BOUNDED OSCILLATION FOR SECOND-ORDER NONLINEAR DELAY DIFFERENTIAL EQUATIONS

Abstract

Two necessary and sufficient conditions for the oscillation of the bounded solutions of the second-order nonlinear delay differential equation $$(a(t)x^{\prime}(t))^{\prime}+q(t)f(x[{\tau}(t)])=0$$ are obtained by constructing the sequence of functions and using inequality technique.

1. Introduction

Consider the second-order nonlinear delay differential equation

The paper assumes the following conditions hold:

We call that x(t) ∈ C1([Tx, ∞), R) (Tx ≥ t0) is the solution of equation (1.1) if a(t)x′(t) ∈ C1([Tx, ∞). R) and x(t) satisfy (1.1) for t ∈ [Tx, ∞). We suppose that every solution of (1.1) can be extended in [t0, +∞). In any infinite interval [T, +∞), we call x(t) is a regular solution of (1.1) if x(t) is not the eventually identically zero. The regular solution of (1.1) is said to be oscillatory in case it has arbitrarily large zero point; otherwise, the solution is said to be nonoscillatory.

For the equation (1.1), if a(t) ≡ 1, the equation (1.1) becomes

For the equation (1.2), if ƒ(x) = x, τ (t) = t, and q(t) = c(t), the equation (1.2) is simplified to be the second-order linear differential equation

There are some oscillation criteria for the equation (1.3), and one of the most important criteria is given by Wintner [1] as follows: If

the equation (1.3) is oscillatory. In 1978, Kamenev [2] improved the result of Wintner. He proved that if

where λ is a constant and λ > 1, the equation (1.3) is oscillatory.

In recent years, the oscillation theory and its application of differential equations have been greatly concerned. For example, you can see the recent monographs [3-5]. In particular, the result on oscillation criteria of second-order differential equation is very rich (see [6-16]), but the most results obtained establish the sufficient condition of the oscillation for differential equations. Generally, the necessary and sufficient condition is difficult to obtain. This article discusses the oscillation for the bounded solution of the equation (1.1), and establishes two necessary and sufficient conditions of oscillation for the bounded solution of (1.1) by constructing a sequence of functions and using inequality technique.

 

2. Main results

Theorem 2.1. Suppose that (H1)−(H3) hold. Then the necessary and sufficient condition of the oscillation for every bounded solution of the equation (1.1) is

Proof. Sufficiency. Suppose that there is nonoscillatory bounded solution x(t) of the equation (1.1). Without loss of generality we assume that x(t) is eventually positive, then there exists t1(t1 ≥ t0) such that as t ≥ t1,

From the equation (1.1), we get

Thus, we can determine that

Actually, if there exists t2(t2 ≥ t1) such that

noticing a(t)x′(t) is nonincreasing, we can obtain

as t ≥ t2, i.e.

Integrating the above formula from t2 to t, by (H1), we get that

This contradicts with that x(t) is eventually positive, so (2.2) holds. Thus, for t ≥ t1, we have

Then there exists t3(t3 ≥ t2) and l(l > 0) such that when t ≥ t3,

Substituting (2.3) into equation (1.1), we get

Multiplying the both ends of the above formula by A(t) , we can get

Because

we can obtain

Substituting the above into (2.5), we get

Integrating the above formula from t3 to t, we obtain

Because x(t) is increasing and bounded, there exists C(C > 0) such that

This contradicts with the condition (2.1). The proof of sufficient section is completed.

Necessity. If

then there is T ≥ t0 such that for t ≥ T, we have

Constructing the sequence of functions such that

for k = 1, 2, · · · , Then

Suppose that

Noticing ƒ(u) is non-decreasing, then

Thus, by mathematical induction, for any positive integer k, we get

Therefore, the limit of the sequence of functions {xk(t)} exists, i.e.

and 1 ≤ x(t) ≤ 2, t ≥ τ (T): Applying Lebesgue control convergence theorem to (2.6), we can get

Derivation of the both sides of the above formula and multiplying them by a(t), we get that for t > T

Continually after derivation of the both sides of the above formula, we get

It is easy to see that x(t) is an eventually positive bounded solution of (1.1), which is contradictory to that every bounded solution of the equation (1.1) is oscillatory. The proof is completed.

Remark 2.1. For linear case of (1.1) (i.e. ƒ(x) = x), oscillation criterion relative to Theorem 2.1 has been obtained in Corollary 2 in [17].

By Theorem 2.1, we can obtain for equation (1.2)

Corollary 2.2. Suppose that (H1)−(H3) hold. Then every bounded solution of equation (1.2) is oscillatory if and only if

Corollary 2.3. Assume that (H1)−(H3) hold. Then every bounded solution of (1.1) is oscillatory if and only if

Proof. Sufficiency. Suppose that there is nonoscillatory bounded solution x(t) of the equation (1.1). Without loss of generality, we assume that x(t) is eventually positive, then there exists t1(t1 ≥ t0) such that

for t ≥ t1. Using arguments similar to ones in the proof of Theorem 2.1, we can get (2.4), i.e.

Integrating the above from t to t + v, we get

Noticing (2.2) and letting v → ∞, we get that

Integrating the above from T(T ≥ t2) to t(t ≥ T), it follows

Let t → ∞ to acquire the limits of both sides of the above. Because x(t) is bounded and increasing, it is easy to get

This is contradictory to the condition (2.8). The proof of sufficiency is completed.

Necessity. Suppose that

and there exists T(T ≥ t0) such that

for t ≥ T. Construct the sequence of functions and let

k = 1, 2, · · · . Similarly to the proof of Theorem 2.1, by the mathematical induction for any positive integer k, we have

So the limit of {xk(t)} exists, i.e.

and 1 ≤ x(t) ≤ 2, t ≥ τ (T): By Lebesgue control convergence theorem to (2.9), it follows that

Derivation of the both sides of the above and multiplying them by a(t), we get that for t > T

Continuing to take the derivatives of the both sides of the above, we can get

Thus x(t) is an eventually positive bounded solution of (1.1), which is contradictory to that every bounded solution of the equation (1.1) is oscillatory. The proof is completed.

By Theorem 2.3, we can get for equation (1.2).

Corollary 2.4. Suppose that (H1)−(

Example 2.5. Consider second-order linear differential equation

Here

The conditions (H1)-(H3), (2.1) and (2.8) are clearly satisfied. Altogether, by Theorems 2.1 and 2.3, every bounded solution of the equation (2.11) is oscillatory. In fact, x(t) = cos ln t is a bounded oscillatory solution of the equation (2.11).