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International Journal of Differential Equations
VolumeΒ 2009, Article IDΒ 714357, 15 pages
http://dx.doi.org/10.1155/2009/714357
Research Article

Oscillation Theorems for Second-Order Damped Nonlinear Differential Equations

1Institute of Applied Mathematics, Shandong University of Technology, Zibo, Shandong 255049, China
2Collage of Mechanical and Electronic Engineering, Shandong University of Science and Technology, Qindao 266510, China

Received 26 September 2008; Revised 28 January 2009; Accepted 23 March 2009

Academic Editor: QingkaiΒ Kong

Copyright Β© 2009 Hui-Zeng Qin and Yongsheng Ren. 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 present new oscillation criteria for the differential equation of the form , where , . Our research is different from most known ones in the sense that H function is not employed in our results, though Riccati's substitution and its generalized forms are used. Our criteria which are established under quite general assumptions are an extension for previous results. In particular, by taking , the above-mentioned equation can be reduced into the various types of equations concerned by people currently.

1. Introduction

The existence of the oscillatory solutions of the nonlinear differential equation with damping,

has received considerable attention from researchers for a long time.

People previously focused on the cases . In recent years, people concerned that may change sign for , regarding work in this area can be seen in literature [1–4].

Recently, Li [5] has extended (1.1) to more general equations of the form

Yamaoka [6] has studied the following class of particular equation:

Tiryaki and Zafer [1] and other authors [2, 7] have considered the following equation of the form

Zheng [8] has discussed the oscillation problem for the following equation:

It is worth noting that (1.1), (1.2), and (1.3) can transform into an undamping equation. For example, the equation

can transform into the undamping equation

where . Although (1.4) and (1.5) can not be transformed into the undamping equation, but from the conditions given by [1, 8, 9], if is changed into or , the above-mentioned equations consistent with (1.6). This shows that under the above conditions, there is no essential difference between (1.4), (1.5), and the undamping equation. We note that the condition must be used for (1.5); however, at this point the condition cannot be guaranteed.

We have removed the condition , considered the oscillation problem for the following equation: applied the results to the above-mentioned equation, and obtained a very good result.

In this paper, we consider the oscillatory behavior of the following differential equation of the form:

where

Today, Riccati transformation, and its generalized forms are one of the most effective method in the oscillatory theory of nonlinear differential equations. Most obvious merits of Riccati's approach is that may change sign in (1.8). For getting the more general results [10, 11], a lot of authors have introduced to a class of Y function

where exists on E and is integral with respect to s. By using this method, peoples have obtained some general results, but its shortcoming is that the property of can be weakened as . We use the method similar to [4], that is, replace the above-mentioned function with . Perhaps the reason that people like to use this method is that integrating by parts with respect to s on can employ .

For (1.9), we make the following assumptions:

(A);(B);(C).

In the paper, a solution of (1.9) is called oscillatory if it has zeros unbounded set. If the solutions are oscillatory, (1.9) is called to be oscillatory equation.

2. Main Theorem

We establish some lemmas which are useful in our discussions.

Lemma 2.1. Let , then

Lemma 2.1 can easily be proved by using the extremum of one variable function. For the sake of convenience, we denote

Theorem 2.2. Assume that holds and there exists , such that If any one of the following two conditions holds, then the solution of (1.9) is oscillatory.
(1)
(2), and

Proof. Let be a nonoscillatory solution of (1.9). Then, there exists such that . Without loss of generality, we may assume that
Define the Riccati Transformation by From conditions (A) and (B), we have Differentiating and applying (1.9) and (2.10), we have
By (2.1), we have Integrating the above inequality from to , we have
Condition (2.4) shows that Without loss of generality, we may assume that , by applying (2.11), we have
Integrating the above inequality from to , we obtain Let We will discuss in the following two cases.
(1) By (2.1) and (2.5), we see that Integrating the above inequality from to , we have But, it is impossible that the above inequality holds.
(2) Observe that and by (2.8), we have so that is monotonic decreasing function for , and if , then Otherwise, if , by (2.15) and (2.9), we have By condition (B), we have Integrating the above inequality from to leads to But, this is impossible. We choose , thus (2.15) has the following form:
When , by considering inequality and (2.10), we have such that However, this is also impossible.
If , then by (2.8), we see that From (2.15), we obtain Integrating the above inequality from to , we have Let ; the above inequality contradicts (2.8); this completes the proof.

Theorem 2.3. Suppose that and If there exists such that where then every solution of (1.9) is oscillatory.

Note
From (2.27), it is easy to obtain the following equation:

Proof. Let be a nonoscillatory solution of (1.9). Then, there exists such that We may assume that
Introduce the Riccati transformation . From conditions (A) and (B), we have Differentiating , and applying (1.9) and the above inequality, leads to By (2.1), we see that The following proof is similar to that in Theorem 2.2, using (2.26), we find that Thus there exists , such that Because is monotonic decreasing function on ; hence By (2.31), we have
By using of weighted mean inequality, we can transform the above inequality into We need to show that Otherwise, if by the above inequality, we have By choosing , such that , integrating the above inequality from to and by (2.27), we can get or Differentiating the above inequality on the interval , we have This is a contradiction to (2.29); hence, we have . According to the above discussion, we have We will discuss in the following two cases.
If there exists such that , by considering inequality and (2.39), we have By choosing , such that , from the above inequality, we have . Inserting it in (2.40), we can get This is contradiction to (2.29).
If for , along with (2.39) and (2.30), we have leading to Integrating the above inequality from to and by (2.40), we can get or Integrating the above inequality on the interval , we have
If the above inequality cannot be satisfied; hence, Inserting it in (2.40), we can get This is contradiction to (2.26).
Hence, we complete the proof of Theorem 2.3.

3. Some Examples

Example 3.1. Let us consider the oscillatory behavior of the following differential equation:
Comparing (3.1) with (1.9), we can find that
Let , (2.4)–(2.7) are transformed into the equations We will discuss inthe following cases.
(1) choosing , provided that , or is satisfied for , then (3.3)–(3.5) hold, and the solution of (3.1) is oscillatory.
(2) , choosing provided that and or then (3.3)–(3.5) hold, the solution of (3.1) is oscillatory.
(3) , we can see that choosing , and therefore, (3.6) and (3.7) are transformed into the following equation: provided that and or then (3.3)-(3.4) and (3.6)-(3.7) hold, the solution of (3.1) is oscillatory.
In particular, we chose , thus the conditions of the case (1) can be satisfied. This is the sufficient condition for all solutions of to be oscillatory.
If we choose , the conditions of the case (2) can be satisfied. Compared with the conditions: in [6], our results are more general.

Example 3.2. Let us consider the oscillatory behavior of the following differential equation:
Comparing (3.9) with (1.9), we can see that . Choosing clearly, the conditions (2.26) and (2.27) of Theorem 6 can be satisfied. Therefore, we may conclude that (3.9) is oscillatory. Example 3.2 is Example 2 of [5]. It is easy to verify that Example 1 of  [5] also satisfies with Theorem 6.

Example 3.3. Let us consider the oscillatory behavior of the following differential equation:
Comparing (3.10) with (1.9), we can see that Thus, we have , so the second condition of Theorem 2.3 in (2.27) is satisfied. Therefore, another condition is and there exists such that If only where M is a sufficiently large constant, then the conditions (3.11) and (3.12) can be satisfied, the solution of (3.10) is oscillatory.
By taking , provided that , (3.11) and (3.12) hold. By the way, we also note that for (3.10), the example given in [3], are not continuous at , and Though (3.11) and (3.12) hold, but it is the general requirement that for fixed is bounded; hence, this example is not appropriate.

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