^{∗} defined by (10.5_{k}) and (10.6_{k}).

By (10.17_{k}) there exist c > 0 and t_{1} ≥ t_{0} such that

∫_{t}^{δ(t)} s^{n−k−1}p(s)ds ≥ c for t ≥ t_{1}.

Therefore for any t ∈ [δ(t_{1}), +∞[ there exist ξ_{1} ∈ [η_{δ}^{∗}(t), t] and ξ_{2} ∈ [t, δ(ξ_{1})] such that

\begin{matrix}∫_{ξ_{1}}^{t} s^{n−k−1}p(s)ds ≥ \frac{c}{4}, ∫_{t}^{ξ_{2}} s^{n−k−1}p(s)ds ≥ \frac{c}{4}, \\ ∫_{ξ_{2}}^{δ(ξ_{1})} s^{n−k−1}p(s)ds ≥ \frac{c}{4}. \end{matrix}  (10.21)

In view of (10.18) and (10.21) we have \frac{δ(ξ_{1})}{ξ_{2}} ≥ exp \{\frac{c}{4r} \} with r =

vrai sup{t^{n−k}p(t) : t ∈ ℝ_{+}}, so according to (10.5_{k}), (10.6_{k}) and (10.21) we obtain

ρ_{k}^{∗}(t) ≥ \frac{c^{2}}{16}t^{k}( exp \{\frac{c}{4r} \} − 1)^{n−1}

whence it follows the validity of (10.20). ■

The following lemma can easily be deduced from Lemma 10.1

Lemma 10.4. Let (10.2) be fulfilled, q ∈ L_{loc}(ℝ_{+}; ℝ_{+}) and for some k ∈ {0, …, n − 1} let

ρ_{k}^{∗}(t) > 0 for t ≥ t_{0}  (10.22_{k})

with ρ_{k}^{∗} defined by (10.5_{k}) let (10.6_{k}). Let, moreover u : [t_{0}, +∞[→ ℝ be a solution of

p(t)|u(δ(t))| ≤ u^{(n)}(t) sign u(t) ≤ q(t)|u(δ(t))|  (10.23)

satisfying (10.1). Then there exists λ > 0 such that

\overline{lim}_{t→+∞}|u^{(n−1)}(t)|exp\{ − λ∫_{t_{0}}^{t} q(s)(ρ_{k}^{∗}(s))^{−1}s^{n−k−1}ds\}<+∞.

Lemma 10.5. Let (10.2), (10.11) and (10.12) be fulfilled, σ ∈ C(ℝ_{+}; ℝ_{+}) and

\overline{lim}_{t→+∞}(σ(t) − t) < +∞.  (10.24)

Let, moreover, u : [t_{0}, +∞[→ ℝ be a solution of (10.3) satisfying (10.1). Then

\overline{lim}_{t→+∞} \frac{|u(σ(t))|}{|u(δ(t))|} < +∞.  (10.25)

Proof. According to (10.11), (10.12) and (10.24) there exist c_{1}, c_{2} ∈]0, +∞[ and t_{1} ≥ t_{0} such that

c_{1} < c_{2}, t + c_{1} ≤ δ(t), σ(t) ≤ t + c_{2} for t ≥ t_{1}.  (10.26)

Due to Lemma 10.2

\overline{lim}_{t→+∞} \frac{|u(δ(t))|}{|u^{(n−1)}(t)|} < +∞,

so, taking into account the nondecreasing character of |u(t)|, by (10.26) we have

\overline{lim}_{t→+∞} \frac{|u(t + c_{1})|}{|u^{(n−1)}(t)|} < +∞.  (10.27)

On the other hand, equality (1.6_{0n}) along with (10.1) implies

|u(t + c_{1})| ≥ \frac{1}{(n − 1)!}|u^{(n−1)}(t + \frac{c_{1}}{2} )|(\frac{c_{1}}{2} )^{n−1} for t ≥ t_{1}. (10.28)

Therefore in view of (10.27) we obtain

\overline{lim}_{t→+∞} \frac{|u^{(n−1)}(t + \frac{c_{1}}{2} )|}{|u^{(n−1)}(t)|} < +∞.  (10.29)

Let k ∈ ℕ satisfy k > \frac{2c_{2}}{c_{1}} . Then by (10.26)

\frac{|u(σ(t))|}{|u(δ(t))|} ≤ \frac{|u(t + k \frac{c_{1}}{2} )|}{|u(t + c_{1})|} = \frac{|u(t + k \frac{c_{1}}{2} )|}{|u^{(n−1)}(t + (k − 1)\frac{c_{1}}{2} )|} ×

×\frac{|u^{(n−1)}(t + \frac{c_{1}}{2} )|}{|u(t + c_{1})|} ∏_{i=2}^{k−1} \frac{|u^{(n−1)}(t + ic_{1})|}{|u^{(n−1)}(t + (i − 1)\frac{c_{1}}{2} )|}

whence according to (10.27)–(10.29) it obviously follows the validity of (10.25). ■

Lemma 10.6. Let (10.2), (10.17_{0}) and (10.18_{0}) be fulfilled, σ ∈ C(ℝ_{+}; ℝ_{+}) and

\overline{lim}_{t→+∞} \frac{σ(t)}{t} < +∞.  (10.30)

Let, moreover, u : [t_{0}, +∞[→ ℝ be a solution of (10.3) satisfying (10.1). Then (10.25) holds.

Proof. By (10.17_{0}), (10.18_{0}) and (10.30) there exist c_{1}, c_{2} ∈]0, +∞[ and t_{1} ≥ t_{0} such that

1 < c_{1} < c_{2}, c_{1}t ≤ δ(t), σ(t) ≤ c_{2}t for t ≥ t_{1}.  (10.31)

Equality (1.6_{0n}) along with (10.1) implies

|u(c, t)| ≥ \frac{1}{(n − 1)!}|u^{(n−1)}(c_{0}t)|(\frac{c_{1} − 1}{2})^{n−1}t^{n−1} for t ≥ t_{1} (10.32)

with c_{0} = \frac{1+c_{1}}{2} . On the other hand, according to Lemma 10.3 and (10.31) we have

\overline{lim}_{t→+∞} \frac{|u(c_{1}t)|}{t^{n−1}|u^{(n−1)}(t)|} < +∞.  (10.33)

Therefore by (10.32) and (10.33)

\overline{lim}_{t→+∞} \frac{|u^{(n−1)}(c_{0}t)|}{|u^{(n−1)}(t)|} < +∞.  (10.34)

Choose k ∈ ℕ such that c_{0}^{k} ≥ c_{2}. Then (10.31) implies

\frac{|u(σ(t))|}{|u(δ(t))|} ≤ \frac{|u(c_{0}^{k}t)|}{|u(c_{1}t)|} = \frac{|u(c_{0}^{k}t|}{|u^{(n−1)}(c_{0}^{k−1}t)|} ×

×\frac{|u^{(n−1)}(c_{0}t)|}{|u(c_{1}t)|} ∏_{i=2}^{k−1} \frac{|u^{(n−1)}(c_{0}^{i}t)|}{|u^{(n−1)}(c_{0}^{i−1}t)|}

whence in view of (10.32)–(10.34) it follows the validity of (10.24). ■

Lemma 10.7. Let F ∈ V (τ) and

F (u)(t) sign u(t) ≤ −φ(|u|)(t) for u ∈ H_{t_{0},τ} , t ≥ t_{0},  (10.35)

where the mapping φ : C(ℝ_{+}; ℝ_{+}) → L_{loc}(ℝ_{+}; ℝ_{+}) is such that

φ(x)(t)≥φ(y)(t), if x, y ∈C(ℝ_{+}; ℝ_{+}), x(s)≥y(s) for s≥τ(t) (10.36)

and for any c > 0

∫_{t}^{+∞} φ(c)(s)ds > 0 for t ∈ ℝ_{+}.  (10.37)

Let, moreover, problem (0.1), (10.1) has a solution. Then the equation

u^{(n)}(t) = φ(u)(t)  (10.38)

has a solution u_{0} : [t_{0}, +∞[→]0, +∞[ satisfying

u_{0}^{(i)}(t) > 0 for t ≥ t_{∗} (i = 0, …, n − 1)  (10.39)

with t_{∗} ≥ t_{0} sufficiently large.

Proof. Let u : [t_{0}, +∞[→ ℝ be a solution of (0.1) satisfying (10.1). Choose t_{1} > t_{0} such that t_{0} ≤ t^{0} = inf{τ(t) : t ∈ [t_{1}, +∞[} and consider the sequence of functions {u_{i}(t)}_{i=1}^{+∞} defined by

u_{1}(t) = |u(t)| for t ≥ t_{0},
u_{i}(t) = \begin{cases}|u(t_{∗})| + \frac{1}{(n−1)!} ∫_{t_{∗}}^{t}(t − s)^{n−1}φ(u_{i−1})(s)ds for t ≥ t_{∗}, \\ |u(t)| for t ∈ [t_{0}, t_{∗}] (i = 2, 3, …) \end{cases}(10.40)

with t_{∗} = max{t^{0}, t_{1}}. By (10.35) and (10.36) this sequence is obviously decreasing. Denote its limit by u_{0}(t). According to (10.40) u_{0} is a solution of (10.38) on [t_{∗}, +∞[. On the other hand, due to (10.37) condition (10.39) is obviously fulfilled. ■

Taking into acount Lemmas 10.3–10.6 one can easily verify the validity of the following lemmas.

Lemma 10.8. Let (10.2) be fulfilled, q ∈ L_{loc}(ℝ_{+}; ℝ_{+}), for some k ∈ {0, …, n − 1} (10.22_{k}) hold and

\overline{lim}_{t→+∞} \frac{1}{t} ∫_{t_{0}}^{t} q(s)s^{n−k−1}(ρ_{k}^{∗}(s))^{−1}ds < +∞  (10.41_{k})

with ρ_{k}^{∗} defined by (10.5_{k}) and (10.6_{k}). Let, moreover, u : [t_{0}, +∞[→ ℝ be a solution of (10.23) satisfying (10.1). Then there exists λ > 0 such that

|u(t)|e^{−λt} → 0 for t → +∞.  (10.42)

Corollary 10.1. Let (10.2), (10.11) and (10.12) be fulfilled, q ∈ L_{loc}(ℝ_{+}; ℝ_{+}) and

\overline{lim}_{t→+∞} \frac{1}{t} ∫_{0}^{t} q(s)ds < +∞.  (10.43)

Let, moreover, u : [t_{0}, +∞[→ ℝ be a solution of (10.23) satisfying (10.1). Then there exists λ > 0 such that (10.42) holds.

Corollary 10.2. Let (10.2) be fulfilled, for some k ∈ {0, …, n−1} (10.17_{k}) and (10.18_{k}) hold, q ∈ L_{loc}(ℝ_{+}; ℝ_{+}) and

\overline{lim}_{t→+∞} \frac{1}{t} ∫_{1}^{t} q(s)s^{n−2k−1}ds < +∞.

Let, moreover, u : [t_{0}, +∞[→ ℝ be a solution of (10.23) satisfying (10.1). Then there exists λ > 0 such that (10.42) holds.

Lemma 10.9. Let (10.2), (10.11) and (10.12) be fulfilled,

\begin{matrix}q_{i} ∈ L_{loc}(ℝ_{+}; ℝ_{+}), σ_{i} ∈ C(ℝ_{+}; ℝ_{+}), \\ \overline{lim}_{t→+∞}(σ_{i}(t) − t) < +∞ (i = 1, …, m) \end{matrix}  (10.44)

and

\overline{lim}_{t→+∞} \frac{1}{t} ∫_{0}^{t} ∑_{i=1}^{m} q_{i}(s)ds < +∞.  (10.45)

Let, moreover, u : [t_{0}, +∞[→ ℝ be a solution of

p(t)|u(δ(t))| ≤ u^{(n)} sign u(t) ≤ ∑_{i=1}^{m} q_{i}(t)|u(σ_{i}(t))|  (10.46)

satisfying (10.1). Then there exists λ > 0 such that (10.42) holds.

Lemma 10.10. Let (10.2) be fulfilled, for some k ∈ {0, …, n − 1} (10.22_{k}) hold and

\overline{lim}_{t→+∞} \frac{1}{ln t} ∫_{t_{0}}^{t} q(s)s^{n−k−1}(ρ_{k}^{∗}(s))^{−1}ds < +∞  (10.47_{k})

with ρ_{k}^{∗}(t) defined by (10.5_{k}) and (10.6_{k}). Let, moreover, u : [t_{0}, +∞[→ ℝ be a solution of (10.23) satisfying (10.1). Then there exists λ > 0 such that

|u(t)|t^{−λ} → 0 for t → +∞.  (10.48)

Corollary 10.3. Let (10.2), (10.17_{0}) and (10.18_{0}) be fulfilled and

\overline{lim}_{t→+∞} \frac{1}{ln t} ∫_{0}^{t} s^{n−1}q(s)ds < +∞.  (10.49)

Let, moreover, u : [t_{0}, +∞[→ ℝ be a solution of (10.23) satisfying (10.1). Then there exists λ > 0 such that (10.48) holds.

Lemma 10.11. Let (10.2), (10.17_{0}) and (10.18_{0}) be fulfilled,

\begin{matrix}q_{i} ∈ L_{loc}(ℝ_{+}; ℝ_{+}), σ_{i} ∈ C(ℝ_{+}; ℝ_{+}), \\ \overline{lim}_{t→+∞} \frac{σ_{i}(t)}{t} < +∞ (i = 1, …, m) \end{matrix}  (10.50)

and

\overline{lim}_{t→+∞} \frac{1}{ln t} ∫_{0}^{t} s^{n−1} ∑_{i=1}^{m} q_{i}(s)ds < +∞.  (10.51)

Let, moreover, u : [t_{0}, +∞[→ ℝ be a solution of (10.46) satisfying (10.1). Then there exists λ > 0 such that (10.48) holds.

§ 11. On monotonically increasing solutions

11.1. Equations with a Linear Minorant. Let

\begin{matrix}σ; \overline{σ} ∈ C(ℝ_{+}; ℝ_{+}), σ(t) ≤ \overline{σ}(t) for t ∈ ℝ_{+}, \\ lim_{t→+∞} σ(t) = +∞, σ ↑ . \end{matrix}  (11.1)

Denote by M^{+}(σ; \overline{σ}) the set of all continuous maps φ : C(ℝ_{+}; ℝ_{+}) → L_{loc}(ℝ_{+}; ℝ_{+}) which for any t ∈ ℝ_{+} satisfy

φ(x)(t) ≥ φ(y)(t) if x, y ∈ C(ℝ_{+}; ℝ_{+}), x(s) ≥ y(s) for s ∈ [σ(t), \overline{σ}(t)], (xy)(t)≥x(σ(t))φ(y)(t) if x, y ∈C(ℝ_{+},]0, +∞[), x(t)↑+∞ as t↑+∞, φ(xy)(t) ≥ x(\overline{σ}(t))φ(y)(t) if x, y ∈ C(ℝ_{+},]0, +∞[), x(t) ↓ 0 as t ↑ +∞.

Lemma 11.1. Let φ ∈ M^{+}(σ; \overline{σ}) and the equation

u^{(n)}(t) = φ(u)(t)  (11.2)

have a solution u : [t_{0}, +∞[→ ℝ satisfying

u^{(i)}(t) > 0 (i = 0, …, n − 1) for t ≥ t_{∗}.  (11.3)

Let, moreover, there exist γ ∈ C(ℝ_{+}; ]0, +∞[), r_{2} > 0 and r_{1} ∈ [0, r_{2}[ such that

γ(t) ↓ 0 for t ↑ +∞, lim_{t→+∞}(γ(t))^{r_{1}}u(t) = +∞,

\underline{lim}_{t→+∞}(γ(t))^{r_{2}}u(t) = 0, (11.4)

\overline{lim}_{t→+∞} \frac{γ(σ(t))}{γ(\overline{σ}(t))} < +∞, lim_{t→+∞} t^{n−1}(γ(t))^{r_{2}} = 0.  (11.5)

Then

\underline{lim}_{t→+∞}(γ(t))^{r_{2}} ∫_{0}^{t}(t − s)^{n−1}φ(θ)(s)ds ≤ c^{r_{2}−r_{1}}(n − 1)!  (11.6)

with θ(t) = (γ(t))^{−r_{2}}.

Proof. Put

ũ(t) = inf {(γ(s))^{r_{1}}|u(s)| : s ≥ t}.  (11.7)

According to (11.1) and (11.7)

ũ(σ(t)) ↑ +∞ for t ↑ +∞  (11.8)

and

\underline{lim}_{t→+∞}ũ(σ(t))(γ(σ(t)))^{r_{2}−r_{1}} = 0.  (11.9)

Taking into account (11.7)–(11.9) and Lemma 7.1 we see that there exists a sequence {t_{k}}_{k=1}^{+∞} such that t_{k} ↑ +∞ as k ↑ +∞ and

ũ(σ(t_{k})) = (γ(σ(t_{k})))^{r_{1}}u(σ(t_{k})), (γ(σ(t_{k})))^{r_{2}−r_{1}} ũ(σ(t_{k})) ≤
≤ (γ(σ(t)))^{r_{2}−r_{1}} ũ(σ(t)) for t_{∗} ≤ t ≤ t_{k}, k = k_{0}, k_{0} + 1, (11.10)

with k_{0} ∈ ℕ and t_{∗} > t_{0} sufficiently large.

On the other hand, in view of (11.3) from (11.2) we obtain

u(σ(t)) ≥ \frac{1}{(n − 1)!} ∫_{t_{∗}}^{σ(t)}(σ(t) − s)^{n−1}φ(u)(s)ds for t ≥ t_{1},

with t_{1} > t_{∗} sufficiently large. Hence by (11.7), (11.10) and the fact that φ ∈ M^{+}(σ; \overline{σ}) we have

u(σ(t_{k})) ≥ \frac{1}{(n − 1)!} ∫_{t_{∗}}^{σ(t_{k})}(σ(t_{k}) − s)^{n−1}ũ(σ(s))φ(θ_{1})(s)ds ≥

≥ \frac{(γ(σ(t_{k})))^{r_{2}}u(σ(t_{k}))}{(n − 1)!} ∫_{t_{∗}}^{σ(t_{k})}(σ(t_{k}) − s)^{n−1} ×

×(γ(σ(s)))^{r_{1}−r_{2}}φ(θ_{1})(s)ds (k = k_{1}, k + 1, …)  (11.11)

with θ_{1}(t) = (γ(t))^{−r_{1}} and t_{k_{1}} ≥ t_{1}.

Take any ε ∈]0, 1[ and choose t^{∗} > t_{∗} such that

\frac{γ(σ(t))}{γ(\overline{σ}(t))} ≤ c + ε for t ≥ t^{∗}.

Then since φ ∈ M^{+}(σ; \overline{σ}), from (11.11) we obtain

(n − 1)! ≥ \overline{lim}_{k→+∞}(γ(σ(t_{k})))^{r_{2}}∫_{t_{∗}}^{σ(t_{k})}(σ(t_{k}) − s)^{n−1}(γ(σ(s)))^{r_{1}−r_{2}}φ(θ_{1})(s)ds≥

≥ \overline{lim}_{k→+∞}(γ(σ(t_{k})))^{r_{2}} ∫_{t^{∗}}^{σ(t_{k})}(σ(t_{k}) − s)^{n−1}(\frac{γ(\overline{σ}(s))}{γ(σ(s))})^{r_{2}−r_{1}}φ(θ)(s)ds ≥

≥ (c + ε)^{r_{1}−r_{2}} \overline{lim}_{k→+∞}(γ(σ(t_{k})))^{r_{2}} ∫_{t^{∗}}^{σ(t_{k})}(σ(t_{k}) − s)^{n−1}φ(θ)(s)ds,

where θ(t) = (γ(t))^{−r_{2}}. Hence, taking onto account the second condition of (11.5),

\underline{lim}_{t→+∞}(γ(t))^{r_{2}} ∫_{0}^{t}(t − s)^{n−1}φ(θ)(s)ds ≤ \overline{lim}_{k→+∞}(γ(σ(t_{k})))^{r_{2}} ×

× ∫_{t^{∗}}^{σ(t_{k})}(σ(t_{k}) − s)^{n−1}φ(θ)(s)ds ≤ (c + ε)^{r_{2}−r_{1}}(n − 1)!.

Since ε is arbitrary, the last inequality implies (11.6). ■

Let t_{0} ∈ ℝ_{+}. Denote by H_{t_{0},τ}^{+} the set of all functions u ∈ C̃_{loc}^{n−1}(ℝ_{+}; ℝ) satifying

u^{(i)}(t)u(t) > 0 (i = 0, …, n − 1), u^{(n)}(t)u(t) ≥ 0 for t ≥ t_{∗},

where t_{∗} = min{t_{0}, τ_{∗}(t_{0})}, τ_{∗}(t) = inf{τ(s) : s ≥ t}.

Theorem 11.1. Let F ∈ V (τ) and for some t_{0} ∈ ℝ_{+}

F (u)(t) sign u(t) ≤ −φ(|u|)(t) for u ∈ H_{t_{0},τ}^{+} , t ≥ t_{0},  (11.12)
p(t)|u(δ(t)|) ≤ φ(|u|)(t) ≤ q(t)|u(δ(t))|for u ∈ H_{t_{0},τ}^{+} , t ≥ t_{0}, (11.13)

where φ ∈ M^{+}(σ; \overline{σ}),

\overline{lim}_{t→+∞}(\overline{σ}(t) − σ(t)) < +∞,  (11.14)
p, q ∈L_{loc}(ℝ_{+}; ℝ_{+}), δ ∈C(ℝ_{+}; ℝ_{+}), δ ↑, δ(t)≥t for t∈ℝ_{+}, (11.15)
∫ ^{+∞} δ(t)^{n−1}p(t)dt = +∞.  (11.16)

Let, moreover, for some k ∈ {0, …, n − 1} (10.22_{k}) and (10.41_{k}) be fulfilled with ρ_{k}^{∗}(t) defined by (10.5_{k}) and (10.6_{k}). Then the condition

inf \{ \underline{lim}_{t→+∞}e^{−λt}∫_{0}^{t}(t−s)^{n−1}φ(θ)(s)ds : λ ∈]0, +∞[\} > (n − 1)! (11.17)

with θ(t) = e^{λt} is sufficient for problem (0.1), (10.1) to have no solution.

Proof. Suppose, on the contrary, that (0.1) has a solution u_{0} : [t_{0}, +∞[→ ℝ satisfying (10.1). According to (11.12) and Lemma 10.7 (11.2) has a solution u : [t_{∗}, +∞[→]0, +∞[ satisfying (11.3). By (10.22_{k}), (10.41_{k}), (11.13), (11.15) and Lemma 10.3 there exists λ > 0 such that lim_{t→+∞} e^{−λt}u(t) = 0.

Denote by Λ the set of all λ satisfying lim_{t→+∞} e^{−λt}u(t) = +∞ and put

λ_{0} = sup Λ. In view of (11.14)–(11.16) it is obvious that 0 ∈ Λ and λ_{0} < +∞. By (11.17) there exist ε > 0 and t_{1} ∈ ℝ_{+} such that

e^{−λt} ∫_{0}^{t}(t − s)^{n−1}φ(θ)(s)ds ≥ (n − 1)! + ε  (11.18)
for t ≥ t_{1}, λ ∈]λ_{0}, λ_{0} + ε].

Choose ε_{2} ∈]0, ε[ and ε_{1} ∈ [0, ε_{1}[ such that

\begin{matrix}λ_{0} − ε_{1} ≥ 0, c^{ε_{2}+ε_{1}}(n − 1)! < (n − 1)! + ε, \\ lim_{t→+∞} e^{−(λ_{0}−ε_{1})t}u(t) = +∞, \underline{lim}_{t→+∞}e^{−(λ_{0}+ε_{2})t}u(t) = 0, \end{matrix} (11.19)

where c = \overline{lim}_{t→+∞}e^{\overline{σ}(t)−σ(t)}. According to (11.19) u is a solution of (11.2)

satisfying all the conditions of Lemma 11.1 with γ(t) = e^{−t}, r_{1} = λ_{0} − ε_{1} and r_{2} = λ_{0} + ε_{2}. Therefore

\underline{lim}_{t→+∞}e^{−(λ_{0}+ε_{2})t} ∫_{0}^{t}(t − s)^{n−1}φ(θ)(s)ds ≤ c^{ε_{2}+ε_{1}}(n − 1)! < (n − 1)! + ε

with θ(t) = e^{(λ_{0}+ε_{2})t}. But this contradicts (11.18). The obtained contradiction proves the theorem. ■

Theorem 11.1′. Let F ∈ V (τ) and conditions (11.12)–(11.15), (10.11), (10.12) and (10.43) be fulfilled, where φ ∈ M^{+}(σ; \overline{σ}). Then (11.17) with θ(t) = e^{λt} is sufficient for problem (0.1), (10.1) to have no solution.

Proof. The validity of the theorem follows from Theorem 11.1 and Corollary 10.2. ■

Theorem 11.2. Let F ∈ V (τ), conditions (11.12) and (11.14) be fulfilled with φ ∈ M^{+}(σ; \overline{σ}), for some t_{0} ∈ ℝ_{+}

p(t)|u(δ(t)|) ≤ φ(|u|)(t) ≤ ∑_{i=1}^{m} q_{i}(t)|u(σ_{i}(t))|  (11.20)
for u ∈ H_{t_{0},τ}^{+} , t ≥ t_{0}

and (10.2), (10.11), (10.12), (10.44) and (10.45) hold. Then (11.1) is sufficient for problem (0.1), (10.1) to have no solution.

Proof. The validity of the theorem follows from Theorem 11.1 and Lemma 10.9. ■

Theorem 11.3. Let F ∈ V (τ) and for some t_{0} ∈ ℝ_{+}

F (u)(t) sign u(t) ≤ − ∑_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} |u(s)|d_{s}r_{i}(s; t)  (11.21)
for u ∈ H_{t_{0},τ}^{+} , t ≥ t_{0},

where

r_{i}(s, t) are measurable, r_{i}(·, t) are nondecreasing,
τ_{i}, σ_{i} ∈ C(ℝ_{+}; ℝ_{+}), τ_{i}(t) ≤ σ_{i}(t) for t ∈ ℝ_{+},

lim_{t→+∞} τ_{i}(t) = +∞ (i = 1, …, m),

(11.22)

\overline{lim}_{t→+∞}(\overline{σ}(t) − σ(t)) < +∞, \overline{lim}_{t→+∞}(\overline{σ}(t) − t) < +∞  (11.23)

with \overline{σ}(t) = max\{σ_{i}(t) : i = 1, …, m\}, σ(t) = inf\{min(τ_{i}(s) : i = 1, …, m) : s ≥ t}. Let, moreover, there exist i_{0} ∈ {1, …, m} and a nondecreasing function δ ∈ C(ℝ_{+}; ℝ_{+}) such that

σ_{i_{0}}(t) > t, max{t, τ_{i_{0}}(t)} ≤ δ(t) ≤ σ_{i_{0}}(t) for t ∈ ℝ_{+}, (11.24)
\underline{lim}_{t→+∞} ∫_{t}^{δ(t)} [r_{i_{0}}(σ_{i_{0}}(s), s) − r_{i_{0}}(δ(s), s)]ds > 0,  (11.25)
vrai sup {[r_{i_{0}}(σ_{i_{0}}(t), t) − r_{i_{0}}(δ(t), t)] : t ∈ ℝ_{+}} < +∞, (11.26)
\overline{lim}_{t→+∞} \frac{1}{t} ∫_{0}^{t} ∑_{i=1}^{m} [r_{i}(σ_{i}(s), s) − r_{i}(τ_{i}(s), s)]ds < +∞.  (11.27)

Then the condition

inf \{ \underline{lim}_{t→+∞}e^{−λt} ∫_{0}^{t}(t − s)^{n−1} ×

× ∑_{i=1}^{m} ∫_{τ_{i}(s)}^{σ_{i}(s)} e^{λξ}d_{ξ}r_{i}(ξ, s)ds : λ ∈]0, +∞[} > (n − 1)! (11.28)

is sufficient for problem (0.1), (10.1) to have no solution.

Proof. It suffices to show that the conditions of Theorem 11.2 are satisfied with

φ(u)(t) = ∑_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} u(s)d_{s}r_{i}(s, t).  (11.29)

It is obvious that φ ∈ M^{+}(σ; \overline{σ}) and (11.14) holds. On the other hand, by (11.24)–(11.27) conditions (11.20), (10.11), (10.12), (10.44) and (10.45) are fulfilled with p(t) = r_{i_{0}}(σ_{i_{0}}(t), t) − r_{i_{0}}(δ(t), t) and q_{i}(t) = r_{i}(σ_{i}(t), t) − r_{i}(τ_{i}(t), t) (i = 1, …, n). Moreover, (11.28) implies (11.17). Therefore the operator defined by (11.29) satisfies all the conditions of Theorem 11.2. ■

Corollary 11.1. Let F ∈ V (τ), conditions (11.21)–(11.27) be fulfilled and for some t_{0} ∈ ℝ_{+}

inf {λ^{−n} vrai inf_{t≥t_{0}} ( ∑_{i=1}^{m}∫_{τ_{i}(t)}^{σ_{i}(t)}e^{λ(ξ−t)}d_{ξ}r_{i}(ξ, t)) : λ ∈]0, +∞[}>1. (11.30)

Then problem (0.1), (10.1) has no solution.

Proof. It suffices to note that (11.30) implies (11.28). ■

Corollary 11.2. Let (11.20)–(11.27) be fulfilled , σ_{i}(t) ≥ t for t ∈ ℝ_{+} (i = 1, …, m) and for some t_{0} ∈ ℝ_{+}

vrai inf_{t≥t_{0}} ( ∑_{i=1}^{m} ∫_{τ_{i}^{∗}(s)}^{σ_{i}(s)}(ξ − t)^{n}d_{ξ}r_{i}(ξ, t)) > (\frac{n}{e})^{n}  (11.31)

with τ_{i}^{∗}(t) = max{t; τ_{i}(t)} (i = 1, …, m). Then problem (0.1), (10.1) has no solution.

Proof. It suffices to note that since e^{x} ≥ x^{n}(\frac{e}{n})^{n} for x ≥ 0, (11.31) implies (11.30). ■

Corollary 11.3. Let F ∈ V (τ) and for some t_{0} ∈ ℝ_{+}

F (u)(t) sign u(t) ≤ − ∑_{i=1}^{m} c_{i} ∫_{t−Δ_{i}}^{t−\overline{Δ}_{i}} |u(s)|ds for u ∈ H_{t_{0},τ}^{+} , t ≥ t_{0}

with c_{i} > 0, Δ_{i}, \overline{Δ}_{i} ∈ ℝ, Δ_{i} > \overline{Δ}_{i} (i = 1, …, m). Let, moreover, \overline{Δ}_{i_{0}} < 0 for some i_{0} ∈ {1, …, m}. Then the condition

inf \{λ^{−n−1} ∑_{i=1}^{m} c_{i}(e^{−λ\overline{Δ}_{i}} − e^{−λΔ_{i}}) : λ ∈]0, +∞[\} > 1 (11.32)

is sufficient for problem (0.1), (10.1) to have no solution.

Corollary 11.3′. Let c_{i} > 0, Δ_{i}, \overline{Δ}_{i} ∈ ℝ, Δ_{i} > \overline{Δ}_{i} (i = 1, …, m) and \overline{Δ}_{i_{0}} < 0 for some i_{0} ∈ \{1, …, m\}. Then (11.32) is necessary and sufficient for the equation

u^{(n)}(t) = ∑_{i=1}^{m} c_{i} ∫_{t−Δ_{i}}^{t−\overline{Δ}_{i}} u(s)ds  (11.33)

to have no solution satisfying (10.1).

Proof. Sufficiency follows from Corollary 11.3. If we suppose that (10.32) is violated, then (11.33) has the solution u(t) = e^{λt} with λ > 0. ■

Theorem 11.4. Let F ∈ V (τ) and for some t_{0} ∈ ℝ_{+}

F (u)(t) sign u(t)≤−∑_{i=1}^{m} p_{i}(t)|u(δ_{i}(t))| for u∈H_{t_{0},τ}^{+} , t≥t_{0} (11.34)

with

\begin{matrix}p_{i} ∈ L_{loc}(ℝ_{+}; ℝ_{+}), δ_{i} ∈ C(ℝ_{+}; ℝ_{+}), \\ lim_{t→+∞} δ_{i}(t) = +∞ (i = 1, …, m), \end{matrix}  (11.35)
\overline{lim}_{t→+∞}(δ_{i}(t) − t) < +∞ (i = 1, …, m).  (11.36)

Let, moreover, there exist i_{0} ∈ {1, …, m} such that δ_{i_{0}}(t) is nondecreasing, δ_{i_{0}}(t) > t for t ∈ ℝ_{+} and

\underline{lim}_{t→+∞} ∫_{t}^{δ_{i_{0}} (t)} p_{i_{0}}(s)ds > 0, vrai sup\{p_{i_{0}}(t) : t ∈ ℝ_{+}\} < +∞. (11.37)

Then the condition

inf \{ \underline{lim}_{t→+∞}e^{−λt} ∫_{0}^{t}(t − s)^{n−1} ×

× ∑_{i=1}^{m} p_{i}(s)e^{λδ_{i}(s)}ds : λ ∈]0, +∞[} > (n − 1)!  (11.38)

is sufficient for problem (0.1), (10.1) to have no solution. Proof. It suffices to note that by (11.34)–(11.38) all the coditions of Theorem 11.3 are satisfied with τ_{i}(t) = δ_{i}(t) − 1, σ_{i}(t) = δ_{i}(t), r_{i}(s, t) = p_{i}(t)e(s − δ_{i}(t)) (i = 1, …, m) (the definition of the function e(t) see on page 104). ■

Corollary 11.4. Let (11.34)–(11.37) be fulfilled and for some t_{0} ∈ ℝ_{+}

inf {λ^{−n} vrai inf_{t≥t_{0}} ∑_{i=1}^{m} p_{i}(t)e^{λ(δ_{i}(t)−t)} : λ ∈]0, +∞[} > 1. (11.39)

Then problem (0.1), (10.1) has no solution. Proof. It suffices to note that (11.39) implies (11.38). ■

Corollary 11.5. Let δ_{i}(t) ≥ t for t ∈ ℝ_{+} (i = 1, …, m), (11.34)–(11.37) be fulfilled and for some t_{0} ∈ ℝ_{+}

vrai inf_{t≥t_{0}} \{ ∑_{i=1}^{m} p_{i}(t)(δ_{i}(t) − t)^{n}\} > (\frac{n}{e})^{n}.

Then problem (0.1), (10.1) has no solution.

Corollary 11.6. Let F ∈ V (τ) and for some t_{0} ∈ ℝ_{+}

F (u)(t) sign u(t) ≤ − ∑_{i=1}^{m} c_{i}|u(t − Δ_{i})| for u ∈ H_{t_{0},τ}^{+} , t ≥ t_{0},

where c_{i} > 0, Δ_{i} ∈ ℝ (i = 1, …, m) and Δ_{i_{0}} < 0 for some i_{0} ∈ {1, …, m}. Then the condition

inf {λ^{−n} ∑_{i=1}^{m} c_{i}e^{−λΔ_{i}} : λ ∈]0, +∞[} > 1  (11.40)

is sufficient for problem (0.1), (10.1) to have no solution.

Corollary 11.7. Let c_{i} > 0, Δ_{i} ∈ ℝ (i = 1, …, m) and Δ_{i_{0}} < 0 for some i_{0} ∈ {1, …, m}. Then (11.40) is necessary and sufficient for the equation

u^{(n)}(t) = ∑_{i=1}^{m} c_{i}u(t − Δ_{i})

to have no solution satisfying (10.1).

Theorem 11.5. Let F ∈ V (τ), conditions (11.12), (11.13), (11.15) and (11.16) be fulfilled with φ ∈ M^{+}(σ; \overline{σ}) and

lim_{t→+∞} \frac{\overline{σ}(t)}{σ(t)} < +∞.  (11.41)

Let, moreover, for some k ∈ {0, …, n−1} (11.22_{k}), (11.47_{k}) hold with ρ_{k}^{∗}(t) defined by (11.5_{k}) and (11.6_{k}). Then the condition

inf \{ \underline{lim}_{t→+∞}t^{−λ}∫_{0}^{t}(t−s)^{n−1}φ(θ)(s)ds : λ∈[n−1, +∞[\}>(n − 1)!, (11.42)

with θ(t) = t^{λ} is sufficient for problem (0.1), (10.1) to have no solution.

Proof. Suppose, on the contrary, that (0.1) has a solution u_{0} : [t_{0}, +∞[→ ℝ satisfying (10.1). According to (11.12) and Lemma 10.7 (11.2) has a solution u : [t_{∗}, +∞[→]0, +∞[ satisfying (11.3). By (10.22_{k}), (10.47_{k}), (11.13) , (11.15) and Lemma 10.10 there exists λ > 0 such that lim_{t→+∞} t^{−λ}u(t) = 0.

Denote by Λ the set af all λ satisfying lim_{t→+∞} t^{−λ}u(t) = +∞ and put λ_{0} =

sup Λ. In view of (11.16) it is obvious that n − 1 ∈ Λ and λ_{0} ∈ [n − 1, +∞[. By (11.42) there exist ε > 0 and t_{1} ∈ ℝ_{+} such that

t^{−λ} ∫_{0}^{t}(t − s)^{n−1}φ(θ)(s)ds ≥ (n − 1)! + ε  (11.43)
for t ≥ t_{1}, λ ∈]λ_{0}, λ_{0} + ε[.

Choose ε_{2} ∈]0, ε[ and ε_{1} ∈ [0, ε_{2}[ such that

\begin{matrix}λ_{0} − ε_{1} ≥ 0, c^{ε_{2}+ε_{1}}(n − 1)! < (n − 1)! + ε, \\ lim_{t→+∞} t^{−(λ_{0}+ε_{1})}u(t) = +∞, \underline{lim}_{t→+∞}t^{−(λ_{0}+ε_{2})}u(t) = 0 \end{matrix} (11.44)

with c = \overline{lim}_{t→+∞} \frac{\overline{σ}(t)}{σ(t)}. According to (11.44) u is a solution of (11.2) satisfying

all the conditions of Lemma 11.1 with γ(t) = t^{−1}, r_{1} = λ_{0} − ε_{1} and r_{2} = λ_{0} + ε_{2}. Therefore we have

\underline{lim}_{t→+∞}t^{−(λ_{0}+ε_{2})} ∫_{0}^{t}(t − s)^{n−1}φ(θ)(s)ds < c^{ε_{2}+ε_{1}}(n − 1)! < (n − 1)! + ε

with θ(t) = t^{λ_{0}+ε_{2}}. But this contradicts (11.43). The obtained contradiction proves the theorem. ■

Theorem 11.5′. Let F ∈ V (τ) and (11.12), (11.13), (11.15), (11.16), (11.41), (10.17_{0}) and (10.18_{0}) be fulfilled with φ ∈ M^{+}(σ; \overline{σ}). Then (11.42) is sufficient for problem (0.1), (10.1) to have no solution.

Proof. The validity of the theorem follows from Theorem 11.5 and Corollary 10.3. ■

Theorem 11.6. Let F ∈ V (τ) and (11.12), (11.21), (11.41), (10.17_{0}), (10.18_{0}), (10.50) and (10.51) be fulfilled with φ ∈ M^{+}(σ; \overline{σ}). Then (11.42) is sufficient for problem (0.1), (10.1) to have no solution.

Proof. The validity of the theorem follows from Theorem 11.5 and Lemma 10.11. ■

Theorem 11.7. Let F ∈ V (τ), conditions (11.21) and (11.22) be fulfilled and

\overline{lim}_{t→+∞} \frac{\overline{σ}(t)}{σ(t)} < +∞, \overline{lim}_{t→+∞} \frac{\overline{σ}(t)}{t} < +∞,  (11.45)

where \overline{σ}(t) = max\{σ_{i}(t) : i = 1, …, m\}, σ(t) = inf\{min(τ_{i}(s) : i = 1, …, m) : s ≥ t},

\overline{lim}_{t→+∞} \frac{1}{ln t} ∫_{0}^{t} s^{n−1} ∑_{i=1}^{m} [r_{i}(σ_{i}(s), s) − r_{i}(τ_{i}(s), s)]ds < +∞. (11.46)

Let, moreover, there exist i_{0} ∈ {1, …, m} and a nondecreasing function δ ∈ C(ℝ_{+}; ℝ_{+}) such that (11.24) holds,

\underline{lim}_{t→+∞} ∫_{t}^{δ(t)} s^{n−1}[r_{i_{0}}(σ_{i_{0}}(s), s) − r_{i_{0}}(δ(s), s)]ds > 0  (11.47)

and

vrai sup {t^{n}[r_{i_{0}}(σ_{i_{0}}(t), t) − r_{i_{0}}(δ(t), t)] : t ∈ ℝ_{+}} < +∞. (11.48)

Then the condition

inf \{ \underline{lim}_{t→+∞}t^{−λ} ∫_{0}^{t}(t − s)^{n−1} ∑_{i=1}^{m} ∫_{τ_{i}(s)}^{σ_{i}(s)} ξ^{λ}d_{ξ}r_{i}(ξ, s)ds :

: λ ∈]n − 1, +∞[} > (n − 1)!  (11.49)

is sufficient for problem (0.1), (10.1) to have no solution.

Proof. As in the proof of Theorem 11.3, we can easily show that the operator defined by (11.29) satisfies all the conditions of Theorem 11.6. This proves the theorem. ■

Corollary 11.8. Let F ∈ V (τ), conditions (11.22), (11.45)–(11.48) be fulfilled, φ ∈ M^{+}(σ, \overline{σ}) and for some t_{0} ∈ ℝ_{+}

inf \{\frac{1}{∏_{i=0}^{n−1}(λ − i)} vrai inf_{t≥t_{0}} (t^{n−λ} ∑_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)} ξ^{λ}d_{ξ}r_{i}(ξ, t)) :

: λ ∈]n − 1, +∞[} > 1.  (11.50)

Then problem (0.1), (10.1) has no solution.

Proof. It suffices to note that (11.50) implies (11.49). ■

Corollary 11.9. Let F ∈ V (τ) and for some t_{0} ∈ ℝ_{+}

F (u)(t)signu(t)≤−t^{−n−1}∑_{i=1}^{m} c_{i}∫_{α_{i}t}^{\overline{α}_{i}t} |u(s)|ds for u ∈ H_{t_{0},τ}^{+} , t ≥ t_{0}

with c_{i} > 0, α_{i}, \overline{α}_{i} ∈]0, +∞[, α_{i} < \overline{α_{i}} (i = 1, …, m). Let, moreover, i_{0} ∈ \{1, …, m\} exist such that \overline{α}_{i_{0}} > 1. Then the condition

inf\{\frac{1}{∏_{i=−1}^{n−1}(λ − i)}∑_{i=1}^{m}c_{i}(\overline{α}_{i} ^{λ+1} − α_{i}^{λ+1}) : λ ∈]n − 1, +∞[\}>1 (11.51)

is sufficient for problem (0.1), (10.1) to have no solution.

Corollary 11.9′. Let c_{i} > 0, α_{i}, \overline{α}_{i} ∈]0, +∞[, α_{i} < \overline{α}_{i} (i = 1, …, m) and \overline{α}_{i_{0}} > 1 for some i_{0} ∈ \{1, …, m\}. Then (10.50) is necessary and sufficient for the equation

u^{(n)} = t^{−n−1} ∑_{i=1}^{m} c_{i} ∫_{α_{i}t}^{\overline{α}_{i}t} u(s)ds  (11.52)

to have no solution satisfying (10.1).

Proof. Sufficiency follows from Corollary 11.9. If we suppose that (11.50) is violated, then (11.52) has the solution u(t) = t^{λ} with λ > n − 1. ■

Theorem 11.8. Let F ∈ V (τ), conditions (11.34) and (11.35) be fulfilled and

\overline{lim}_{t→+∞} \frac{\overline{σ}(t)}{σ(t)} < +∞, \overline{lim}_{t→+∞} \frac{σ(t)}{t} < +∞  (11.53)

with \overline{σ}(t) = max\{δ_{i}(t) : i = 1, …, m\}, σ(t) = inf\{min(δ_{i}(s) : i = 1, …, m) : s ≥ t}. Let, moreover, i_{0} ∈ {1, …, m} exist such that δ_{i_{0}}(t) is nondecreasing,

\underline{lim}_{t→+∞} ∫_{t}^{δ_{i_{0}} (t)} s^{n−1}p_{i_{0}}(s)ds > 0,  (11.54)
vrai sup{t^{n}p_{i_{0}}(t) : t ∈ ℝ_{+}} < +∞.  (11.55)

Then the condition

inf \{ \underline{lim}_{t→+∞}t^{−λ} ∫_{0}^{t}(t − s)^{n−1} ∑_{i=1}^{m} p_{i}(s)δ_{i}^{λ}(s)ds :

: λ ∈]n − 1, +∞[} > (n − 1)!  (11.56)

is sufficient for problem (0.1), (10.1) to have no solution.

Proof. It suffices to note that by (11.34), (11.35) and (11.53)-(11.56) all the conditions of Theorem 11.6 are satisfied with τ_{i}(t) = δ_{i}(t) − 1, σ_{i}(t) = δ_{i}(t) and r_{i}(s, t) = p_{i}(t)e(s − δ_{i}(t)) (the definition of the function e see on page 104). ■

Corollary 11.10. Let F ∈ V (τ), conditions (11.34), (11.35) and (11.53)– (11.56) be fulfilled and for some t_{0} ∈ ℝ_{+}

inf \{\frac{1}{∏_{i=0}^{n−1}(λ − i)} vrai inf_{t≥t_{0}} (t^{n−λ} ∑_{i=1}^{m} p_{i}(t)δ_{i}^{λ}(t)) :

: λ ∈]n − 1, +∞[} > 1.  (11.57)

Then problem (0.1), (10.1) has no solution.

Proof. It suffices to note that (11.57) implies (11.56). ■

Corollary 11.11. Let F ∈ V (τ), for some t_{0} ∈ ℝ_{+}

F (u)(t) sign u(t) ≤ −t^{−n} ∑_{i=1}^{m} c_{i}|u(α_{i}t)| for u ∈ H_{t_{0},τ}^{+} , t ≥ t_{0}

with c_{i} > 0, α_{i} > 0 (i = 1, …, m) and let α_{i_{0}} > 1 for some i_{0} ∈ {1, …, m}. Then the condition

inf \{\frac{1}{∏_{i=0}^{n−1}(λ − i)} ∑_{i=1}^{m} c_{i}α_{i}^{λ} : λ ∈]n − 1, +∞[\} > 1  (11.58)

is sufficient for problem (0.1), (10.1) to have no solution.

Corollary 11.11′. Let c_{i} > 0, α_{i} > 0 (i = 1, …, m) and α_{i_{0}} > 1 for some i_{0} ∈ {1, …, m}. Then (11.58) is necessary and sufficient for the equation

u^{(n)}(t) = t^{−n} ∑_{i=1}^{m} c_{i}u(α_{i}t)

to have no solution satisfying (10.1).

11.2. Differential Inequalities with Deviating Arguments.

Theorem 11.9. Let n > 1, F ∈ V (τ), for any t_{0} ∈ ℝ_{+}

F (u)(t) sign u(t)≤−p(t)|u(δ(t))| for u ∈ H_{t_{0},τ}^{+} , t ≥ t_{0}, (11.59)

(10.2) be fulfilled and for some k ∈ {1, …, n − 1} (10.17_{k}), (10.18_{k}) hold. Then problem (0.1), (10.1) has no solution.

Proof. Suppose that (0.1) has a solution u satisfying (10.1). Then by (11.58), (11.17_{k}), (10.18_{k}) and Lemma 10.3

\overline{lim}_{t→+∞} \frac{t^{2k+1−n}|u(δ(t))|}{|u^{(n−1)}(t)|} < +∞  (11.60)

and

\underline{lim}_{t→+∞} \frac{δ(t)}{t} > 1.  (11.61)

On the other hand, by (10.1) from (1.6_{0n}) we have

|u(δ(t))| ≥ \frac{1}{(n − 1)!}(δ(t) − t)^{n−1}|u^{(n−1)}(t)| for t ≥ t_{1},

where t_{1} > t_{0} is sufficiently large. Hence in view of (11.61) we obtain

\underline{lim}_{t→+∞} \frac{|u(δ(t))|t^{1−n}}{|u^{(n−1)}(t)|} > 0.

Since k ≥ 1, the last inequality contradicts (11.60). The obtained contradiction proves the theorem. ■

Analogously can be proved

Theorem 11.10. Let F ∈ V (τ), conditions (11.59), (10.17_{0}) and (10.18_{0}) be fulfilled and

\overline{lim}_{t→+∞} \frac{δ(t)}{t} = +∞.

Then problem (0.1), (10.1) has no solution.

Theorem 11.11. Let F ∈ V (τ), conditions (10.2) and (11.59) be fulfilled and for some k ∈ ℕ and i ∈ {0, …, n − 1}

\overline{lim}_{t→+∞} ∫_{t}^{δ(t)} p(s)(δ(t) − s)^{n−i−1}(δ(s) − δ(t))^{i}g_{k}(δ(s), δ(t))ds >

> i!(n − i − 1)!,  (11.62)

where

g_{k}(t, s) = exp \{\frac{1}{(n − 1)!} ∫_{s}^{t}(t − ξ)^{n−1}p(ξ)ψ_{k}(ξ)dξ\},  (11.63)

ψ_{1}(t) = 1, ψ_{j}(t) = exp \{\frac{1}{(n − 1)!} ∫_{t}^{δ(t)}(δ(t) − ξ)^{n−1} ×

×p(ξ)ψ_{j−1}(ξ)dξ} (j = 2, …, k).  (11.64)

Then problem (0.1), (10.1) has no solution.

Proof. Suppose that (0.1) has a solution u : [t_{0}, +∞[→ ℝ satisfying (10.1). From (0.1) by (11.59) we have

\frac{1}{(n − 1)!} ∫_{s}^{δ(t)} \frac{(δ(t) − ξ)^{n−1}|u^{(n)}(ξ)|}{u_{0}(δ(t), ξ)}dξ ≥ \frac{1}{(n − 1)!} ×

× ∫_{s}^{δ(t)}(δ(t) − ξ)^{n−1}p(ξ)\frac{|u(δ(ξ))|}{u_{0}(δ(t), ξ)}dξ for t_{0} ≤ s ≤ δ(t), (11.65)

where

u_{0}(t, s) = ∑_{j=0}^{n−1} \frac{(t − s)^{j}}{j!} |u^{(j)}(s)|.

Since

\frac{du_{0}(δ(t), ξ)}{dξ} = \frac{(δ(t) − ξ)^{n−1}}{(n − 1)!} |u^{(n)}(ξ)|, u_{0}(δ(t), δ(t)) = |u(δ(t))|,

in view of (11.65) we obtain

ln \frac{|u(δ(t))|}{u_{0}(δ(t), s)} ≥ \frac{1}{(n − 1)!} ∫_{s}^{δ(t)}(δ(t) − ξ)^{n−1}p(ξ)\frac{|u(δ(ξ))|}{u_{0}(δ(t), ξ)}dξ.

As u_{0}(δ(ξ), ξ) ≥ u_{0}(δ(t), ξ) for t_{0} ≤ t ≤ ξ ≤ δ(t), the last inequality implies

\frac{|u(δ(t))|}{u_{0}(δ(t), s)} ≥ exp \{\frac{1}{(n − 1)!} ∫_{s}^{δ(t)}(δ(t) − ξ)^{n−1}p(ξ)\frac{|u(δ(ξ))|}{u_{0}(δ(ξ), ξ)}dξ\}

for t_{0} ≤ t ≤ s ≤ δ(t),

whence we easily conclude that

|u(δ(t))| ≥ exp \{\frac{1}{(n − 1)!} ∫_{s}^{δ(t)}(δ(t) − ξ)^{n−1}p(ξ)ψ_{k}(ξ)dξ\}u_{0}(δ(t), s)

for t_{0} ≤ t ≤ s ≤ δ(t)

with ψ_{k}(t) defined by (11.64). Therefore according to (10.1) and (11.59) from (0.1) we have

|u^{(i)}(δ(t))| ≥ \frac{1}{(n − i − 1)!} ∫_{t}^{δ(t)}(δ(t) − ξ)^{n−i−1}p(ξ) ×

×g_{k}(δ(ξ), δ(t))u_{0}(δ(ξ), δ(t))dξ  (11.66)

with g_{k}(t, s) defined by (11.63). Since

u_{0}(δ(ξ), δ(t)) ≥ \frac{|u^{(i)}(δ(t))|}{i!}(δ(ξ) − δ(t))^{i} for t_{0} ≤ t ≤ ξ,

from (11.66) we finally obtain

|u^{(i)}(δ(t))| ≥ \frac{1}{i!(n − i − 1)!} ∫_{t}^{δ(t)}(δ(t) − ξ)^{n−i−1} ×

×(δ(ξ) − δ(t))^{i}g_{k}(δ(ξ), δ(t))dξ|u^{(i)}(δ(t))|

which contradicts (11.62). The obtained contradiction proves the theorem. ■

Corollary 11.12. Let F ∈ V (τ), conditions (10.2) and (11.59) be fulfilled and for some i ∈ {0, …, n − 1}

\overline{lim}_{t→+∞} ∫_{t}^{δ(t)} p(s)(δ(t) − s)^{n−i−1}(δ(s) − δ(t))^{i}ds > i!(n − i − 1)!. (11.67)

Then problem (0.1), (10.1) has no solution.

Remark 11.1. (11.67) cannot be replaced by

\overline{lim}_{t→+∞} ∫_{t}^{δ(t)} p(s)(δ(t) − s)^{n−i−1}(δ(s) − δ(t))^{i}ds >

> i!(n − i − 1)!(1 − ε)^{n},  (11.68)

however small ε ∈]0, 1] would be.

Indeed, let ε ∈]0, 1]. Choose n_{0} ∈ ℕ, Δ > 0 and c > 0 such that

\sqrt[n]{n!} < \frac{(1 + ε)n}{e} for n ≥ n_{0}, cΔ^{n} = (1 − ε)^{n}n!.

Then the equation

u^{(n)}(t) = cu(t + Δ)

has the solution u(t) = e^{λt} with λ > 0. On the other hand, (11.68) holds with p(t) = c and δ(t) = t + Δ.

Theorem 11.12. Let F ∈ V (τ), conditions (10.2) and (11.59) be fulfilled and

\underline{lim}_{t→+∞} ∫_{t}^{δ(t)} p(s)(δ(t) − s)^{n−1}ds > \frac{(n − 1)!}{e}.  (11.69)

Then problem (0.1), (10.1) has no solution.

This theorem can be prooved analogously to Theorem 6.2.

11.3. Nonlinear Equations.

Theorem 11.13. Let F ∈ V (τ) and for some t_{0} ∈ ℝ_{+}

F (u)(t) sign u(t)≤−φ(u)(t) sign u(t)≤0 for u ∈ H_{t_{0},τ}^{+} , t≥t_{0}, (11.70)

where φ ∈ M(σ), σ(t) ≥ t for t ∈ ℝ_{+}. Then the condition

φ̃_{n} ∈ M_{2}^{β}(σ)  (11.71)

with

β(t) = t, φ̃_{n}(u)(t) = \frac{1}{(n − 1)!}φ((σ(t) − t)^{n−1}u(t))(t) (11.72)

is sufficient for problem (0.1), (10.1) to have no solution. Proof. Suppose that (0.1) has a solution u : [t_{0}, +∞[→ ℝ satisfying (10.1). Then from (1.6_{0n}) we have

|u(σ(t))| ≥ \frac{|u^{(n−1)}(t)|}{(n − 1)!}(σ(t) − t)^{n−1} for t ≥ t_{0}.

Therefore, taking into account the fact that φ ∈ M(σ), by (11.70) we obtain

x(t) ≥ ∫_{t_{0}}^{t} |φ̃_{n}(sign u(t_{0})x)(s)|ds,

where x(t) = \frac{1}{(n−1)!}|u^{(n−1)}(t)| > 0 for t ≥ t_{0}. But this contradicts (11.71), which proves the validity of the theorem. ■

Analogously can be proved

Theorem 11.12′. Let F ∈ V (τ) and for some t_{0} ∈ ℝ_{+}

F (u)(t) sign u(t) ≤ −φ(t, u(δ_{1}(t)), …, u(δ_{m}(t))) sign u(t) ≤ 0

for u ∈ H_{t_{0},τ}^{+} , t ≥ t_{0},

where δ_{i} ∈ C(ℝ_{+}; ℝ_{+}), δ_{i}(t) ≥ t for t ∈ ℝ_{+} (i = 1, …, m), φ ∈ K_{loc}(ℝ_{+} × ℝ^{m}; ℝ) and

φ(t, x_{1}, …, x_{m}) sign x_{1} ≥ φ(t, y_{1} …, y_{m}) sign y_{1} ≥ 0
for t ∈ ℝ_{+}, x_{i}x_{1} > 0, x_{i}y_{i} > 0, |x_{i}| ≥ |y_{i}| (i = 1, …, m).

Then condition (11.71) with

φ̃_{n}(u)(t) = \frac{1}{(n − 1)!}φ(t, (δ_{1}(t) − t)^{n−1}u(t), …, (δ_{m}(t) − t)^{n−1}u(t))

is sufficient for problem (0.1), (10.1) to have no solution.

Theorem 11.14. Let F ∈ V (τ), (11.70) be fulfilled with φ ∈ M(σ), the function σ ∈ C(ℝ_{+}; ℝ_{+}) be nondecreasing and σ(t) ≥ t for t ∈ ℝ_{+}. Then condition (11.71) with

φ̃_{n}(u)(t)= \frac{(σ(t) − t)^{n−1}}{(n − 1)!} |φ(u(t))(t)|  (11.73)

is sufficient for problem (0.1), (10.1) to have no solution.

Proof. Suppose that (0.1) has a solution u : [t_{0}, +∞[→ ℝ satisfying (10.1). Then by (10.1), (11.70) and the fact that φ ∈ M(σ) we obtain

|u(σ(t))| ≥ \frac{1}{(n − 1)!} ∫_{t_{0}}^{σ(t)}(σ(t) − s)^{n−1}|u^{(n)}(s)|ds ≥

≥ \frac{1}{(n − 1)!} ∫_{t_{0}}^{t}(σ(s) − s)^{n−1}|φ( sign u(t_{0})||u(σ(s))|)ds for t ≥ t_{0}.

Therefore

x(t) ≥ ∫_{t_{0}}^{t} |φ̃_{n}( sign u(t_{0})x)(s)|ds for t ≥ t_{0}

with x(t) = |u(σ(t))|. But this contradicts (11.71). The obtained contradiction proves the theorem. ■

CHAPTER 5

SPECIFIC PROPERTIES OF FUNCTIONAL

DIFFERENTIAL EQUATIONS

In chapter 5 the oscillatory properties which are specific for functional differential equations are studied. These properties have no analogues for ordinary differential equations.

§ 12. Equations with property Ã

Everywhere in this section it will be assumed that the inequality

(−1)^{n+1}F (u)(t)u(t) ≥ 0 for u ∈ H_{t_{0},τ} , t ≥ t_{0}  (12.1)

holds for some t_{0} ∈ ℝ_{+}.

12.1. Nonlinear Equations.

Theorem 12.1. Let F ∈ V (τ), (12.1) be fulfilled for some t_{0} ∈ ℝ_{+} and

|F (u)(t)| ≥ |φ(u)(t)| for u ∈ H_{t_{0},τ} , t ≥ t_{0}  (12.2)

with φ ∈ M(τ, σ), where σ(t) ≤ t for t ∈ ℝ_{+}. Then, if n is odd, the conditions (3.3_{n−1}) and

φ̃_{0} ∈ M_{1}(τ, σ)  (12.3)

are sufficient for (0.1) to have property Ã, and if n is even, such are conditions (3.11), (3.3_{n−2}) and (12.3), where φ̃_{n−1}, φ̃_{n−2} and φ̃_{0} are defined, respectively, by (3.12), (3.14) and (9.86).

Proof. Suppose that u : [t_{0}, +∞[→ ℝ is a nonoscillatory proper solution of (0.1). According to Lemma 1.2 and (12.1) l ∈ {0, …, n} exists such that l is even and (2.14_{l}) holds. By Theorems 3.1′, 3.2′ and 9.15 and conditions (3.2_{n−1}), (3.2_{n−2}) and (12.3) we have l ∉ {0, …, n − 1}. Therefore, n is even and l = n, so in view of (3.13) (0.5) is fulfilled. ■

Taking into account Theorems 3.1′′, 3.2′′ and 9.15, we can analoguosly prove the following

Theorem 12.1′. Let F ∈ V (τ) and conditions (12.1) and (12.2) be fulfilled with φ ∈ M(τ, σ), where σ(t) ≤ t for t ∈ ℝ_{+}. Then, if n is odd, conditions (3.16), (12.3) are sufficient for (0.1) to have property Ã, while if n is even, such are conditions (3.11), (3.18) and (12.3), where φ̃_{n−1}, φ̃_{n−2} and φ̃_{0} are defined, respectively, by (3.17), (3.18) and (9.15).

Theorem 12.2. Let F ∈ V (τ), (12.1) be fulfilled for some t_{0} and

|F (u)(t)| ≥ ∫_{τ_{1}(t)}^{σ_{1}(t)} |u(s)|^{λ}d_{s}r(s, t) for u ∈ H_{t_{0},τ} , t ≥ t_{0}, (12.4)

where

0 < λ < 1, τ_{1}, σ_{1} ∈ C(ℝ_{+}; ℝ_{+}), τ_{1}(t) ≤ σ_{1}(t) ≤ t for t ∈ ℝ_{+}, lim_{t→+∞}τ_{1}(t)=+∞, r(s, t) is measurable, r(·, t) is nondecreasing.

(12.5)

Let, moreover,

∫ ^{+∞} ∫_{τ_{1}(t)}^{σ_{1}(t)} s^{λ(n−1)}d_{s}r(s, t)dt = +∞

and

∫ ^{+∞} t^{(n−1)(1−λ)}(t − σ_{1}(t))^{λ(n−1)}(r(σ_{1}(t), t) − r(τ_{1}(t), t))dt = +∞.

Then (0.1) has the property Ã.

Theorem 12.2′. Let F ∈ V (τ), conditions (12.1) and (12.5) be fulfilled for some t_{0} ∈ ℝ_{+} and

∫_{τ_{1}(t)}^{σ_{1}(t)} |u(s)|^{λ}d_{s}r(s, t) ≤ |F (u)(t)| ≤ δ ∫_{τ_{1}(t)}^{σ_{1}(t)} |u(s)|^{λ}d_{s}r(s, t)
for u ∈ H_{t_{0},τ} , t ≥ t_{0},

with 0 < λ < 1 and δ ∈ [1, +∞[. Let, moreover,

\overline{lim}_{t→+∞} \frac{σ_{1}(t)}{τ_{1}(t)} < +∞, \underline{lim}_{t→+∞} \frac{t^{1−λ}(t − σ_{1}(t))^{λ}}{(τ_{1}(t))^{λ}} > 0.

Then the condition ∫ ^{+∞} τ_{1}^{λ(n−1)}(t)(r(σ_{1}(t), t) − r(τ_{1}(t), t))dt = +∞

is necessary and sufficient for (0.1) to have property Ã.

The validity of Theorems 12.2 and 12.2′ follows from Corollaries 4.1 and 4.2, 9.16. 12.2. Equations With a Linear Minorant.

Theorem 12.3. Let F ∈ V (τ), conditions (12.1), (8.2) and (8.3) be fulfilled for some t_{0} and

|F (u)(t)| ≥ p(t)|u(δ(t))| for u ∈ H_{t_{0},τ} , t ≥ t_{0}.  (12.6)

Let, moreover,

∫ ^{+∞}(δ(t))^{n−1−ε}p(t)dt = +∞  (12.7)

with ε > 0 and either (9.73) or (9.75) hold. Then (0.1) has property Ã.

Proof. The validity of the theorem follows from Corollaries 3.12 and 9.16 and Theorem 9.12. ■

Theorem 12.4. Let F ∈ V (τ), condition (12.1) be fulfilled for some t_{0} ∈ ℝ_{+} and

|F (u)(t)| ≥ ∑_{i=1}^{m} c_{i} ∫_{t−Δ_{i}}^{t−\overline{Δ}_{i}} |u(s)|ds for u ∈ H_{t_{0},τ} , t ≥ t_{0}, (12.8)

where c_{i} > 0, Δ_{i}, \overline{Δ}_{i} ∈ ℝ, \overline{Δ}_{i} < Δ_{i} (i = 1, …, m) and Δ_{i_{0}} > 0 for some i_{0} ∈ {1, …, m}. Then (9.28) is sufficient for (0.1) to have the property Ã.

Proof. The validity of the theorem immediately follows from Corollaries 3.1 and 9.2. ■

Theorem 12.4′. Let c_{i} > 0, Δ_{i}, \overline{Δ}_{i} ∈ ℝ, \overline{Δ}_{i} < Δ_{i} (i = 1, …, m) and Δ_{i_{0}} > 0 for some i_{0} ∈ {1, …, m}. Then (9.28) is necessary and sufficient for (9.29) to have property Ã.

Theorem 12.5. Let F ∈ V (τ), condition (12.1) be fulfilled and

|F (u)(t)| ≥ ∑_{i=1}^{m} c_{i}|u(t − Δ_{i})| for u ∈ H_{t_{0},τ} , t ≥ t_{0},  (12.9)

where c_{i} > 0, Δ_{i} ∈ ℝ (i = 1, …, m) and Δ_{i_{0}} > 0 for some i_{0} ∈ {1, …, m}. Then (9.37) is sufficient for (0.1) to have property Ã.

Proof. The thoerem follows from Corollaries 3.1 and 9.4. ■

Theorem 12.6. Let F ∈ V (τ), condition (12.1) be fulfilled and

|F (u)(t)| ≥ t^{−1−n} ∑_{i=1}^{m} c_{i} ∫_{α_{i}t}^{\overline{α}_{i}t} |u(s)|ds for u ∈ H_{t_{0},τ} , t ≥ t_{0}, (12.10)

where c_{i} > 0, α_{i}, \overline{α}_{i} ∈]0, +∞[, α_{i} < \overline{α}_{i} (i = 1, …, m) and α_{i_{0}} < 1 for some i_{0} ∈ {1, …, m}. Let, moreover, (9.49) hold and

∑_{i=1}^{m} c_{i}(\overline{α} _{i}^{λ+1} − α_{i}^{λ+1}) > − ∏_{i=−1}^{n−1}(i − λ) for λ ∈ [1, n − 1].^{18} (12.11)

Then (0.1) has property Ã.

Proof. The theorem follows from Corollary 9.7 and Theorems 7.4 and 7.5. ■

18When n = 1, 2, (12.11) is unnecessary.

Corollary 12.1. Let F ∈ V (τ) and (12.1) and (12.10) be fulfilled with c_{i} > 0, α_{i}, \overline{α}_{i} ∈]0, 1], α_{i} < \overline{α}_{i} (i = 1, …, m). Then the conditions (9.49) and

∑_{i=1}^{m} c_{i}(\overline{α} _{i}^{λ+1} − α_{i}^{λ+1}) > − ∏_{i=−1}^{n−1}(i − λ) for λ ∈ [n − 3, n − 1] (12.12)

are sufficient for (0.1) to have property Ã.

Theorem 12.6′. Let c_{i} > 0, α_{i}, \overline{α}_{i} ∈]0, +∞[, α_{i} < \overline{α}_{i} (i = 1, …, m) and α_{i_{0}} < 1 for some i_{0} ∈ {1, …, m}. Then (9.49) and (12.11) are necessary and sufficient for the equation

u^{(n)}(t) + (−1)^{n+1} ∑_{i=1}^{m} c_{i}t^{−n−1} ∫_{α_{i}t}^{\overline{α}_{i}t} u(s)ds = 0  (12.13)

to have property Ã.

Proof. The sufficiency follows from Theorem 12.6. If we assume that either (9.49) or (12.11) is violated, then equation (12.13) has the solution u(t) = t^{λ}, where either λ < 0 or λ ∈]1, n − 1[. ■

Corollary 12.6′. Let c_{i} > 0, α_{i}, \overline{α}_{i} ∈]0, 1] and α_{i} < \overline{α}_{i} (i = 1, …, m). Then (9.49) and (12.12) are necessary and sufficient for equation (12.13) to have property Ã.

Theorem 12.7. Let F ∈ V (τ), condition (12.1) be fulfilled for some t_{0} ∈ ℝ_{+} and

|F (u)(t)| ≥ ∑_{i=1}^{m} c_{i} ∫_{α_{i}t}^{\overline{α}_{i}t} s^{−n−1}|u(s)|ds for u ∈ H_{t_{0},τ} , t ≥ t_{0}, (12.14)

where c_{i} > 0, α_{i}, \overline{α}_{i} ∈]0, ∞[, α_{i} < \overline{α}_{i} (i = 1, …, m) and α_{i_{0}} < 1 for some i_{0} ∈ {1, …, m}. Then conditions (9.46) and

∑_{i=1}^{m} c_{i}(α_{i}^{λ−n} − \overline{α}_{i}^{λ−n}) > ∏_{i=0}^{n}(i − λ) for λ ∈ [1, n − 1] (12.15)

are sufficient for (0.1) to have property Ã.

Proof. The theorem follows from Corollary 9.6 and Theorems 7.4 and 7.5. ■

Corollary 12.2. Let (12.1) and (12.14) be fulfilled with c_{i} > 0, α_{i}, \overline{α}_{i} ∈ ]0, 1] and α_{i} < \overline{α}_{i} (i = 1, …, m). Then conditions (9.46) and

∑_{i=1}^{m} c_{i}(α_{i}^{λ−n} − \overline{α}_{i}^{λ−n}) > ∏_{i=0}^{n}(i − λ) for λ ∈ [n − 3, n − 1] (12.16)

are sufficient for (0.1) to have property Ã.

Theorem 12.7′. Let c_{i} > 0, α_{i}, \overline{α}_{i} ∈]0, +∞[, α_{i} < \overline{α}_{i} (i = 1, …, m) and α_{i_{0}} < 1 for some i_{0} ∈ {1, …, m}. Then (9.46) and (12.15) are necessary and sufficient for (9.47) to have property Ã.

Corolary 12.2′. Let c_{i} > 0, α_{i}, \overline{α}_{i} ∈]0, 1] and α_{i} < \overline{α}_{i} (i = 1, …, m). Then (9.46) and (12.16) are necessary and sufficient for (9.47) to have property Ã.

Theorem 12.8. Let F ∈ V (τ), condition (12.1) be fulfilled for some t_{0} ∈ ℝ_{+} and

|F (u)(t)| ≥ t^{−n} ∑_{i=1}^{m} c_{i}|u(α_{i}t)| for u ∈ H_{t_{0},τ} , t ≥ t_{0}, (12.17)

where c_{i} > 0, α_{i} > 0 (i = 1, …, m) and α_{i_{0}} < 1 for some i_{0} ∈ {1, …, m}. Then conditions (9.53) and

∑_{i=1}^{m} c_{i}α_{i}^{λ} > ∏_{i=0}^{n−1}(i − λ) for λ ∈ [1, n − 1]  (12.18)

are sufficient for (0.1) to have property Ã.

Proof. The theorem follows from Corollaries 7.1, 7.5 and 9.9. ■

Corollary 12.3. Let (12.1) and (12.17) be fulfilled, where c_{i} > 0, α_{i} ∈]0, 1] (i = 1, …, m) and α_{i_{0}} < 1 for some i_{0} ∈ {1, ….m}. Then conditions (9.53) and

∑_{i=1}^{m} c_{i}α_{i}^{λ} > ∏_{i=0}^{n−1}(i − λ) for λ ∈ [n − 3, n − 1]  (12.19)

are sufficient for (0.1) to have property Ã.

Theorem 12.8′. Let c_{i} > 0, α_{i} > 0 (i = 1, …, m) and α_{i_{0}} < 1 for some i_{0} ∈ {1, …, m}. Then (9.53) and (12.18) are necessary and sufficient for the equation

u^{(n)}(t) + (−1)^{n+1}t^{−n} ∑_{i=1}^{m} c_{i}u(α_{i}t) = 0  (12.20)

to have property Ã.

Corollary 12.3′. Let c_{i} > 0, α_{i} ∈]0, 1[ (i = 1, …, m) and α_{i_{0}} < 0 for some i_{0} ∈ {1, …, m}. Then (9.53) and (12.19) are necessary and sufficient for (12.20) to have property Ã.

§ 13. Equations with property B̃

13.1. Nonlinear Equations.

Theorem 13.1. Let F ∈ V (τ) and (0.3) and (12.2) be fulfilled, where

φ ∈ M(σ), σ is nondecreasing, σ(t) ≥ t for t ∈ ℝ_{+}.  (13.1)

Let, moreover, (3.3_{n−2}) and (11.71) hold, where φ̃_{n} is defined by either (11.72) or (11.73) and

φ̃_{n−2}(u)(t) = \frac{t}{(n − 1)!} φ(t^{n−2}u(t))(t).  (13.2)

Then (0.1) has property B̃.

Proof. Suppose that u : [t_{0}, +∞[→ ℝ is a proper nonoscillatory solution of (0.1). By Lemma 1.1 there exists l ∈ {0, …, n} such that l + n is even and (2.14_{l}) holds. According to (13.2) and Theorem 3.2′ l ∉ {1, …, n − 1}. On the other hand, in view of (11.72)–(11.74) and Theorems 11.12 and 11.13 l ≠ n. Therefore l = 0 which is possible only if n is even, so using (3.3_{n−2}) we can easily show that (0.4) is fulfilled. ■

Taking into account Theorems 3.2′′, 11.12 and 11.13, we can analogously prove the following

Theorem 13.1′. Let F ∈ V (τ) and (0.3), (12.2) and (13.1) be fulfilled with φ ∈ M(σ, \overline{σ}). Let, moreover, (3.18) and (11.71) hold where φ̃_{n} is defined by either (11.72) or (11.73) and

φ̃_{n−2}(u)(t) = \frac{t}{(n − 1)!} φ(ψ_{t}(u))(t),

ψ_{t}(u)(s) = [\overline{σ}(t)]^{2−n}t^{n−2}u(t)s^{n−2} for s ∈ ℝ_{+}.

Then (0.1) has property B̃.

Theorem 13.2. Let F ∈ V (τ) and (0.3), (12.2) and (13.1) be fulfilled. Let, moreover, (11.71) and (3.7_{1}) hold where β(t) = t, φ̃_{n} is defined by either (11.72) or (11.73) and

φ̃_{1}(u)(t) = \frac{t^{n−1}}{(n − 1)!} φ(u(t))(t).

Then (0.1) has property B̃.

Theorem 13.2′. Let F ∈ V (τ) and (0.3), (12.2) and (13.1) be fulfilled with φ ∈ M(σ, \overline{σ}). Let, moreover, (3.22) and (11.71) hold where β(t) = t, φ̃_{n} is defined by either (11.72) or (11.73) and

φ̃_{1}(u)(t) = \frac{t^{n−1}}{(n − 1)!} φ(ψ_{t}(u))(t), ψ_{t}(u)(s) = [\overline{σ}(t)]^{1−n}u(t)s^{n−1}.

Then equation (0.1) has property B̃.

The validity of Theorem 13.2 (Theorem 13.2′) follows from Theorems 3.3′, 11.12 and 11.13 (Theorems 3.5, 11.12 and 11.13).

13.2. Equations with a Linear Minorant.

Theorem 13.3. Let F ∈ V (τ) and (0.3), (10.2) and (12.6) be fulfilled. Then (12.7) and either (11.68) or (11.70) are sufficient for (0.1) to have property B̃.

Proof. The theorem follows from Corollaries 3.12, 11.12 and Theorem 11.11. ■

Theorem 13.4. Let F ∈ V (τ) and (0.3), (12.8) be fulfilled, wh where c_{i} > 0, Δ_{i}, \overline{Δ}_{i} ∈ ℝ, \overline{Δ}_{i} < Δ_{i} (i = 1, …, m) and \overline{Δ}_{i_{0}} < 0 for some i_{0} ∈ {1, …, m}. Then (11.32) is sufficient for (0.1) to have property B̃.

Proof. The theorem follows from Corollaries 3.1 and 11.3. ■

Theorem 13.4′. Let c_{i} > 0, Δ_{i}, \overline{Δ}_{i} ∈ ℝ, Δ_{i} > \overline{Δ}_{i} (i = 1, …, m) and \overline{Δ}_{i_{0}} < 0 for some i_{0} ∈ \{1, …, m\}. Then (11.32) is necessary and sufficient for (11.33) to have property B̃.

Theorem 13.5. Let F ∈ V (τ) and (0.3), (12.9) be fulfilled, where c_{i} > 0, Δ_{i} ∈ ℝ (i = 1, …, m) and Δ_{i_{0}} < 0 for some i_{0} ∈ {1, …, m}. Then (11.40) is sufficient for (0.1) to have property B̃.

The validity of Theorem 13.5 follows from Corollaries 3.1 and 11.6.

Theorem 13.6. Let F ∈ V (τ) and (0.3) and (12.10) be fulfilled where c_{i} > 0, α_{i}, \overline{α}_{i} ∈]0, +∞[, α_{i} < \overline{α}_{i} (i = 1, …, m) and \overline{α}_{i_{0}} > 1 for some i_{0} ∈ {1, …, m}. Then the condition

∑_{i=1}^{m} c_{i}(\overline{α} _{i}^{λ+1} − α_{i}^{λ+1}) > ∏_{i=−1}^{n−1}(λ − i) for λ ∈ ℝ_{+}  (13.3)

is sufficient for (0.1) to have property B̃.

The validity of Theorem 13.6 follows from Theorem 7.4 and Corollary 11.9.

Theorem 13.6′. Let c_{i} > 0, α_{i}, \overline{α}_{i} ∈]0, +∞[, α_{i} < \overline{α}_{i} (i = 1, …, m) and \overline{α}_{i_{0}} > 1 for some i_{0} ∈ \{1, ….m\}. Then (13.3) is necessary and sufficient for (11.51) to have property B̃.

Corollary 13.1. Let F ∈ V (τ) and (0.3), (12.10) be fulfilled, where c_{i} > 0 and 1 ≤ α_{i} < \overline{α}_{i} (i = 1, …, m). Then the condition

∑_{i=1}^{m} c_{i}(\overline{α} _{i}^{λ+1} − α_{i}^{λ+1}) > ∏_{i=−1}^{n−1}(λ − i) for λ ∈ [0, 2] ∪ [n − 1, +∞[ (13.4)

is sufficient for (0.1) to have property B̃.

Corollary 13.1′. Let c_{i} > 0, 1 ≤ α_{i} < \overline{α}_{i} (i = 1, …, m). Then (13.4) is necessary and sufficient for (11.51) to have property B̃.

Theorem 13.7. Let F ∈ V (τ) and (0.3) and (12.17) be fulfilled where, c_{i} > 0, α_{i} > 0 (i = 1, …, m) and α_{i_{0}} > 1 for some i_{0} ∈ {1, …, m}. Then the condition

∑_{i=1}^{m} c_{i}α_{i}^{λ} > ∏_{i=0}^{n−1}(λ − i) for λ ∈ ℝ_{+}  (13.5)

is sufficient for (0.1) to have property B̃.

The validity of Theorem 13.7 follows from Corollaries 7.5 and 11.11.

Theorem 13.7′. Let c_{i} > 0, α_{i} > 0 (i = 1, …, m) and α_{i_{0}} > 1 for some i_{0} ∈ {1, …, m}. Then (13.5) is necessary and sufficient for the equation

u^{(n)}(t) = t^{−n} ∑_{i=1}^{m} c_{i}u(α_{i}t)  (13.6)

to have property B̃.

Corollary 13.2. Let F ∈ V (τ) and (0.3), (12.17) be fulfilled, where c_{i} > 0, α_{i} ≥ 1 (i = 1, …, m) and α_{i_{0}} > 1 for some i_{0} ∈ {1, …, m}. Then the condition

∑_{i=1}^{m} c_{i}α_{i}^{λ} > ∏_{i=0}^{n−1}(λ − i) for λ ∈ [0, 2] ∪ [n − 1, +∞[  (13.7)

is sufficient for (0.1) to have property B̃.

Corollary 13.2′. Let c_{i} > 0, α_{i} ≥ 1 (i = 1, …, m) and α_{i_{0}} > 1 for some i_{0} ∈ {1, …, m}. Then (13.7) is necessary and sufficient for (13.6) to have property B̃.

§ 14. Oscillatory equations

Sufficient conditions for oscillation of any proper solution of (0.1) in the case of even n were given in §§3,4,6,7, and in the case of odd n in §12. If n is odd and (0.3) holds, the analogous question was studied in §13. In this section we establish sufficient conditions for any proper solution of (0.1) to be oscillatory in the case when n is even and (0.3) is fulfilled. Throughout this section, without mentioning it specially, it will be assumed that n is even.

14.1. Equations with a Linear Minorant.

Theorem 14.1. Let F ∈ V (τ) and (0.3) and (12.8) be fulfilled, where c_{i} > 0, Δ_{i}, \overline{Δ}_{i} ∈ ℝ, \overline{Δ}_{i} < Δ_{i} (i = 1, …, m) and Δ_{i_{0}} > 0, \overline{Δ}_{i_{1}} < 0 for some i_{0}, i_{1} ∈ {1, …, m}. Then (9.28) and (11.32) are sufficient for every proper solution of (0.1) to be oscillatory.

The validity of the theorem follows from Corollaries 3.1, 9.2, and 11.3.

Theorem 14.1′. Let c_{i} > 0, Δ_{i}, \overline{Δ}_{i} ∈ ℝ, \overline{Δ}_{i} < Δ_{i} (i = 1, …, m) and Δ_{i_{0}} > 0, \overline{Δ}_{i_{1}} < 0 for some i_{0}, i_{1} ∈ \{1, …, m\}. Then (9.38) and (11.32) are necessary and sufficient for every proper solution of equation (11.33) to be oscillatory.

Proof. Sufficiency follows from Theorem 14.1. If we assume that (9.32) ((11.32)) is violated, then (11.33) has the solution u(t) = e^{λt} with λ < 0 (λ > 0). ■

Theorem 14.2. Let F ∈ V (τ) and (0.3) and (12.9) be fulfilled, where m ≥ 2, c_{i} > 0, Δ_{i} ∈ ℝ (i = 1, …, m) and Δ_{i_{0}} > 0, Δ_{i_{1}} < 0 for some i_{0}, i_{1} ∈ {1, …, m}. Then (9.37) and (11.40) are sufficient for every proper solution of (0.1) to be oscillatory.

The validity of the theorem follows from Corollaries 3.1, 9.4 and 11.6.

Theorem 14.3. Let F ∈ V (τ) and (0.3) and (12.10) be fulfilled where c_{i} > 0, α_{i}, \overline{α}_{i} ∈]0, +∞[, α_{i} < \overline{α}_{i} (i = 1, …, m) and α_{i_{0}} < 1, \overline{α}_{i_{1}} > 1 for some i_{0}, i_{1} ∈ {1, …, m}. Then (9.49) and (13.3) are sufficient for every proper solution of (0.1) to be oscillatory.

The validity if the theorem follows from Corollary 9.7 and Theorem 13.6.

Theorem 14.3′. Let c_{i} > 0, α_{i}, \overline{α}_{i} ∈]0, +∞[, α_{i} < \overline{α}_{i} (i = 1, …, m) and α_{i_{0}} < 1, \overline{α}_{i_{1}} > 1 for some i_{0}, i_{1} ∈ \{1, …, m\}. Then (9.49) and (13.3) are necessary and sufficient for every proper solution of (11.51) to be oscillatory.

Theorem 14.4. Let F ∈ V (τ) and (0.3) and (12.17) be fulfillled where m ≥ 2, c_{i} > 0, α_{i} > 0 (i = 1, …, m) and α_{i_{0}} < 1, α_{i_{1}} > 1 for some i_{0}, i_{1} ∈ {1, …, m}. Then (13.5) and (9.53) are sufficient for every proper solution of (0.1) to be oscillatory.

The validity of the theorem follows from Corollary 9.9 and Theorem 13.7.

Theorem 14.4′. Let m ≥ 2, c_{i} > 0, α_{i} > 0 (i = 1, …, m) and α_{i_{0}} < 1, α_{i_{1}} > 1 for some i_{0}, i_{1} ∈ {1, …, m}. Then (13.5) and (9.53) are necessary and sufficient for every proper solution of (13.6) to be oscillatory.

14.2. Equations of the Emden-Fowler Type. In this subsection we consider the equation

u^{(n)}(t) = ∫_{τ_{1}(t)}^{σ_{1}(t)} |u(s)|^{λ_{1}} sign u(s)d_{s}r_{1}(s, t) +

+ ∫_{τ_{2}(t)}^{σ_{2}(t)} |u(s)|^{λ_{2}} sign u(s)d_{s}r_{2}(s, t),  (14.1)

where λ_{i} > 0, τ_{i}, σ_{i} ∈ C(ℝ_{+}; ℝ_{+}), τ_{i}(t) ≤ σ_{i}(t) for t ∈ ℝ_{+}, r_{i}(s, t) are measurable and r_{i}(·, t) are nondecreasing (i = 1, 2).

The results of the previous sections enable us to obtain sufficient conditions for every proper solution of (14.1) to be oscillatory in the case when n is even. To illustrate this fact several theorems are given below.

Theorem 14.5. Let λ_{1} = 1, λ_{2} < 1 and there exist t_{0} ∈ ℝ_{+} and nondecreasing functions δ_{i} ∈ C(ℝ_{+}; ℝ_{+}) (i = 1, 2) such that

τ_{1}(t) ≤ δ_{1}(t) < t, t < δ_{2}(t) ≤ σ_{1}(t) for t ≥ t_{0},  (14.2)

\underline{lim}_{t→+∞} ∫_{δ_{1}(t)}^{t} p_{1}(s)s^{n−1}ds > 0, \underline{lim}_{t→+∞} ∫_{t}^{δ_{2}(t)} p_{2}(s)s^{n−1}ds > 0

vrai sup{t^{n}p_{i}(t) : t ∈ ℝ_{+}} < +∞ (i = 1, 2),

(14.3)

where

p_{1}(t)=r_{1}(δ_{1}(t), t)−r_{1}(τ_{1}(t), t), p_{2}(t)=r_{1}(σ_{1}(t), t)−r_{1}(δ_{2}(t), t). (14.4)

Let, moreover,

inf \{ \underline{lim}_{t→+∞}t^{λ} ∫_{t}^{+∞}(s − t)^{n−1} ∫_{τ_{1}(s)}^{σ_{1}(s)} ξ^{−λ}d_{ξ}r_{1}(ξ, s)ds :

: λ ∈]0, +∞[} > (n − 1)!,  (14.5)

inf \{ \underline{lim}_{t→+∞}t^{−λ} ∫_{0}^{t}(t − s)^{n−1} ∫_{τ_{1}(s)}^{σ_{1}(s)} ξ^{λ}d_{ξ}r_{1}(ξ, s)ds :

: λ ∈]n − 1, +∞[} > (n − 1)!  (14.6)

and

\overline{lim}_{t→+∞} \frac{σ_{2}(t)}{t} < +∞, ∫ ^{+∞} t ∫_{τ_{2}(t)}^{σ_{2}(t)} s^{λ_{2}(n−1)}d_{s}r_{2}(s, t)dt = +∞. (14.7)

Then every proper solution of (14.1) is oscillatory.

Theorem 14.6. Let λ_{1} = 1, λ_{2} < 1 and (14.7) be fulfilled. Let, moreover, there exist t_{0} ∈ ℝ_{+} and nondecreasing functions δ_{i} ∈ C(ℝ_{+}; ℝ_{+}) (i = 1, 2) such that (14.2) holds and

\begin{matrix}\underline{lim}_{t→+∞} ∫_{δ_{1}(t)}^{t}(s − δ_{1}(t))^{n−1}p_{1}(s)ds > \frac{(n − 1)!}{e}, \\ \underline{lim}_{t→+∞} ∫_{t}^{δ_{2}(t)}(δ_{2}(t) − s)^{n−1}p_{2}(s)ds > \frac{(n − 1)!}{e} \end{matrix}  (14.8)

with p_{i} (i = 1, 2) defined by (14.4). Then every proper solution of (14.1) is oscillatory.

Theorem 14.7. Let λ_{1} = 1, λ_{2} > 1, either (14.2)–(14.6) or (14.8) be fulfilled and

∫ ^{+∞}[β(t)]^{n−1}[σ_{2}(t)]^{(2−n)λ_{2}} ∫_{τ_{2}(t)}^{σ_{2}(t)} s^{λ_{2}(n−2)}d_{s}r_{2}(s, t)dt = +∞ (14.9)

with β(t) = min{t, σ_{2}(t)}. Then every proper solution of (14.1) is oscillatory.

Theorem 14.8. Let λ_{1} < 1, λ_{2} > 1, σ_{1}(t) ≤ t, τ_{2}(t) ≥ t for t ∈ ℝ_{+} and

∫ ^{+∞} t^{λ_{1}(n−1)}(t − σ_{1}(t))^{λ_{1}(n−1)}[r_{1}(σ_{1}(t), t) − r_{1}(τ_{1}(t), t)]dt = +∞,

∫ ^{+∞}(τ_{2}(t) − t)^{λ_{2}(n−1)}[r_{2}(σ_{2}(t), t) − r_{2}(τ_{2}(t), t)]dt = +∞.

Let, moreover, (14.9) hold. Then every proper solution of (14.1) is oscillatory.

§ 15. Existence of an oscillatory solution.

15.1. Existence of a Proper Solution. Suppose that t_{0} > 0 and F ∈ V (τ; σ) with

τ(t) ≤ σ(t) < t for t ≥ t_{0}.  (15.1)

Consider the following Cauchy problem for (0.1)

u(t) = φ(t) for t ∈ [τ_{0}, t_{0}], u^{(i)}(t_{0}) = c_{i} (i = 1, …, n − 1), (15.2)

where τ_{0} = inf{τ(t) : t ∈ [t_{0}, +∞[}, φ ∈ C([τ_{0}, t_{0}]; ℝ), and c_{i} ∈ ℝ (i = 1, …, n − 1).

Lemma 15.1. Let for any t ≥ t_{0}

F (u)(t) ≠ 0 for u ∈ C(ℝ_{+}; ℝ) satisfying,

u(s) ≥ 0, u(s) ≢ 0 (u(s) ≤ 0, u(s) ≢ 0) for s ∈ [τ(t), σ(t)]. (15.3)

Let, moreover, φ(t) ≢ 0 for t ∈ [τ_{0}, t_{0}]. Then problem (0.1), (15.2) has a unique proper solution.

Proof. According to (15.1) problem (0.1), (15.2) has a unique solution u_{0} : [t_{0}, +∞[→ ℝ. Suppose that u_{0} is not proper. Then we can find t_{1} ∈ [t_{0}, +∞[ and t_{2}, t_{3} ∈ [t_{1}, +∞[ such that t_{2} < t_{3}, u_{0}(t) ≡ 0 for t ∈ [t_{1}, +∞[, u_{0}(s) ≢ 0 for s ∈ [τ(t), σ(t)] for any t ∈ [t_{2}, t_{3}] and either u_{0}(t) ≥ 0 or u_{0}(t) ≤ 0 for t ∈ [τ_{1}, t_{1}] where τ_{1} = min{τ(t) : t ∈ [t_{2}, t_{3}]}. Therefore by (15.3) from (0.1) we get

0 = u_{0}^{(n−1)}(t_{2}) − u_{o}^{(n−1)}(t_{3}) = ∫_{t_{2}}^{t_{3}} F (u_{0})(s)ds ≠ 0.

The obtained contradiction proves the lemma. ■

15.2. Existence of a Monotonically Increasing Solution. Suppose that F ∈ V (τ; σ) where τ, σ satisfy (15.1). For (0.1) consider the following Cauchy problem

\begin{matrix}u(t) = φ(t) for t ∈ [τ_{0}, t_{0}], φ(t_{0}) = 0, \\ u^{(i)}(t_{0}) = 0 (i = 1, …, n − 2), u^{(n−1)}(t_{0}) = γ. \end{matrix}  (15.4)

By (15.1) it is obvious that problem (0.1), (15.4) has a unique solution u(·, γ) which depends continuously on the parameter γ.

Define the sets Γ_{1} and Γ_{2} as follows:

γ̃ ∈ Γ_{1} iff the inequality γ ≥ γ̃ implies

lim_{t→+∞} u^{(i)}(t; γ) = +∞ (i = 0, …, n − 2), lim_{t→+∞} u^{(n−1)}(t; γ) > 0. (15.5)

γ̃ ∈ Γ_{2} iff the inequality γ ≤ γ̃ implies

lim_{t→+∞} u^{(i)}(t; γ) = −∞ (i = 0, …, n − 2), lim_{t→+∞} u^{(n−1)}(t; γ) < 0.

Suppose that t_{1} ∈ [t_{0}, η_{τ} (t_{0})], where η_{τ} (t) = sup{s : τ(t) < t}. Define the sets E_{i}(t_{1}) (i = 1, 2, 3) as follows:

E_{1}(t_{1}) = {t : t ∈ [t_{0}, t_{1}], τ(t) < t_{0}, σ(t) > t_{0}},
E_{2}(t_{1}) = {t : t ∈ [t_{0}, t_{1}], σ(t) ≤ t_{0}},
E_{3}(t_{1}) = {t : t ∈ [t_{0}, t_{1}], σ(t) ≥ t_{0}}.

Lemma 15.2. Let F ∈ V (τ; σ), conditions (0.3) and (15.1) be fulfilled and for any t_{1} ∈ [t_{0}, η_{τ} (t_{0})] and u ∈ C(ℝ_{+}; ℝ) satisfying u(t) = φ(t) for t ∈ [τ_{0}, t_{0}], u(t) ≠ 0 for t ∈]t_{0}, t_{1}] we have

∫_{E_{1}(t_{1})} F (u)(s)ds ≤ ∫_{E_{1}(η_{τ} (t_{0}))} |F (θ)(s)|ds,  (15.6)

where

θ(t) = \begin{cases}0 for t ∈]t_{0}, +∞], \\ φ(t) for t ∈ [τ_{0}, t_{0}[. \end{cases}

Then the sets Γ_{1} and Γ_{2} are nonempty and

γ_{2} ≤ γ_{1}, γ_{1}, γ_{2} ∉ Γ_{1} ∪ Γ_{2},

where γ_{2} = sup Γ_{2}, γ_{1} = inf Γ_{1}.

Proof. Denote

γ_{0} = ∫_{E_{2}(η_{τ} (t_{0}))} |F (φ)(s)|ds + ∫_{E_{1}(η_{τ} (t_{0}))} |F (θ)(s)|ds.  (15.7)

Show that if γ > γ_{0} then γ ∈ Γ_{1}. First prove that

u^{(i)}(t; γ) > 0 for t ∈]t_{0}, η_{τ} (t_{0})] (i = 0, …, n − 1).  (15.8)

Indeed, otherwise we can find t_{1} ∈]t_{0}, η_{τ} (t_{0})] such that

u^{(i)}(t; γ) > 0 for t ∈]t_{0}, t_{1}[ (i = 0, …, n − 1), u^{(n−1)}(t_{1}; γ) = 0.

Then (0.3), (15.6) and (15.7) imply

u^{(n−1)}(t_{1}; γ) = γ − ∫_{t_{0}}^{t_{1}} F (u)(s)ds = γ −
− ∫_{E_{1}(t_{1})} F (u)(s)ds − ∫_{E_{2}(t_{1})} F (u)(s)ds − ∫_{E_{3}(t_{1})} F (u)(s)ds ≥
≥ γ − ∫_{E_{2}(η_{σ}(t_{0}))} |F (φ)(s)|ds − ∫_{E_{1}(η_{σ}(t_{0}))} |F (θ)(s)|ds > 0.

The obtained contradiction shows that (15.8) is true. Therefore by (0.3) u(t; γ) satisfies (15.5), hence γ ∈ Γ_{1}.

Analogously it can be shown that if γ < −γ_{0}, then γ ∈ Γ_{2}. The nonemptiness of Γ_{1} and Γ_{2} and the fact that γ_{2} ≤ γ_{1} are thus proved.

Now prove that γ_{1} ∉ Γ_{1}. Indeed, otherwise we can find t_{γ_{1}} ∈ ℝ_{+} such that

u^{(i)}(t; γ_{1}) > 0 for t ∈ [t_{γ_{1}}, +∞[ (i = 0, …, n − 1).

Therefore there exists ε > 0 exists such that if γ ∈ [γ_{1} − ε, γ_{1}], then

u^{(i)}(t; γ) > 0 for t ∈ [t_{γ_{1}}, η_{τ} (t_{γ_{1}})] (i = 0, …, n − 1).

By (0.3) this means that u(t; γ) satisfies (15.5) whence it follows that γ_{1}−ε ∈ Γ_{1}. But this contradicts the definition of γ_{1}, so γ_{1} ∉ Γ_{1}. Analogously we can show that γ_{1}, γ_{2} ∉ Γ_{1} ∪ Γ_{2}. ■

15.3. Existence of a Proper Oscillatory Solution.

Theorem 15.1. Let n be even, F ∈ V (τ, σ), conditions (0.2), (15.1) and (15.3) be fulfilled and (0.1) have property A. Then (0.1) has an oscillatory solution.

Proof. By Lemma 15.1 (0.1) has a proper solution. Since n is even and the equation has the property A, this solution has to be oscillatory. ■

Theorem 15.2. Let n be odd,conditions (0.3), (15.1) and (15.3) be fulfilled and suppose that for some t_{0} > 0 and φ ∈ C([τ_{0}, t_{0}]; ℝ), satisfying φ(t) ≢ 0 for t ∈ [τ_{0}, t_{0}] and φ(t_{0}) = 0 inequality (15.6) holds. Let, moreover, (0.1) have property B. Then it has an oscillatory solution as well as solutions satisfying (0.5).

Proof. The existence of proper solutions satisfying (0.5) follows from Lemma 15.2 and the definition of property B. Besides, Lemmas 15.1 and 15.2 imply the existence of a proper solution not satisfying (0.5). Since n is odd and (0.1) has property B, this solution has to be oscillatory. ■

Theorem 15.3. Let F ∈ V (τ; σ), conditions (12.1), (15.1) and (15.3) be fulfilled and suppose that for some t_{0} > 0 and φ ∈ C([τ_{0}, t_{0}]; ℝ), satisfying φ(t) ≢ 0 for t ∈ [τ_{0}, t_{0}] and φ(t_{0}) = 0 inequality (15.6) holds. Let, moreover, equation (0.1) have property Ã. Then (0.1) has an oscillatory solution. Moreover, if n is even, along with the oscillatory solution it has a proper solutions satisfying (0.5).

Proof. By Lemma 15.1 (0.1) has a proper solution. Since (0.1) has property Ã, this solution has to be oscillatory when n is odd. Suppose now that n is even. Then according to Lemma 15.2 (0.1) has proper solutions satisfying (0.5). Besides, by Lemmas 15.1 and 15.2 this equation has a proper solution not satisfying (0.5). Since (0.1) has property Ã and n is even, this solution has to be oscillatory. ■

CHAPTER 6

SECOND ORDER DELAY EQUATIONS

§ 16. On a Singular Boundary Value Problem

In this section we shall establish the sufficient conditions for the unique solvability and oscillation of solutions of the following boundary value problem:

u′′(t) = ∑_{i=1}^{m} p_{i}(t)u(δ_{i}(t)),  (16.1)
u(t) = φ(t) for t ∈ [τ_{0}, 0], \underline{lim}_{t→+∞}|u(t)| < +∞,  (16.2)

where

p_{i} ∈ L_{loc}(ℝ_{+}; ℝ_{+}), δ_{i} ∈ C(ℝ_{+}, ℝ), δ_{i}(t) ≤ t

for t ∈ ℝ_{+}, lim_{t→+∞} δ_{i}(t) = +∞ (i = 1, …, m),

φ ∈ C([τ_{0}, 0]; ℝ), τ_{0} = min{(inf δ_{i}(t) : t ∈ ℝ_{+}) : i = 1, …, m}.

(16.3)

Theorem 16.1. If

\overline{lim}_{t→+∞} ∫_{t}^{η(t)}(s − t) ∑_{i=1}^{m} p_{i}(s)ds < +∞,  (16.4)

where

\begin{matrix}η(t) = max\{η_{δ_{i}}(t) : i = 1, …, m\}, \\ η_{δ_{i}}(t) = sup\{s : δ_{i}(s) < t\} (i = 1, …, m), \end{matrix}  (16.5)

then problem (16.1), (16.2) has a unique solution.

To the theorem we need some auxiliary assertions.

For any γ ∈ ℝ denote by u(·; γ) the solution of (16.1) satisfying

u(t) = φ(t) for t ∈ [τ_{0}, 0], u′(0) = γ.

Define the sets Γ_{1}, Γ_{2} ⊂ ℝ as follows:

γ ∈ Γ_{1} ⇔ lim_{t→+∞} u(t; γ) = +∞, lim_{t→+∞} u′(t; γ) > 0,

γ ∈ Γ_{2} ⇔ lim_{t→+∞} u(t; γ) = −∞, lim_{t→+∞} u′(t; γ) < 0.

Lemma 16.1. Let conditions (16.3) be fulfilled. Then Γ_{1} and Γ_{2} are nonempty and

γ_{2} ≤ γ_{1}, γ ∈ [γ_{2}, γ_{1}] ⇒ γ ∉ Γ_{1} ∪ Γ_{2},

where γ_{1} = inf Γ_{1}, γ_{2} = sup Γ_{2}. Moreover, if γ_{2} < γ_{1}, then problem (16.1), (16.2) has an infinite set of solutions and each of them is oscillatory and unbounded.

Proof. The nonemptiness of Γ_{1} and Γ_{2}, as well as the inequality γ_{2} ≤ γ_{1} can be proved similarly to Lemma 15.2. If γ_{2} < γ_{1}, then for any γ ∈ [γ_{2}, γ_{1}] we readily find that u(·; γ) is oscillatory and unbounded and ℝ = Γ_{1} ∪ Γ_{2} ∪ [γ_{2}, γ_{1}]. ■

Lemma 16.2. Let conditions (16.3) be fulfilled, γ_{2} < γ_{1} and γ ∈ [γ_{2}, γ_{1}]. Then there exists γ^{∗} ∈]γ_{1}, +∞[ such that

u(t; γ^{∗}) > |u(t; γ)|, u′(t; γ^{∗}) > |u′(t; γ)|, for t ≥ 1.  (16.6)

Proof. Introduce the notation

c_{0} = max{|φ(t)| : t ∈ [τ_{0}, 0]} + max{|u(t; γ)| + |u′(t; γ)| : t ∈ [0, η(1)]},

γ^{∗} = 2c_{0} + c_{0} ∫_{0}^{η(1)} ∑_{i=1}^{m} p_{i}(s)ds.

In the first place show that

u′(t; γ^{∗}) ≥ 2c_{0} for 0 ≤ t ≤ η(1).  (16.7)

Assume the contrary. Then for some t_{0} ∈ [0, η(1)] we have

u′(t; γ^{∗}) > 0 for 0 ≤ t ≤ t_{0},
u(t; γ^{∗}) > −c_{0} for 0 ≤ t ≤ t_{0}, and u′(t_{0}; γ^{∗}) < 2c_{0}

which is impossible because (16.1) implies

u′(t_{0}; γ^{∗}) = γ^{∗}+∫_{0}^{t_{0}} ∑_{i=1}^{m} p_{i}(s)u(δ_{i}(s); γ^{∗})ds ≥ γ^{∗}−c_{0} ∫_{0}^{η(1)} ∑_{i=1}^{m} p_{i}(s)ds = 2c_{0}.

Therefore (16.7) is valid.

By (16.7) we have

u′(η(1); γ^{∗}) > |u′(η(1); γ)|, u(t; γ^{∗}) ≥ 2c_{0}t + φ(0) > |u(t; γ)|
for 1 ≤ t ≤ η(1).

Keeping in mind that p_{i}(t) ≥ 0 for t ∈ ℝ_{+} (i = 1, …, m), from the latter inequality we obtain (16.6). ■

Proof of Theorem 16.1. Due to Lemma 16.1 it suffices to show that γ_{2} = γ_{1}. Assume on the contrary that γ_{2} < γ_{1} and take γ ∈]γ_{2}, γ_{1}[. By Lemma 16.2 there exists γ^{∗} ∈]γ_{1}, +∞[ such that (16.6) holds. We have to show that for any sufficiently large t

u(t; γ^{∗}) ≥ (1 + \frac{1}{ρ})|u(t; γ)|,  (16.8)

where ρ ∈]0, +∞[ satisfies

ρ ≥ ∫_{t}^{η(t)}(s − t) ∑_{i=1}^{m} p_{i}(s)ds for t ∈ ℝ_{+}.  (16.9)

u(·; γ) is oscillatory by Lemma 16.1. Let t_{0} ∈ [η(1), +∞[ be a zero of u(·; γ) and s_{0} ∈]t_{0}, +∞[ be any point which is not a zero of u(·; γ). For the sake of definiteness assume that u(s_{0}; γ) > 0. Then we can choose s_{1} ∈ [t_{0}, s_{0}[, s_{2} ∈]s_{0}, +∞[ and s̃ ∈]s_{1}, s_{2}[ such that

\begin{matrix}u(s_{1}; γ) = u(s_{2}; γ) = u′(s̃; γ) = 0, \\ u(s̃; γ) ≥ u(s; γ) > 0 for s_{1} < s < s_{2}. \end{matrix}  (16.10)

It is easy to see that s̃ < η(s_{1}). Moreover, using (16.6) and (16.9), from the equality

u(s̃; γ) = − ∫ _{s} ; s_{1}(s − s_{1}) ∑_{i=1}^{m} p_{i}(s)u(δ_{i}(s); γ)ds

we obtain

u(s̃; γ) = −( ∑_{i=1}^{m} ∫_{E′_{i}}(s − s_{1})p_{i}(s)u(δ_{i}(s); γ)ds +

+∑_{i=1}^{m}∫_{E′′_{i}}(s − s_{1})p_{i}(s)u(δ_{i}(s); γ)ds)≤− ∑_{i=1}^{m}∫_{E′′_{i}}(s − s_{1})p_{i}(s)u(δ_{i}(s); γ)ds≤

≤ u(s_{1}; γ^{∗}) ∫_{s_{1}}^{η(s_{1})}(s − s_{1}) ∑_{i=1}^{m} p_{i}(s)ds ≤ ρu(s_{1}; γ^{∗}),

where E′_{i} = {s ∈ [s_{1}, s̃] : δ_{i}(s) ≥ s_{1}}, E′′_{i} = {s ∈ [s_{1}, s̃] : δ_{i}(s) < s_{1}} (i = 1, …, m). Now by (16.6) and (16.10) we have

u(s_{0}; γ^{∗}) = u(s_{1}; γ^{∗}) + u′(s_{1}; γ^{∗})(s_{0} − s_{1}) + ∫_{s_{1}}^{s_{0}}(s_{0} − s) ×

× ∑_{i=1}^{m} p_{i}(s)u(δ_{i}(s); γ^{∗})ds ≥ \frac{1}{ρ}u(s̃; γ) +

+u′(s_{1}; γ)(s_{0} − s_{1}) + ∫_{s_{1}}^{s_{0}}(s_{0} − s) ∑_{i=1}^{m} p_{i}(s)u(δ_{i}(s); γ)ds =

= \frac{1}{ρ}u(s̃; γ) + u(s_{0}; γ) ≥ (1 + \frac{1}{ρ})u(s_{0}; γ).

Since s_{0} has been chosen arbitrarily we hence conclude that (16.8) is valid on [t_{0}, +∞[.

Let t_{1} ∈ [η(t_{0}), +∞[ and u′(t_{1}; γ) = 0. By (16.6) and (16.8)

u′(t; γ^{∗}) = u′(t_{1}; γ^{∗}) + ∫_{t_{1}}^{t} ∑_{i=1}^{m} p_{i}(s)u(δ_{i}(s); γ^{∗})ds > (1 + \frac{1}{ρ}) ×

×∫_{t_{1}}^{t} ∑_{i=1}^{m}p_{i}(s)|u(δ_{i}(s); γ)|ds≥(1 + \frac{1}{ρ})|u′(t; γ)| for t≥t_{1}. (16.11)

Using (16.8) and (16.1) we can show by induction that there exists a sequence of points t_{i} ∈ ℝ_{+} (i = 1, 2, …) such that for any i ∈ ℕ we have the equalities

u(t; γ^{∗}) ≥ (1 + \frac{1}{ρ})^{i}|u(t; γ)|, u′(t; γ^{∗}) ≥ (1 + \frac{1}{ρ})^{i}|u′(t; γ)|

for t ≥ t_{i}, (i = 1, 2, …).

Therefore

lim_{t→+∞} \frac{|u(t; γ)|}{u(t; γ^{∗})} = 0.  (16.12)

Put γ_{0} = (γ + γ_{2})/2 and c = −(γ − γ_{0})/(γ^{∗} − γ). Clearly, u(t; γ_{0}) = (1 − c)u(t; γ) + cu(t; γ_{∗}). Hence by (16.12) we have

lim_{t→+∞} \frac{u(t; γ_{0})}{u(t; γ^{∗})} = c < 0

which is impossible since γ_{0} ∈]γ_{2}, γ_{1}[ and by Lemma 16.1 u(·; γ_{0}) is an oscillatory solution. The obtained contradiction proves the theorem. ■

Corollary 16.1. Let p ∈ L_{loc}(ℝ_{+}; ℝ_{+}), δ ∈ C(ℝ_{+}; ℝ), δ(t) ≤ t for t ∈ ℝ_{+}, lim_{t→+∞} δ(t) = +∞,

vrai sup{p(t) : t ∈ ℝ_{+}} < +∞, sup{t − δ(t) : t ∈ ℝ_{+}} < +∞

and

∫ ^{+∞} p(t)dt = +∞.

Then the boundary value problem

u′′(t) = p(t)u(δ(t)), u(t) = φ(t) for t ∈ [τ_{0}, 0], \underline{lim}_{t→+∞}|u(t)| < +∞,

where τ_{0} = inf{δ(t) : t ∈ ℝ_{+}} and φ ∈ C([τ_{0}, 0]; ℝ), has a unique solution.

Corollary 16.1 was proved in [102] under the additional restriction ∫^{+∞} p(t)dt = +∞ which, as Theorem 16.1 shows, is quite unnecessary.

Theorem 16.2. In addition to (16.3) and (16.4), let the condtion

∫ ^{+∞} t ∑_{i=1}^{m} p_{i}(t)dt = +∞.  (16.13)

be also fulfilled. Then problem (16.1), (16.2) has a unique solution u : ℝ_{+} → ℝ satisfying

\underline{lim}_{t→+∞}|u(t)| = 0.  (16.14)

Proof. The existence and uniqueness of a solution are provided by Theorem 16.1. Show that (16.14) is fulfilled. Assume the contrary. Then there exist c ∈]0, +∞[ and t_{0} ∈ ℝ_{+} such that

|u(t)| ≥ c, u(t)u′(t) ≤ 0 for t ≥ t_{0}.

Multiplying both sides of (16.1) by (t − t_{1}) sign u(t_{1}), where t_{1} = η(t_{0}), and integrating, we obtain

|u(t_{1})| = (t − t_{1})|u′(t_{1})| + |u(t)|+ ∫_{t_{1}}^{t}(s − t_{1}) ∑_{i=1}^{m} p_{i}(s)|u(δ_{i}(s))|ds
for t ≥ t_{1}

whence it follows that

∫_{t_{1}}^{+∞}(s − t_{1}) ∑_{i=1}^{m} p_{i}(s)ds ≤ \frac{|u(t_{1})|}{c}

which contradicts (16.13). The obtained contradiction proves the validity of (16.14). ■

Theorem 16.3. Let

p_{i} ∈ L_{loc}(ℝ_{+}; ]0, +∞[) (i = 1, …, m), φ(t) ≢ 0

for t ∈ [τ_{0}, 0], δ_{i}(t) < t for t ∈ ℝ_{+}

(i = 1, …, m), \underline{lim}_{t→+∞}(δ_{i}(t) − t) > −∞,

vrai sup{p_{i}(t) : t ∈ ℝ_{+}} < +∞ (i = 1, …, m).

(16.15)

Then problem (16.1), (16.2) has a unique proper solution. Moreover, if δ_{i} (i = 1, …, m) are nondecreasing and

\underline{lim}_{t→+∞} ∫_{δ_{i}(t)}^{t} p_{i}(s)ds > 0 (i = 1, …, m),  (16.16)

inf\{ \underline{lim}_{t→+∞}e^{λt}∫_{t}^{+∞}(s − t) ∑_{i=1}^{m} p_{i}(s)e^{−λδ_{i}(s)}ds : λ ∈]0, +∞[\}>1, (16.17)

then this solution is oscillatory.

Proof. By virtue of (16.15), Lemma 15.1 and Theorem 16.1 problem (16.1), (16.2) has a unique proper solution u : ℝ_{+} → ℝ. Thus to complete the proof it suffices to show that u is oscillatory. Assume the contrary. Then there exists t_{0} ∈ ℝ_{+} such that

(−1)^{i}u^{(i)}(t)u(t) > 0 for t ≥ t_{0} (i = 0, 1).  (16.18)

On the other hand, by Theorem 9.3′ and (16.15)–(16.17) equation (16.1) has no solution satisfying (16.18). The obtained contradiction proves the theorem. ■

Corollary 16.2. Let (16.15) be fulfilled. Then problem (16.1), (16.2) has a unique proper solution. Moreover, if (16.16) holds and for some t_{0} ∈ ℝ_{+} we have

vrai inf \{ ∑_{i=1}^{m} p_{i}(t)(t − δ_{i}(t))^{2} : t ∈ [t_{0}, +∞[\} > (\frac{2}{e})^{2}, (16.19)

then this solution is ocillatory.

Proof. It suffices to note that (16.19) implies (16.17). ■

Corollary 16.2′. Let p_{i}, Δ_{i} ∈]0, +∞[ (i = 1, …, m), φ ∈ C([−Δ, 0]; ℝ) and φ(t) ≢ 0 on [−Δ, 0], where Δ = max{Δ_{i} : i = 1, …, m}. Then the problem

u′′(t) = ∑_{i=1}^{m} p_{i}u(t − Δ_{i}),  (16.20)
u(t) = φ(t) for − Δ ≤ t ≤ 0, \underline{lim}_{t→+∞}|u(t)| < +∞,  (16.21)

has a unique proper solution. Moreover, if

inf \{\frac{1}{λ^{2}} ∑_{i=1}^{m} p_{i}e^{λΔ_{i}} : λ ∈]0, +∞[\} > 1,

then this solution is oscillatory.

Theorem 16.4. Let p_{i} ∈ L_{loc}(ℝ_{+}; ]0, +∞[) (i = 1, …, m), φ(t) ≢ 0 for τ_{0} ≤ t ≤ 0,

δ_{i}(t) < t for t ∈ ℝ_{+}, vrai sup{t^{2}p_{i}(t) : t ∈ ℝ_{+}} < +∞

\underline{lim}_{t→+∞} \frac{δ_{i}(t)}{t} > 0 (i = 1, …, m). (16.22)

Then problem (16.1), (16.2) has a unique proper solution. Moreover, if δ_{i} (i = 1, …, m) are nondecreasing and

\underline{lim}_{t→+∞} ∫_{δ_{i}(t)}^{t} sp_{i}(s)ds > 0 (i = 1, …, m),  (16.23)

inf \{ \underline{lim}_{t→+∞}t^{λ} ∫_{t}^{+∞}(s − t) ∑_{i=1}^{m} p_{i}(s)δ_{i}^{−λ}(s)ds : λ ∈]0, +∞[\} > 1, (16.24)

then this solution is oscillatory.

Proof. Following Lemma 15.1, Theorem 16.1 and (16.22), problem (16.1), (16.22) has a unique proper solution u : ℝ_{+} → ℝ. If we assume that u is nonoscillatory, then there exists t_{0} ∈ ℝ_{+} such that (16.18) is fulfilled. But by Corollary 9.9 and (16.22)–(16.24) equation (16.1) has no solution satisfying (16.18). The obtained contradiction proves the theorem. ■

Corollary 16.3. Let (16.22) be fulfilled. Then problem (16.1), (16.2) has a unique proper solution. Moreover, if δ_{i} (i = 1, …, m) are nondecreasing, (16.23) holds and for some t_{0} ∈ ℝ_{+} we have

inf \{\frac{1}{λ(λ + 1)} vrai inf_{t≥t_{0}} t^{2+λ} ∑_{i=1}^{m} p_{i}(t)δ_{i}^{−λ}(t) : λ ∈]0, +∞[\} > 1, (16.25)

then this solution is oscillatory.

Corollary 16.3′. Let p_{i} ∈]0, +∞[ and α_{i} ∈]0, 1[ (i = 1, …, m). Then the problem

u′′(t) = ∑_{i=1}^{m} \frac{p_{i}}{(t + 1)^{2}} u(α_{i}t), u(0) = c_{0} ≠ 0 \underline{lim}_{t→+∞}|u(t)| < +∞,

has a unique proper solution. Moreover, if

inf \{\frac{1}{λ(λ + 1)} ∑_{i=1}^{m} p_{i}α_{i}^{−λ} : λ ∈]0, +∞[\} > 1,

then this solution is oscillatory.

Theorem 16.5. Let

p_{i} ∈ L_{loc}(ℝ_{+}; ]0, +∞[), δ_{i}(t) < t for t ∈ ℝ_{+},

\underline{lim}_{t→+∞} \frac{ln δ_{i}(t)}{ln t} > 0 (i = 1, …, m),

φ(t) ≢ 0 for τ_{0} ≤ t ≤ 0,
vrai sup{t^{2} ln t p_{i}(t) : t ∈ [1, +∞[} < +∞ (i = 1, …, m).

(16.26)

Then problem (16.1), (16.2) has a unique proper solution. Moreover, if δ_{i} (i = 1, …, m) are nondecreasing, (16.23) is fulfilled and for any k ∈ ℕ we have

inf \{ \underline{lim}_{t→+∞}(ln t)^{λ} ∫_{t}^{+∞}(s − t) ×

× ∑_{i=1}^{m} p_{i}(s)(ln δ_{i}(s))^{−λ} : λ ∈]0, k]} > 1,  (16.27)

then this solution is oscillatory.

Proof. By Lemma 15.1, Theorem 16.1 and (16.26) problem (16.1), (16.2) has a unique proper solution. If we assume that u is nonoscillatory, then there exists t_{0} ∈ ℝ_{+} such that (16.18) is valid which is impossible by Theorem 9.10′, (61.23), (16.26) and (16.27). ■

Corollary 16.4. Let (16.26) be fulfilled. Then problem (16.1), (16.2) has a unique proper solution. Moreover, if δ_{i} (i = 1, …, m) are nondecreasing functions, (16.23) holds and for any k ∈ ℕ there exists t_{k} ∈ [1, +∞[ such that

inf \{\frac{1}{λ} vrai inf_{t≥t_{k}} t^{2}(ln t)^{λ+1} ∑_{i=1}^{m} p_{i}(t)(ln δ_{i}(t))^{−λ} : λ ∈]0, k]\} > 1, (16.28)

then this solution is oscillatory.

Proof. It suffices to note that (6.28) implies (6.27). ■

Corollary 16.4′. Let p_{i} ∈]0, +∞[ and α_{i} ∈]0, 1[ (i = 1, …, m). Then the problem

u′′(t)=∑_{i=1}^{m} \frac{p_{i}}{(t + 1)^{2} ln(t + 2)}u(t^{α_{i}}), u(0)=c_{0} ≠0, \underline{lim}_{t→+∞}|u(t)|<+∞

has a unique proper solution. Moreover, if

inf \{\frac{1}{λ} ∑_{i=1}^{m} p_{i}α_{i}^{−λ} : λ ∈]0, +∞[\} > 1,

then this solution is oscillatory.

§ 17. Existence of Bounded Solutions

In this section we shall establish the sufficient conditions for the existence of bounded solutions of the equation

u′′(t) = f(t, u(δ_{1}(t)), …, u(δ_{m}(t)))  (17.1)

where f ∈ K_{loc}(ℝ_{+} × ℝ^{m}; ℝ) and

δ_{i}(t) < t for t ∈ ℝ_{+}, lim_{t→+∞} δ_{i}(t) = +∞ (i = 1, …, m). (17.2)

Theorem 17.1. Let

0 ≤ f(t, x_{1}, …, x_{m}) sign x_{1} ≤ p(t) ∏_{i=0}^{m} |x_{i}|^{λ_{i}}  (17.3)
for t ∈ ℝ_{+}, (x_{1}, …, x_{m}) ∈ ℝ^{m},

where λ_{i} ∈]0, 1[ (i = 1, …, m), ∑_{i=1}^{m} λ_{i} < 1 and p ∈ L_{loc}(ℝ_{+}; ℝ_{+}). Moreover, if

\overline{lim}_{t→+∞} ∫_{t}^{η_{1}(t)}(s − t)p(s)ds < +∞,  (17.4)

where η_{1}(t) = sup{s : δ_{1}(s) < t}, then every oscillatory solution of (17.1) is bounded.

Proof. Let u : [t_{0}, +∞[→ ℝ be an oscillatory solution of (17.1). By (17.3)

u′′(t) sign u(δ_{1}(t)) ≥ 0 for t ≥ t_{∗}  (17.5)

and

|u′′(t)| ≤ p(t) ∏_{i=1}^{m} |u(δ_{i}(t))|^{λ_{i}} for t ≥ t_{∗},  (17.6)

where t_{∗} ∈ [t_{0}, +∞[ is sufficiently large.

There are two possibilities: either \overline{lim}_{t→+∞}|u(t)| ≤ |u(t^{∗})| for some t^{∗} ∈

[t_{∗}, +∞[ or \overline{lim}_{t→+∞}|u(t)| > |u(t)| for t ≥ t_{∗}. In the first case the validity of the theorem is obvious.

Consider the second case. We find that there exist the increasing sequences {t_{k}}_{k=1}^{+∞} and {t̃_{k}}_{k=1}^{+∞} tending to +∞ and satisfying

t_{k} < t̃_{k}, u(t_{k}) = u′(t̃_{k}) = 0, u(t) ≠ 0,  (17.7)
for t_{k} < t ≤ t̃_{k} (k = 1, 2, …)

and

|u(t)| ≤ |u(t̃_{k})| for t_{∗} ≤ t ≤ t̃_{k} (k = 1, 2, …).  (17.8)

Show that

η_{1}(t_{k}) ≥ t̃_{k} (k = 1, 2, …).  (17.9)

Indeed, assuming that η_{1}(t_{k}) < t̃_{k} for some k ∈ ℕ, by (17.5) and (17.7) we obtain

u′(t)u(t) ≥ 0, |u(t)| ≥ |u(t̃_{k})| for t ≥ t̃_{k}

which is impossible because u is oscillatory.

Inequalities (17.6), (17.8) and (17.9) together with the equality

u(t̃_{k}) = − ∫ _{t_{k}} ; t_{k}(s − t_{k})u′′(s)ds

imply

|u(t̃_{k})|≤∫ _{t_{k}} ; t_{k}(s − t_{k})p(s) ∏_{i=1}^{m} |u(δ_{i}(s))|^{λ_{i}}ds≤|u(t̃_{k})|^{λ}∫ _{t_{k}} ; t_{k}(s − t_{k})p(s)ds≤

≤ |u(t̃_{k})|^{λ} ∫_{t_{k}}^{η_{1}(t_{k})}(s − t_{k})p(s)ds (k = k_{0}, k_{0} + 1, …),

where t_{k_{0}} ≥ η(t_{∗}). Therefore

|u(t̃_{k})|^{1−λ} ≤ ∫_{t_{k}}^{η_{1}(t_{k})}(s − t_{k})p(s)ds (k = k_{0}, k_{0} + 1, …)

with λ = ∑_{i=1}^{m} λ_{i} < 1. Hence by (17.4) and (17.8) we conclude that u is bounded. ■

For equation (17.1) consider the initial value problem

u(t) = φ(t) for τ_{0} ≤ t ≤ 0, u′(0) = γ  (17.10)

and the boundary value problem

u(t) = φ(t) for τ_{0} ≤ t ≤ 0, \overline{lim}_{t→+∞}|u(t)| < +∞,  (17.11)

where φ ∈ C([τ_{0}, 0]; ℝ]), φ(0) = 0, γ ∈ ℝ, τ_{0} = min{inf_{t≥0} δ_{i}(t) : i = 1, …, m}.

By (17.2) it is clear that for any γ ∈ ℝ problem (17.1), (17.10) has a unique solution u(·; γ) : ℝ_{+} → ℝ depending continuously on the parameter γ. By Lemma 15.2 there exists γ ∈ ℝ such that u(·; γ) either is oscillatory or satisfies

u(t, γ)u′(t, γ) ≤ 0 for t ≥ t_{γ},

where t_{γ} ∈ ℝ_{+} is sufficiently large.

Theorem 17.2. Let (17.3) and (17.4) be fulfilled. Then problem (17.1), (17.11) is solvable. Moreover, if φ(t) ≢ 0 on [τ_{0}, 0] and

\begin{matrix}f(t, x_{1}, …, x_{m}) sign x_{1} > 0 for x_{1} ≠ 0, \\ t ∈ ℝ_{+}, (x_{1}, …, x_{m}) ∈ ℝ^{m}, \end{matrix}  (17.12)

then every solution of this problem is proper.

Proof. The first part of the asserion of the theorem follows from Theorem 17.1. As to the second part, its proof can be obtained from Lemma 15.1 due to (17.2) and (17.12). ■

Theorem 17.3. Let (17.2)–(17.4) and (17.12) be fulfilled, λ_{i} ∈]0, 1[ (i = 1, …, m), ∑_{i=1}^{m} λ_{i} < 1, p ∈ L_{loc}(ℝ_{+}; ]0, +∞[), φ(t) ≢ 0 on [τ_{0}, 0] and φ(0) = 0. Then problem (17.1), (17.11) has a proper solution. Moreover, if

f(t, x_{1}, …, x_{m}) sign x_{1} ≥ q(t) ∏_{i=1}^{m} |x_{i}|^{λ_{i}} for t ∈ ℝ_{+},  (17.13)
(x_{1}, …, x_{m}) ∈ ℝ^{m}

and

∫_{0}^{+∞} t^{1−λ}q(t) ∏_{i=1}^{m}[t − δ_{i}(t)]^{λ_{i}}dt = +∞,  (17.14)

where q ∈ L_{loc}(ℝ_{+}; ]0, +∞[), then every solution of this problem is oscillatory.

Proof. The fact that problem (17.1), (17.11) has a proper solution u : ℝ_{+} → ℝ follows from theorem 17.2. To complete the proof it suffices to show that u is oscillatory. Indeed, otherwise there would exist t_{0} ∈ ℝ_{+} such that (16.18) be satisfied which would be impossible by (17.13), (17.14) and Theorem 9.15′. ■

Corollary 17.1. Let p, Δ ∈]0, +∞[, λ ∈]0, 1[. Then the equation

u′′(t) = p|u(t − Δ)|^{λ} sign u(t − Δ)

has a proper bounded solution. Moreover, every oscillatory solution of this equation is bounded.

Corollary 17.2. Let δ ∈ C(ℝ_{+}; ℝ), δ(t) < t for t ∈ ℝ_{+}, λ ∈]0, 1[, α ∈]0, 1], p ∈]0, +∞[ and

\overline{lim}_{t→+∞} \frac{δ(t)}{t} < 1, \underline{lim}_{t→+∞} \frac{δ(t)}{t^{α}} > 0.

Then the equation

u′′(t) = \frac{p}{(t + 1)^{2} ln(t + 2)}|u(δ(t))|^{λ} sign u(δ(t))

has a proper bounded solution. Moreover, every oscillatory solution of this equation is bounded.

Similarly to Theorem 17.1 one can prove

Theorem 17.4. Let conditions (16.3) be fulfilled and for any sufficiently large t there hold

∫_{t}^{η(t)}(s − t) ∑_{i=1}^{m} p_{i}(s)ds < 1,  (17.15)

where η is defined by (16.5). Then every oscillatory solution of (16.1) is bounded.

Theorem 17.5. Let (16.15) and (17.15) be fulfilled. Then problem (16.1), (17.11) has a unique proper solution. Moreover, if δ_{i} (i = 1, …, m) are nondereasing, (16.16) and (16.17) hold and

\underline{lim}_{t→+∞}(δ_{i}(t) − t) > −∞, vrai sup\{p_{i}(t) : t ∈ ℝ_{+}\} < +∞ (17.16)
(i = 1, …, m),

then this solution is oscillatory.

Proof. The fact that problem (16.1), (17.1) has a unique proper solution u : ℝ_{+} → ℝ follows from Lemma 16.1 and Theorems 16.3 and 17.4. As to the second part of the assertion of the theorem, as while proving Theorem 16.3, the oscillation of u is obtained by Theorem 9.3′, (16.16), (16.17) and (17.6). ■

Corollary 17.3. Let conditions (16.15) and (17.15) be fulfilled. Then problem (16.1), (17.11) has a unique proper solution. Moreover, if δ_{i} (i = 1, …, m) are nondereasing and (16.16), (16.19) and (17.16) hold, then this solution is oscillatory.

Corollary 17.4. Let Δ, p ∈]0, +∞[ and

\frac{2}{e^{2}} < \frac{pΔ^{2}}{2} < \frac{19 + 2\sqrt{34}}{18}.

Then the problem

u′′(t) = pu(t − Δ), u(t) = φ(t) for t ∈ [−Δ, 0], lim_{t→+∞} u(t) = 0

where φ ∈ C([−Δ, 0]; ℝ) and φ(t) ≢ 0 on [−Δ, 0], has a unique solution and this solution is oscillatory.

Proof. The uniqueness and oscillation of a solution satisfying \overline{lim}_{t→+∞}|u(t)| <

+∞ are obtained by Theorem 17.5. The fact that this solution tends to zero as t → +∞ follows from Theorem 77 in [86]. ■

Theorem 17.6. Let (16.15) and (17.15) be fulfilled. Then problem (16.1), (17.11) has a unique solution. Moreover, if δ_{i} (i = 1, …, m) are nondecreasing and (16.22)–(16.24) hold, then this solution is oscillatory.

Proof. The first part of the above assertion follows from Theorem 17.5. The second part is proved by Corollary 9.9. using (16.22)–(16.24). ■

Corollary 17.5. Let (16.15) and (17.15) be fulfilled so that problem (16.1), (17.11) has a unique solution. Moreover, if δ_{i} (i = 1, …, m) are nondecreasing and (16.22), (16.23), (16.25) hold, then this solution is oscillatory.

Corollary 17.6. Let p ∈]0, +∞[, α ∈]0, 1[ and

p(| ln α| + α − 1) < 1,  (17.17)
p(\sqrt{4 + ln^{2} α} − 2)e^{\frac{\sqrt{4+ln^{2} α}−| ln α|}{2}} > \frac{1}{e}.  (17.18)

Then the problem

u′′(t) = \frac{p}{(t + 1)^{2}} u(αt), u(0) = c_{0} ≠ 0, \overline{lim}_{t→+∞}|u(t)| < +∞

has a unique solution and this solution is oscillatory.

Remark 17.1. One can easily verify that there really exist p ∈]0, +∞[ and α ∈]0, 1[ satisfying both (17.17) and (17.18).

Theorem 17.7. Let (16.15) and (17.15) be fulfilled. Then problem (16.1), (17.11) has a unique proper solution. Moreover, if δ_{i} (i = 1, …, m) are nondecreasing functions and (16.23), (16.26) and 16.27 hold, then this solition is oscillatory.

Proof. The existence and uniqueness follow from Theorem 17.5. As to the oscillation, it is proved by Theorem 9.10′ using (16.23), (6.26) and (16.27). ■

Corollary 17.7. Let (16.15), (17.15) be fulfilled. Then problem (16.1), (17.11) has a unique proper solution. Moreover, this solution is oscillatory provided that (16.23), (16.26) and (16.28) hold.

Corollary 17.8. Let p ∈]0, +∞[, α ∈]0, 1[ and

\frac{1}{e} < p| ln α| < 1.

Then the problem

u′′(t) = \frac{p}{(t + 1)^{2} ln(t + 2)}u(t^{α}),

u(0) = c_{0} ≠ 0, \overline{lim}_{t→+∞}|u(t)| < +∞

has a unique solution and this solution is oscillatory.

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(Received 2.09.1994)

Author’s address: I. Vekua Institute of Applied Mathematics Tbilisi State University 2, University St., Tbilisi 380043 Republic of Georgia

CONTENTS

R. Koplatadze On Oscillatory Properties Of Solutions Of Functional Differential Equations . . . . . . . . . . . . . . . . . . . . . 1

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.1. Subject of Investigation and a Brief Survey

of the Obtained Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 0.2. Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 Chapter I. Essentially Nonlinear Equations With

Properties A and B

§1. Some Auxiliary Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1. On Some Classes of Nonoscillatory Functions . . . . . . . . . . . . 7 1.2. On Some Classes of Mappings from

C(ℝ_{+}; ℝ) into L_{loc}(ℝ_{+}; ℝ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

§2. Comparison Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1. Minorant Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2. Superposition Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

§3. Sufficient Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23 3.1. Ineffective Sufficient Conditions . . . . . . . . . . . . . . . . . . . . . . . . .23 3.2. Effective Sufficient Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 29

§4. Necessary and Sufficient Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Chapter II. Properties A And B Of Equations With

A Linear Minorant

§5. Linear Differential Inequalities with a Deviating Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.1. Auxiliary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46 5.2. On Solutions of Differential Inequalities . . . . . . . . . . . . . . . . . 51

§6. Linear Differential Inequalities with a Deviating Argument and Property A (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6.1. Equations with Property A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6.2. Equations with Property B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

§7. Equations with a Linear Minorant Having Properties A and B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74 7.1. Some Auxiliary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74 7.2. Functional Differential Equations with a Linear

Minorant Having Properties A and B . . . . . . . . . . . . . . . . . . . 79 7.3. Sufficient Conditions for the Existence

of a Nonoscillatory Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Chapter III. On Kneser-type Solutions §8. Some Auxiliary Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 §9. On the Existence of Kneser-type Solutions . . . . . . . . . . . . . . . . . . . . 98

9.1. Functional Differential Equations with Linear Minorant . 98 9.2. Linear Inequalities with Deviated Arguments . . . . . . . . . . 114 9.3. Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118

Chapter IV. Monotonically Increasing Solutions §10. Auxiliary Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 §11. On Monotonically Increasing Solutions . . . . . . . . . . . . . . . . . . . . . . . 128

11.1. Equations with a Linear Minorant . . . . . . . . . . . . . . . . . . . . . 128 11.2. Differential Inequalities with Deviating Arguments . . . . . 138 11.3. Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142

Chapter V. Specific Properties Of Functional differential equations

§12. Equations with Property Ã . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 12.1. Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144 12.2. Equations with a Linear Minorant . . . . . . . . . . . . . . . . . . . . . 145

§13. Equations with the Property B̃ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 13.1. Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .149 13.2. Equations with a Linear Minorant . . . . . . . . . . . . . . . . . . . . . 150

§14. Oscillatory Equations with a Linear Minorant . . . . . . . . . . . . . . . . 152 14.1. Equations with a Linear Minorant . . . . . . . . . . . . . . . . . . . . . 152 14.2. Equations of the Emden-Fowler Type . . . . . . . . . . . . . . . . . .153

§15. Existence of an Oscillatory Solution . . . . . . . . . . . . . . . . . . . . . . . . . .154 15.1. Existence of a Proper Solution . . . . . . . . . . . . . . . . . . . . . . . . .154 15.2. Existence of a Monotonically Increasing Solution . . . . . . .155 15.3. Existence of a Proper Oscillatory Solution . . . . . . . . . . . . . 157

Chapter VI. Second Order Delay Equations §16. On a Singular Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . 158 §17. Existence of Bounded Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171