Discrete Dynamics in Nature and Society

VolumeΒ 2010Β (2010), Article IDΒ 796256, 9 pages

http://dx.doi.org/10.1155/2010/796256

## New Oscillation Criteria for Second-Order Delay Differential Equations with Mixed Nonlinearities

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

Received 10 January 2010; Accepted 17 June 2010

Academic Editor: BinggenΒ Zhang

Copyright Β© 2010 Yuzhen Bai and Lihua Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We establish new oscillation criteria for second-order delay differential equations with mixed nonlinearities of the form , where , , , and are continuous functions defined on , and , , and . No restriction is imposed on the potentials , , and to be nonnegative. These oscillation criteria extend and improve the results given in the recent papers. An interesting example illustrating the sharpness of our results is also provided.

#### 1. Introduction

We consider the second-order delay differential equations containing mixed nonlinearities of the form

In what follows we assume that , , , and

As usual, a solution of (1.1) is called oscillatory if it is defined on some ray with and has arbitrary large zeros, otherwise, it is called nonoscillatory. Equation (1.1) is called oscillatory if all of its extendible solutions are oscillatory.

Recently, Mustafa [1] has studied the oscillatory solutions of certain forced Emden-Fowler like equations Sun and Wong [2], as well as Sun and Meng [3] have established oscillation criteria for the second-order equation Later in [4], Li and Chen have extended (1.4) to the delay differential equation As it is indicated in [2, 3], further research on the oscillation of equations of mixed type is necessary as such equations arise in mathematical modeling, for example, in the growth of bacteria population with competitive species. In this paper, we will continue in the direction to study the oscillatory properties of (1.1). We will employ the method in study of Kong in [5] and the arithmetic-geometric mean inequality (see [6]) to establish the interval oscillation criteria for the unforced (1.1) and forced (1.1), which extend and improve the known results. Our results are generalizations of the main results in [3, 4]. We also give an example to illustrate the sharpness of our main results.

#### 2. Main Results

We need the following lemma proved in [2, 3] for our subsequent discussion.

Lemma 2.1. *For any given -tuple satisfying there corresponds an n-tuple such that
**
which also satisfies either
**
or
*

For a given set of exponents satisfying Lemma 2.1 ensures the existence of an -tuple such that either (a) and (b) hold or (a) and (c) hold. When and in the first case, we have where can be any positive number satisfying This will ensure that , , and conditions (a) and (b) are satisfied. In the second case, we simply solve (a) and (c) and obtain

Following Philos [7], we say that a continuous function belongs to a function class denoted by if for and has continuous partial derivatives and in Set

Based on Lemma 2.1, we have the following interval criterion for oscillation of (1.1).

Theorem 2.2. *If, for any there exist , , , and such that **
and there exist such that
**
for where , are defined as in (2.3), are positive constants satisfying (a) and (b) in Lemma 2.1, and
**
then (1.1) is oscillatory.*

*Proof. *Let be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that for all where depends on the solution and , When is eventually negative, the proof follows the same argument by using the interval instead of Choose such that , for and

From (1.1), we have that for If not, there exists such that Because
we have Integrating from to , we obtain
Noting the assumption (1.2), we have for sufficient large This is a contradiction with From (2.7) and the conditions , , we obtain for

Employing the convexity of , we obtain

Define

Recall the arithmetic-geometric mean inequality
where and , , are chosen according to the given as in Lemma 2.1 satisfying (a) and (b). Let
We have

Multiplying both sides of (2.13) by and integrating by parts, we find that
That is,

On the other hand, multiplying both sides of (2.13) by and integrating by parts, we can easily obtain

Equations (2.15) and (2.16) yield
which contradicts (2.5) for The proof of Theorem 2.2 is complete.

*Remark 2.3. *When , the conditions for , and (1.2) can be removed. Therefore, Theorem 2.2 reduces to Theorem in [3].

*Remark 2.4. *When , Theorem 2.2 reduces to Theorem in [4] for which the conditions for and (1.2) are needed. There are some mistakes in the proof of Theorem in [4].

The following theorem gives an oscillation criterion for the unforced (1.1).

Theorem 2.5. *If, for any there exist , and such that , and for and there exists such that
**
where
** are positive constants satisfying (a) and (c) in Lemma 2.1, and , are defined as in (2.3), then the unforced (1.1) is oscillatory.*

*Proof. *Let be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that for all where depends on the solution and , Similar to the proof in Theorem 2.2, we can obtain
Define
Recall the arithmetic-geometric mean inequality
where , are chosen according to the given as in Lemma 2.1 satisfying (a) and (c). Let
We can obtain

Multiplying both sides of (2.24) by and integrating by parts, we obtain
It follows that
On the other hand, multiplying both sides of (2.24) by and integrating by parts, we have
Equations (2.26) and (2.27) yield
which contradicts (2.24). The proof of Theorem 2.5 is complete.

*Remark 2.6. *When , the conditions for , and (1.2) can be removed. Therefore, Theorem 2.5 reduces to Theorem in [3].

*Remark 2.7. *When , Theorem 2.5 reduces to Theorem in [4] for which the conditions for , and (1.2) are needed.

#### 3. Example

In this section, we provide an example to illustrate our results.

Consider the following equation: where , , are positive constants, , and Here

According to the direct computation, we have where can be any positive number satisfying and , satisfy (2.1). For any we can choose for and By simple computation, we obtain , From Theorem 2.2, we have that (3.1) is oscillatory if If by simple computation, we obtain for From Theorem 2.2, we have that (3.1) is oscillatory if

#### Acknowledgments

The authors would like to thank the referees for their valuable comments which have led to an improvement of the presentation of this paper. This project is supported by the National Natural Science Foundation of China (10771118) and STPF of University in Shandong Province of China (J09LA04).

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