Mem. Differential Equations Math. Phys. 20(2000), 145–148

L. Kokilashvili

ON A BOUNDARY VALUE PROBLEM WITH A WEIGHTED CONDITION AT INFINITY FOR EVEN ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

(Reported on November 22, 1999)

Consider the nonlinear ordinary differential equation

u^{(n)} + ∑_{k=1}^{n−1} p_{k}(t)u^{(k)} = f(t, u, u′, …, u^{(n−1)})  (1)

on the infinite interval [a, +∞[, where n ≥ 2 is an even number, 1 < a < +∞, each of the functions p_{k} : [a, +∞[→ ℝ for k ∈ {1, …, n − 1} is locally absolutely continuous together with its derivatives up to the order k−1 inclusive (i.e., the functions p_{k}^{(i)} (i = 0, …, k−1) are absolutely continuous on any finite segment contained in [a, +∞[), and the function f : [a, +∞[×ℝ^{n} → ℝ satisfies the local Carathéodory conditions.

Let ν < −1, n_{0} = \frac{n}{2} . Consider the problem on the existence of a solution of the equation (1) satisfying the boundary conditions

\begin{matrix}u^{(i)}(a) = φ_{i}(u(a), u′(a), …, u^{(n−1)}(a)) (i = 0, …, n_{0} − 1), \\ ∫_{a}^{+∞}t^{ν}|u^{(j)}(t)|^{2}dt < +∞ (j = 0, … n_{0}), \end{matrix}  (2)

where the functions φ_{i} : ℝ^{n} → ℝ (i = 0, …, n_{0} − 1) are continuous and satisfy the condition

∑_{i=0}^{n_{0}−1} |φ_{i}(x_{0}, x_{1}, …, x_{n−1})| ≤ c_{1}(1 + ∑_{i=0}^{n_{0}−1} |x_{i}|)^{−ϑ}  (3)

on ℝ^{n}, where c > 0 and ϑ ∈ [0, 1].

In the case where p_{k}(t) ≡ 0, problems of such type were investigated by I. Kiguradze [1]. The author has recently studied the case where ν = 0 (see [5]). Our interest to the problem (1), (2) is two-fold. First, to supplement the results of [1] and generalize those of [5] which correspond to the case of an even n. Second, to supplement in certain cases some results appearing in the qualitative theory.

Below the use will be made of the following notation:

ℝ is the set of real numbers;

ℝ^{n} the n-dimensional Euclidean space;

μ_{i}^{k}(i = 1, 2, …; k = 2i, 2i+1, …) are real constants defined by the recurrence relation

μ_{0}^{i+1} = 1/2, μ_{i}^{2i} = 1, μ_{i+1}^{k} = μ_{i+1}^{k−1} + μ_{i}^{k−2} (i = 0, 1, …; k = 2i + 3, …).

2000 Mathematics Subject Classification. 34B40.

Key words and phrases. Nonlinear ordinary differential equation, boundary value problem, weighted condition at infinity, solvability.

Everywhere below it will be assumed that the function f satisfies the condition

|f(t, x_{0}, x_{1}, …, x_{n−1})| ≤ h(t, |x_{0}|, |x_{1}|, …, |x_{n_{0}−1}|)  (4)

on [a, +∞[×ℝ^{n}, where the function h : [a, +∞[×ℝ_{+}^{n_{0}} → ℝ_{+} is locally summable in the first argument, nondecreasing in the last n_{0} arguments, and for any ρ_{0} > 0 satisfies the condition

lim sup_{\begin{matrix}t→a \\ ρ→+∞ \end{matrix}} \frac{1}{ρ^{2}} | ∫_{a}^{t} h(τ, ρ_{0}, ρ, …, ρ) dτ|^{1−ϑ} < +∞.  (5)

Theorem 1. Let the inequality

(−1)^{n_{0}−1}f(t, x_{0}, x_{1}, …, x_{n−1}) sgn x_{0} ≥ − ∑_{i=0}^{n_{0}−1} α_{i}(t)|x_{i}| − α(t)

hold on [a, +∞[×ℝ^{n}, where the function α_{0} : [a, +∞[→ ℝ and α_{i} : [a, +∞[→ ℝ_{+} (i = 1, …, n_{0} − 1) are locally summable, and α : [a, +∞[→ ℝ is measurable and satisfies the condition

∫_{a}^{+∞}t^{ν}α^{2}(t)dt < +∞.

Moreover, let there exist the constants γ_{i} ≥ 0 (i = 1, …, n_{0} − 1), δ > 0 and η > max{[ν(ν − 1)]^{n_{0}−1}, 2^{(n_{0}−1)(n_{0}−2)}} satisfying

1 − ∑_{i=1}^{n_{0}−1} γ_{i} ∏_{k=i}^{n_{0}−1} η^{\frac{1}{k}} ≥ δ

and such that the inequalities

t^{−ν} ∑_{k=2i}^{n}(−1)^{n+k−i−1}μ_{i}^{k}[t^{ν}p_{k}(t)]^{(k−2i)} + α_{i}(t) ≤ γ_{i} (i = 1, …, n_{0} − 1),

t^{−ν} ∑_{k=1}^{n}(−1)^{n_{0}+k}μ_{0}^{k}[t^{ν}p_{k}(t)]^{(k)} − ∑_{i=0}^{n_{0}−1} α_{i}(t) ≥ ∑_{i=1}^{n_{0}−1} γ_{i}( ∑_{k=i}^{n_{0}−1} 2^{k−1}η^{1+ \frac{1}{2} +···+ \frac{1}{k}} ) + δ

hold on [a, +∞[, where p_{n}(t) ≡ 0. Then the problem (1), (2) is solvable.

Corollary 1. Let the inequality (−1)^{n_{0}−1}f(t, x_{0}, x_{1}, …, x_{n−1}) sgn x_{0} ≥ γ(t)|x_{0}|^{λ} be fulfilled on [a, +∞[×ℝ^{n}, where λ > 1 and the function γ : [a, +∞[→]0, +∞[ is measurable and satisfies the condition

∫_{a}^{+∞}t^{ν}[γ(t)]^{− \frac{2}{λ−1}} dt < +∞.

Moreover, let there exist a constant r > 0 such that the inequalities

t^{−ν} ∑_{k=2i}^{n}(−1)^{n_{0}+k−i−1}μ_{i}^{k}[t^{ν}p_{k}(t)]^{(k−2i)} < r (i = 1, …, n_{0} − 1),

t^{−ν} ∑_{k=1}^{n}(−1)^{n_{0}+k}μ_{0}^{k}[t^{ν}p_{k}(t)]^{(k)} > −r

hold on [a, +∞[, where p_{n}(t) ≡ 1. Then the problem (1), (2) is solvable.

From these results and also from the existence of a so-called proper solution of (1) (i.e., a nontrivial solution of (1) defined in some neighbourhood at infinity) we obtain its asymptotic behaviour.

Theorem 2. Assume that all the hypotheses of Theorem 1 (Corollary 1) hold. Then for arbitrary continuous functions φ_{i}(i = 0, …, n_{0} −1) satisfying (3) on ℝ^{n}, where c > 0 and ϑ ∈ [0, 1], there exists at least one proper solution of the equation (1) satisfying the initial conditions

u^{(i)}(a) = φ_{i}(u(a), u′(a), …, u^{(n−1)}(a)) (i = 0, …, n_{0} − 1),  (6)

and possessing the following asymptotic property:

lim_{t→+∞} t^{\frac{ν}{2}} |u^{(i)}(t)| = 0 (i = 0, …, n_{0} − 1).  (7)

Corollary 2. Assume that all the hypotheses of Corollary 1, except that of the restriction (5), are satisfied. Then the equation (1) has an n_{0}-parametric family of proper solutions possessing the asymptotic property (7).

These results generalize those obtained in [5] and complement the results of [1] concerning the case of even n.

On the other hand, Theorem 2 provides us with sufficient conditions on the existence of proper oscillatory (i.e., having a sequence of zeros converging at infinity) solutions of the equation

u^{(n)} + u^{(n−2)} = f(t, u, u′, …, u^{(n−1)})  (1_{1})

which appear from qualitative theory (see Corollary 1.1[2], p. 208). Therefore, the result below fills in a certain way the gap in [4] (see Theorem 2, p. 39).

Corollary 3. Let n = 2n_{0} ≥ 4, and along with (4) let the inequality

γt^{μ}|x_{0}|^{λ} ≤ (−1)^{n_{0}−1}f(t, x_{0}, x_{1}, …, x_{n−1}) sgn x_{0}

hold on [a, +∞[×ℝ^{n}, where λ > 1, μ ≥ 2−n, γ > 0, the function h : [a, +∞[×ℝ_{+}^{n_{0}} → ℝ_{+} is locally summable in the first argument, nondecreasing in the last n_{0} arguments and for any ρ_{0} > 0 satisfies (5), where c ≥ 0 and ϑ ∈ [0, 1]. Then for arbitrary continuous functions φ_{i} : ℝ^{n} → ℝ (i = 0, …, n_{0} − 1) satisfying (3) on ℝ^{n} there exists at least one proper solution of the problem (1_{1}), (6) such that

lim_{t→+∞} t^{\frac{μ}{λ−1} − \frac{1+ε}{2}} |u^{(i)}(t)| = 0 (i = 0, …, n_{0} − 1).

Moreover, in the case n_{0} is even, every proper solution is oscillatory.

The last result is new even for the Emden-Fowler type equation

u^{(n)} + u^{(n−2)} = p(t)|u|^{λ} sgn u  (1_{2})

and gives an answer to some open problems of the oscillation theory. For example, as early as in 1992, I. Kiguradze [2] (see Corollary 1. 6) proved that if n ≥ 4 is even, λ > 1 and p : [0, +∞[→] − ∞, 0] is locally summable, then the condition

∫_{0}^{+∞}t^{n−3}p(t)dt = −∞

is necessary and sufficient for every proper solution of (1_{2}) to be oscillatory. However, the question on the existence of at least one proper solution of (1_{2}) remained open. Clearly, Corollary 3 implies

Corollary 4. Let n = 2n_{0}, n_{0} be even, and let the inequality p(t) ≤ −γt^{2−n} on [a, +∞[, where γ > 0. Then the equation (1_{2}) has an n_{0}-parametric family of proper oscillatory solutions.

Finally, we consider the generalized Emden-Fowler equation

u^{(n)} + ∑_{k=0}^{n−1} p_{k}(t)u^{(k)} = −δ(t)|u|^{λ} sgn u,  (8)

where n ≥ 2, λ > 1 and δ : [a, +∞[→ [0, +∞[ is measurable. From the above reasoning we answer the question on the existence of proper solutions of (8). Moreover, using a result of T. Chanturia (see Theorem 1.9 in [3], p. 50), we obtain the following sufficient conditions for the existence of proper oscillatory solutions of (8).

Corollary 5. Let n = 2n_{0} and n_{0} > 1 be even. Assume that all hypotheses of Corollary 1 are satisfied. Moreover, let the functions p_{k} : [a, +∞[→ ℝ (k = 0, …, n − 1) be summable and the condition

lim inf_{t→+∞} p_{0}(t) > 0

be fulfilled. Then for arbitrary continuous functions φ_{i} : ℝ^{n} → ℝ (i = 0, …, n_{0} − 1) satisfying (3) on ℝ^{n}, there exists at least one proper oscillatory solution of (8) satisfying the initial condition (6) and possessing the asymptotic property (7).

References

1. I. Kiguradze, On a boundary value problem with a condition at infinity for higher order ordinary differential equations. (Russian) In: Trudy Vsesojuznigo Simposiuma po Diff. Uravnenijam v Chastnykh Proizvodnykh, Tbilis. Univ. Press, Tbilisi, 1986, 91–105.

2. I. Kiguradze, An oscillation criterion for a class of ordinary differential equations. (Russian) Differentsial’nye Uravneniya 28(1992), No. 2, 207–219.

3. I. Kiguradze and T. Chanturia, Asymptotic properties of solutions of nonautonomous ordinary differential equations. Kluwer Academic Publishers, Dordrecht-Bos-ton-London, 1992.

4. L. Kokilashvili, On the existence of proper solutions of high order nonlinear ordinary differential equations. Fasc. Math. 26(1996), 37–60.

5. L. Kokilashvili, On a certain boundary value problem for nonlinear ordinary differential equations. Mem. Differential Equations Math. Phys. 13(1998), 145–147.

Author’s address:

Faculty of Mechanics and Mathematics

I. Javakhishvili Tbilisi State University

2, University St., Tbilisi 380043

Georgia