Mem. Differential Equations Math. Phys. 13(1998), 143–147

R. Koplatadze

COMPARISON THEOREMS FOR DEVIATED DIFFERENTIAL EQUATIONS WITH PROPERTY B

(Reported on September 21, 1998)

1. Comparrison Theorems

Consider the deviated differential equations

u^{(n)}(t) − p(t) u(τ(t)) = 0,  (1.1)
v^{(n)}(t) − q(t) v(σ(t)) = 0,  (1.2)

where p, q ∈ L_{loc}(R_{+}; R_{+}); τ, σ ∈ C(R_{+}; R_{+}), lim_{t→+∞} τ(t) = lim_{t→+∞} σ(t) = +∞ and σ, τ

are nondecreasing.

Definition 1.1. We say that the equation (1.1) has Property B if any of its proper solutions either is oscillatory or satisfies

|u^{(i)}(t)| ↑ +∞ for t ↑ +∞ (i = 0, …, n − 1),  (1.3)

when n is odd, and either is oscillatory or satisfies either (1.3) or

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

when n is even.

For ordinary differential equations T. Chanturia [1] obtained integral comparison theorems. In particular, he proved the following

Theorem 1.1 (T. Chanturia). Let τ(t) ≡ σ(t) ≡ t for t ∈ R_{+}. Let

∫_{t}^{+∞}s^{n−2}p(s) ds ≥ ∫_{t}^{+∞}s^{n−2}q(s) ds for t ∈ R_{+}

be fulfilled if n is even, while

∫_{t}^{+∞}s^{n−3}p(s) ds ≥ ∫_{t}^{+∞}s^{n−3}q(s) ds for t ∈ R_{+}

be fulfilled if n is odd. Let, moreover, the equation (1.2) have Property B. Then the equation (1.1) also has Property B.

Basing on this theorem, T. Chanturia obtained the following effective criterion for an ordinary differential equation to have Property B.

1991 Mathematics Subject Classification. 34K15.

Key words and phrases. Advanced differential equation, Property B, comparison theorem.

Theorem 1.2 (T. Chanturia). Let τ(t) ≡ σ(t) ≡ t for t ∈ R_{+}. Let, moreover,

lim inf_{t→+∞} t ∫_{t}^{+∞}s^{n−2}p(s) ds > M_{n}^{∗}

be fulfilled if n is odd, while

lim inf_{t→+∞} t^{2} ∫_{t}^{+∞}s^{n−3}p(s) ds > \frac{1}{2} M_{n}^{∗}

be fulfilled if n is even, where M_{n}^{∗} is the greater among the local maxima of the polynomial x(x − 1) ⋯ (x − n + 1). Then the equation (1.1) has Property B.

Below we formulate integral comparison theorems concerning Property B for the deviated differential equations (1.1) and (1.2). They enable us, using some results from [2], to obtain new effective criteia for (1.1) to have Property B. Analogous results for Property A are given in [3].

Theorem 1.3. Let σ(t) ≤ τ(t) ≤ t and

∫_{0}^{+∞}τ^{n−1}(t)p(t) dt = +∞.^{∗}  (1.4)

Let

∫_{t}^{+∞}sτ^{n−3}(s)p(s) ds ≥ ∫_{t}^{+∞}sσ^{n−3}(s)q(s) ds for t ≥ t_{0}  (1.5)

be fulfilled if n is even, while (1.5) and

∫_{t}^{+∞}s^{n−2}p(s) ds ≥ ∫_{t}^{+∞}s^{n−2}q(s) ds for t ≥ t_{0}  (1.6)

be fulfilled if n is odd. Let, moreover, the equation (1.2) have Property B. Then the equation (1.1) also has Property B.

Remark. It is clear that in the case where τ(t) ≡ σ(t) ≡ t and n is even, Theorem 1.3 is a refinement of Theorem 1.1.

Theorem 1.4. Let τ(t) ≥ max{t, σ(t)} and

∫_{0}^{+∞}t^{n−1}p(t) dt = +∞.  (1.7)

Let

∫_{t}^{+∞}s^{n−3}τ(s)p(s) ds ≥ ∫_{t}^{+∞}s^{n−3}σ(s)q(s) ds for t ≥ t_{0}

∗As is known, (1.4) is necessary for (1.1) to have Property B. be fulfilled if n is even, while (1.6) with t_{0} ∈ R_{+} be fulfilled if n is odd. Let, moreover, the equation (1.2) have Property B. Then the equation (1.1) also has Property B.

Theorem 1.5. Let τ(t) ≤ min{t, σ(t)}, (1.4) hold and for any sufficiently large t_{0} t_{1} = t_{1}(t_{0}) ≥ t_{0} can be found such that

∫_{t_{0}}^{t} s^{2}τ^{n−2}(s)p(s) ds ≥ ∫_{t_{0}}^{t} s^{2}σ^{n−2}(s)q(s) ds for t ≥ t_{1}  (1.8)

be fulfilled if n is even, while (1.8) and

∫_{t_{0}}^{t} s^{n−1}τ(s)p(s) ds ≥ ∫_{t_{0}}^{t} s^{n−1}σ(s)q(s) ds for t ≥ t_{1}

be fulfilled if n is odd. Let, moreover, the equation (1.2) have Property B. Then the equation (1.1) also has Property B.

Theorem 1.6. Let t ≤ τ(t) ≤ σ(t), (1.7) hold and for any sufficiently large t_{0} t_{1} = t_{1}(t_{0}) ≥ t_{0} can be found such that

∫_{t_{0}}^{t} s^{n−2}τ^{2}(s)p(s) ds ≥ ∫_{t_{0}}^{t} s^{n−2}σ^{2}(s)q(s) ds for t ≥ t_{1}

be fulfilled if n is even, while

∫_{t_{0}}^{t} s^{n−1}τ(s)p(s) ds ≥ ∫_{t_{0}}^{t} s^{n−1}σ(s)q(s) ds for t ≥ t_{1}

be fulfilled if n is odd. Let, moreover, the equation (1.2) have Property B. Then the equation (1.1) also has Property B.

2. Effective Criteria for Property B

Let l ∈ {1, …, n − 1}, α ∈]0, +∞[. Denote

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

Theorem 2.1. Let α ∈]0, 1] and αt ≤ τ(t) ≤ t for t ∈ R_{+}. Let

lim inf_{t→+∞} t ∫_{t}^{+∞}sτ^{n−3}(s)p(s) ds > α^{n−3}M_{n−2}^{∗}(α)  (2.1)

be fulfilled if n is even, while (2.1) and

lim inf_{t→+∞} t ∫_{t}^{+∞}s^{n−2}p(s) ds > M_{1}^{∗}(α)

be fulfilled if n is odd. Then the equation (1.1) has Propetry B.

Remark. Even in the case of ordinary differential equations (τ(t) ≡ t), Theorem 2.1 is a refinement of Theorem 1.2.

Theorem 2.2. Let α ∈ [1, +∞[ and τ(t) ≥ αt. Let (1.7) and

lim inf_{t→+∞} t ∫_{t}^{+∞}s^{n−3}τ(s)p(s) ds > αM_{2}^{∗}(α)

be fulfilled if n is even, while

lim inf_{t→+∞} t ∫_{t}^{+∞}s^{n−2}p(s) ds > M_{1}^{∗}(α)

be fulfilled if n is odd. Then the equation (1.1) has Property B.

Theorem 2.3. Let α ∈]0, 1], τ(t) ≤ αt for t ∈ R_{+} and (1.7) hold. Let

lim inf_{t→+∞} t^{−1} ∫_{0}^{t} s^{2}τ^{n−2}(s)p(s) ds > α^{n−2}M_{n−2}^{∗}(α)  (2.2)

be fulfilled if n is even, while

lim inf_{t→+∞} t^{−1} ∫_{0}^{t} s^{n−1}τ(s)p(s) ds > αM_{1}^{∗}(α)

be fulfilled if n is odd. Then the equation (1.1) has Property B.

Theorem 2.4. Let α ∈ [1, +∞[, t ≤ τ(t) ≤ αt. Let

lim inf_{t→+∞} t^{−1} ∫_{0}^{t} s^{n−2}τ^{2}(s)p(s) ds > α^{2}M_{2}^{∗}(α)

be fulfilled if n is even, while

lim inf_{t→+∞} t^{−1} ∫_{0}^{t} s^{n−1}τ(s)p(s) ds > αM_{1}^{∗}(α)

be fulfilled if n is odd. Then the equation (1.1) has Property B.

Remark. If in Theorems 2.1–2.4 one replace the strict inequalities with the nonstrict ones, they will cease to be true. More than that, none of the hypotheses of Theorems 2.1–2.4 can be omitted without affecting their validity.

Theorem 2.5. Let σ(t) ≤ τ(t) ≤ t for t ∈ R_{+}, \frac{σ(t)}{t} ↓ 0 as t ↑ +∞, (1.4) hold and

∫^{+∞}\frac{dt}{tσ(t)} < +∞, lim inf_{t→+∞} φ(t) ∫_{t}^{+∞}sτ^{n−3}(s)p(s) ds > 0,

where φ(t) = ( ∫_{t}^{+∞} \frac{ds}{sσ(s)})^{−1}. Then the equation (1.1) has Property B.

Corollary 2.1. Let α ∈]0, 1[, t^{α} ≤ τ(t) ≤ t for t ∈ R_{+}, (1.4) hold and

lim inf_{t→+∞} t^{α} ∫_{t}^{+∞}sτ^{n−3}(s)p(s)ds > 0.

Then the equation (1.1) has Property B.

Theorem 2.6. Let t ≤ τ(t) ≤ σ(t), \frac{σ(t)}{t} ↑ +∞ for t ↑ +∞ and (1.7) hold. Let, moreover,

lim inf_{t→+∞} φ(t) ∫_{0}^{t} s^{n−2}τ^{2}(s)p(s) ds > 0

if n is even, while

lim inf_{t→+∞} φ(t) ∫_{0}^{t} s^{n−1}τ(s)p(s) ds > 0

if n is odd, where φ(t) = ( ∫_{0}^{t} \frac{σ(s)}{s} ds)^{−1}. Then the equation (1.1) has Property B.

Corollary 2.2. Let α ∈]1, +∞[, t ≤ τ(t) ≤ t^{α} and (1.7) hold. Let, moreover,

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

be fulfilled if n is even, while

lim inf_{t→+∞} t^{−α} ∫_{0}^{t} s^{n−1}τ(s)p(s) ds > 0

be fulfilled if n is even. Then the equation (1.1) has Property B.

Acknowledgement

This work was supported by Grant No. 1.6/1998 of the Georgian Academy of Sciences.

References

1. I. Kiguradze and T. Chanturia, Asymptotic properties of solutions of nonautonomous ordinary differential equations. Kluwer Academic Publishers, Dordrecht–Boston–London, 1993.

2. R. Koplatadze, On oscillatory properties of solutions of functional differential equations. Mem. Differental Equations Math. Phys. 3(1994), 1–177.

3. R. Koplatadze, Comparison theorems for deviated differential equations with Property A. Mem. Differental Equations Math. Phys. 15(1998), 141–144.

Author’s address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, M. Aleksidze St., Tbilisi 380093

Georgia