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Abstract and Applied Analysis
Volume 2009, Article ID 897058, 12 pages
http://dx.doi.org/10.1155/2009/897058
Research Article

Oscillation Criteria for a Class of Second-Order Nonlinear Differential Equations with Damping Term

1Center of Nuclear Energy Economy and Management, University of South China, Hengyang 421001, China
2School of Mathematics and Physics, University of South China, Hengyang 421001, China

Received 31 August 2009; Accepted 29 September 2009

Academic Editor: Yong Zhou

Copyright © 2009 Zigen Ouyang et al. 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

A class of second-order nonlinear differential equations with damping term are investigated in this paper. By using a new method, we obtain some new sufficient conditions for the oscillation of the above equation, and some references are extended in this paper. Examples are inserted to illustrate this result.

1. Introduction

Consider the following second-order nonlinear differential equations with damping term: where , , is a positive constant, and is a continuous real-valued function on the real line and satisfies for . We restrict our attention to those solutions of (1.1) which exist on some half line and satisfy for any .

Recently, there are many authors who have investigated the oscillation for second-order differential equations [19], Li [10] and Zhao investigated oscillation criteria for the following equation: where is a quotient of odd positive integer. It is obvious that (1.2) is a special case of (1.1). In fact, the conditions of Theorem in [10] are too complex.

More recently, Rogovchenko and Tuncay [11] have obtained oscillation criteria of the following:

Motivated by the above discussions, we investigate the oscillation of (1.1) in this paper; our oscillatory conditions and the proof of the main results are more simple than those of Theorem in [10].

A solution of (1.1) is oscillatory if and only if it has arbitrarily large zeros; otherwise, it is nonoscillatory. Equation (1.1) is oscillatory if and only if every solution of (1.1) is oscillatory.

The paper is arranged as follows. In Section 2, we will establish our main results. Finally, examples are given to illustrate our results.

2. Main Results

To obtain our results, we introduce a lemma as follows.

Lemma 2.1 (see [2, 3]). Let the function be such that for each fixed , , the function is nondecreasing. Further, let be a given function and satisfies that and is the minimal (maximal) solution of Then for all .

Now, we give our main results.

Theorem 2.2. Assume that , , and hold. Suppose that there exists a positive function such that Then every solution of (1.1) is oscillatory.

Proof. Assume that (1.1) has a nonoscillatory solution . Without loss of generality, suppose that it is an eventually positive solution (if it is an eventually negative solution, the proof is similar), that is, for all .
We consider the following three cases.
Case 1. Suppose that is oscillatory. Then there exists such that . From (1.1), we have which means that it follows that for all , which contradicts to the assumption that is oscillatory.Case 2. Suppose that . From (1.1), we obtain then there exists an and a , such that it follows which means that , this contradicts the assumption that .Case 3. Suppose that . Let , then in view of (1.1), we obtain noticing that integrating the above from to , we get Using (2.3), (2.4), and , we have this is a contradiction, the proof is complete.

Remark 2.3. If we replace , by , , , Theorem 2.2 holds also.

Theorem 2.4. Assume that holds. Suppose also that and such that (2.3) holds. Then every solution of (1.1) is oscillatory.

Proof. To the contrary, (1.1) has a nonoscillatory solution . Without loss of generality, we assume that is an eventually positive solution. Let , then for and in view of (1.1) and (2.15), we obtain since integrating the above from to , we have In view of (2.3), there exists a constant and such that which means that Because that is positive, then (2.22) implies , or equivalently . Let thus (2.22) can be changed as Define Then, for any fixed and , is nondecreasing in . Let be the minimal solution of the equation Applying Lemma 2.1, we obtain Dividing both sides of (2.26) by and deriving both sides of (2.26), it follows On the other hand, Combining (2.28) and (2.29), it means So , . From (2.27), we obtain Integrating both sides of the above from to , we have Letting in (2.32), and using (2.16), it follows that , which contradicts to that is eventually positive. The proof is complete.

In the following, we always suppose that and it satisfies the following two conditions:

() for , is a bounded function;(), is a bounded function.

Theorem 2.5. Assume that , hold, and or Suppose further that there exists a function that satisfies (), (), and such that where Then every solution of (1.1) is oscillatory.

Proof. For the sake of contradiction, (1.1) has a nonoscillatory solution . Without loss of generality, we may assume that for all .
Define Deriving (2.39), we get Multiplying (2.40) by , it follows We consider the following three cases.
Case 1 ( is oscillatory). Then there exists a sequence , as and such that , Integrating both sides of (2.41) from to , we obtain that is which contradicts (2.35).Case 2 ( is eventually positive). Integrating both sides of (2.41) from to , we obtain which also contradicts to (2.35).Case 3 ( is eventually negative). If , then there exists a sequence , , that satisfies as and such that . Because is a bounded function, then there exists a such that , According to (2.41), we obtain Using (2.35) and taking limit as , it is easy to show that which is obviously a contradiction.
If , . From the definition of , combining (2.36) and (2.39), it follows that and , which means that . Owing to , , or , and , using the similar method of the proof of Case 2 in Theorem 2.2, we will derive a contradiction. Then the proof is complete.

Theorem 2.6. Assume that (2.36) holds, , , and or Suppose further that there exists a function that satisfies (), () and such that where and is defined in (2.38). Then every solution of (1.1) is oscillatory.

Proof. For the sake of contradiction, (1.1) has a nonoscillatory solution. Without loss of generality, we may assume that (1.1) has an eventually positive solution (if it has an eventually negative solution, the proof is similar), then there exists a such that for all . Define The rest of the proof is similar to Theorem 2.5. The proof is complete.

Theorem 2.7. Assume that (2.36) holds, , and or Suppose further that there exists a function that satisfies (), () and such that where where is defined in (2.38). Then every solution of (1.1) is oscillatory.

Proof. For the sake of contradiction, (1.1) has a nonoscillatory solution . Without loss of generality, we may assume that for all .
Define Noting that for , so for . Deriving (2.56), we obtain Multiplying (2.57) by , we get The rest of the proof is similar to Theorem 2.5; the proof is complete.

Theorem 2.8. Assume that (2.36) holds, , and or Suppose further that there exists a function satisfies (), () and such that where where is defined in (2.38). Then every solution of (1.1) is oscillatory.

Proof. For the sake of contradiction, (1.1) has a nonoscillatory solution . Without loss of generality, we may assume that for all .
Define The rest of the proof is similar to Theorem 2.5; the proof is complete.

3. Examples

Example 3.1. Consider the following delay differential equation: It is obvious that , , , , and . It is difficult to distinguish whether every solution of (3.1) is oscillatory by Theorem  3.2 of [10].
By taking , then From Theorem 2.2 or Theorem 2.4, it is easy to show that (3.1) is oscillatory.
In fact, is such an oscillatory solution.

Example 3.2. Consider the following differential equation: It is obvious that , , , , and .
We are taking , . By a simple calculation, it is easy to show that , , , and .
From Theorem 2.5 or Theorem 2.6, it follows that (3.3) is oscillatory.
In fact, is such an oscillatory solution.

Acknowledgments

This work was supported by the Natural Science Foundation of Hunan Province under Grant no. 07JJ3130, the Doctor Foundation of University of South China under Grant no. 5-XQD-2006-9, and the Subject Lead Foundation of University of South China under Grant no. 2007XQD13.

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