# Oscillation Criteria for Second-Order Neutral Delay Dynamic Equations with Mixed Nonlinearities

- Ethiraju Thandapani
^{1}, - Veeraraghavan Piramanantham
^{2}and - Sandra Pinelas
^{3}Email author

**2011**:513757

https://doi.org/10.1155/2011/513757

© Ethiraju Thandapani et al. 2011

**Received: **20 September 2010

**Accepted: **23 January 2011

**Published: **23 February 2011

## Abstract

This paper is concerned with some oscillation criteria for the second order neutral delay dynamic equations with mixed nonlinearities of the form where and with . Further the results obtained here generalize and complement to the results obtained by Han et al. (2010). Examples are provided to illustrate the results.

## 1. Introduction

Since the introduction of time scale calculus by Stefan Hilger in 1988, there has been great interest in studying the qualitative behavior of dynamic equations on time scales, see, for example, [1–3] and the references cited therein. In the last few years, the research activity concerning the oscillation and nonoscillation of solutions of ordinary and neutral dynamic equations on time scales has been received considerable attention, see, for example, [4–8] and the references cited therein. Moreover the oscillatory behavior of solutions of second order differential and dynamic equations with mixed nonlinearities is discussed in [9–16].

on time scale , where , is a quotient of odd positive integers such that , , are real valued rd-continuous functions defined on such that , , and .

where is a quotient of odd positive integers, , are positive real valued rd-continuous functions on , is a nonnegative real valued rd-continuous function on and established sufficient conditions for the oscillation of all solutions of (1.2) using Ricatti transformation.

where , is a quotient of odd positive integers, , are real valued nonnegative rd-continuous functions on such that , and .

where , , , are quotients of odd positive integers such that and , , , , and are real valued rd-continuous functions on .

where , are quotients of odd positive integers such that , , , , and are real valued rd-continuous functions on .

where is a time scale, and , and this includes all the equations (1.1)–(1.5) as special cases.

By a proper solution of (1.6) on we mean a function which has a property that and satisfies (1.6) on . For the existence and uniqueness of solutions of the equations of the form (1.6), refer to the monograph [2]. As usual, we define a proper solution of (1.6) which is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is known as nonoscillatory.

Throughout the paper, we assume the following conditions:

(C_{1})the functions
are nondecreasing right-dense continuous and satisfy
,
,
with
,
, and
for
;

(C_{2})
is a nonnegative real valued rd-continuous function on
such that
;

(C_{3})
and
,
are positive real valued rd-continuous functions on
with
;

(C_{4})
,
,
are positive constants such that
.

Since we are interested in the oscillatory behavior of the solutions of (1.6), we may assume that the time scale is not bounded above, that is, we take it as .

The paper is organized as follows. In Section 2, we present some oscillation criteria for (1.6) using the averaging technique and the generalized Riccati transformation, and in Section 3, we provide some examples to illustrate the results.

## 2. Oscillation Results

In this section, we obtain some oscillation criteria for (1.6) using the following lemmas. Lemma 2.1 is an extension of Lemma 1 of [13].

Lemma 2.1.

where is a positive and delta differentiable function on .

Lemma 2.2 (see [23]).

Lemma 2.3.

Since the proof of Lemma 2.3 is similar to that of Lemma 2.1 in [6], we omit the details.

Lemma 2.4.

Proof.

which implies that is strictly increasing on . Pick so that and for . Then , and , so that for .

and we have that is strictly decreasing on .

Theorem 2.5.

for all sufficiently large where , and . Then every solution of (1.6) is oscillatory.

Proof.

which leads to a contradiction to condition (2.18). The proof is now complete.

By different choices of and , we obtain some sufficient conditions for the solutions of (1.6) to be oscillatory. For instance, , and , in Theorem 2.5, we obtain the following corollaries:

Corollary 2.6.

where is as in Theorem 2.5. Then every solution of (1.6) is oscillatory.

Corollary 2.7.

where is as in Theorem 2.5. Then every solution of (1.6) is oscillatory.

Next we establish some Philos-type oscillation criteria for (1.6).

Theorem 2.8.

where is same as in Theorem 2.5. Then every solution of (1.6) is oscillatory.

Proof.

which contradicts condition (2.35). This completes the proof.

Finally in this section we establish some oscillation criteria for (1.6) when the condition (1.8) holds.

Theorem 2.9.

where holds, then every solution of (1.6) either oscillates or converges to zero as .

Proof.

Assume to the contrary that there is a nonoscillatory solution such that , , , and for for some . From Lemma 2.3 we can easily see that either eventually or eventually.

If eventually, then the proof is the same as in Theorem 2.5, and therefore we consider the case .

*t*, it follows that the limit of exists, say . Clearly . We claim that . Otherwise, there exists such that and . From (1.6) we have

which contradicts condition (2.43). Therefore , and there exists a positive constant such that and . Since is bounded, and . Clearly . From the definition of , we find that ; hence and . This completes proof of the theorem.

Remark 2.10.

If , , or , , and , , then Theorem 2.5 reduces to a result obtained in [20] or [24], respecively. If , or , and , or , and , , then the results established here complement to the results of [5, 9, 15] respectively.

## 3. Examples

## Declarations

### Acknowledgment

The authors thank the referees for their constructive suggestions and corrections which improved the content of the paper.

## Authors’ Affiliations

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