Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 539213, 10 pages

doi:10.1155/2012/539213

Research Article

Generalized Variational Oscillation Principles for Second-Order Differential Equations with Mixed-Nonlinearities

Jing Shao,1,2 Fanwei Meng,¹ and Xinqin Pang²

¹School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

²Department of Mathematics, Jining University, Shandong, Qufu 273155, China

Received 14 March 2012; Accepted 4 June 2012

Academic Editor: Mingshu Peng

Copyright © 2012 Jing Shao 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.

Using generalized variational principle and Riccati technique, new oscillation criteria are established for forced second-order differential equation with mixed nonlinearities, which improve and generalize some recent papers in the literature.

1. Introduction

In this paper, we consider the second-order forced differential equation with mixed nonlinearities:

where with and are real numbers, (), and might change signs.

In this paper, we are concerned with the nonhomogeneous equation (1.1). By a solution of (1.1), we mean that a function (, where depends on the particular solution) which has the property and satisfies (1.1). A nontrivial solution of (1.1) is called oscillatory if it has arbitrarily large zeros; otherwise, it is said to be nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.

When , we have the following second-order half-linear differential equation without or with forcing term:

There are a lot of papers involved oscillation (see [1–6]) for these equations since the foundation work of Elbert [2]. In paper [1], using Leighton’s variational principle (see [3]) for (1.3), the following result was obtained by Li and Cheng.

Theorem 1.1. Suppose that for any , there exist such that for and for . Let for . If there exist and a positive, nondecreasing function such that

for . Then, (1.3) is oscillatory.

Unfortunately, Theorem 1.1 cannot be applied to the case where , since for , the term will appear as a denominator in (1.4) so that the requirement will cause trouble. This certainly calls for investigation of oscillation criteria that can handle with such cases.

When , (1.2) and (1.3) are reduced to the linear differential equation:

In paper [7], Wong proved the following result for (1.6).

Theorem 1.2. Suppose that for any , there exist such that for and for . Let for . If there exists such that

then (1.6) is oscillatory.

On the other hand, among the oscillation criteria, Komkov [8] gave a generalized Leighton’s variational principle, which also can be applied to oscillation for (1.5).

Theorem 1.3. Suppose that there exist a function defined on and a function such that is not constant on , , is continuous,

and for . Then, every solution of (1.5) must vanish on .

We note that when , the left-hand side of (1.8) is the energy functional related to (1.5).

When , , (1.1) turns into the quasilinear differential equation:

where with and being constants. In paper [9], using the generalized variational principle, Shao proved the following result for (1.9).

Theorem 1.4. Assume that for any , there exist such that

Let and nonnegative functions satisfying , are continuous and for , . If there exists a positive function such that

for . Then (1.9) is oscillatory, where

with the convention that .

Recently, using Riccati transformation, the following oscillation criteria were given for (1.1) by Zheng et al. [10].

Theorem 1.5. Assume that for any , there exist such that for and

Let for . If there exist and a positive function such that

for . Then (1.1) is oscillatory, where

with the convention that .

The purpose of this paper is to obtain new oscillation criteria for (1.1) based on generalized variational principles. Roughly, if the existence of a “positive” solution of a functional relation implies the “positivity” of an associated functional over a set of “admissible” functions, then we say that a variational oscillation principle is valid. For instance, in Theorem 1.1, is admissible, and the functional is

Our emphasis will be directed towards oscillation criteria that are closely related to the generalized energy functional (the generalization of -degree energy functional) for half-linear equations (see [4, 11–13] for more details on these functionals), which improve the results mentioned above. Examples will also be given to illustrate the effectiveness of our main results.

2. Main Results

Firstly, we give an inequality, which is a transformation of Young’s inequality.

Lemma 2.1 (see [14]). Suppose that and are nonnegative, then

where equality holds if and only if .

Now, we will give our main results.

Theorem 2.2. Assume that for any , there exist such that

Let and nonnegative functions satisfying , are continuous and for . If there exists a positive function such that

for , where is defined as (1.15) with the convention that . Then, (1.1) is oscillatory.

Proof. Suppose to the contrary that there is a nontrivial nonoscillatory solution . We assume that on for some . Set

Then differentiating (2.5) and making use of (1.1), it follows that for all ,

By the assumptions, we can choose for so that on the interval , with and , or on the interval , with and . For given or , set , , we have , , where . So, obtains it minimum on and

So on the interval or , (2.6) and (2.2) imply that satisfies

Multiplying through (2.8) and integrating (2.8) from to , using the fact that , we obtain

Let

by Lemma 2.1 and (2.9), we have

which contradicts with (2.3). This completes the proof of Theorem 2.2.

Corollary 2.3. If in Theorem 2.2, and (2.3) is replaced by

for . Then, (1.1) is oscillatory.

If we choose in Corollary 2.3, then we have the following corollary.

Corollary 2.4. Suppose that for any , there exist such that (2.2) is true. Let for . If there exist such that

for . Then, (1.3) is oscillatory.

Remark 2.5. Corollary 2.4 is closely related to the -degree functional (1.8), so Theorem 2.2, Corollaries 2.3, and 2.4 are generalizations of Theorem 1.2, and improvement of Theorem 1.1 since the positive constant in Theorem 2.2 and Corollary 2.3 can be selected as any number lying in . We note further that in most cases, oscillation criteria are obtained using the same auxiliary function on and , we note that such functions can be selected differently.

Remark 2.6. If , then Theorem 2.2 reduces to Theorem 1.5, and if , , Theorem 2.2 reduces to Theorem 1.4. So Theorem 2.2 and Corollary 2.3 are generalizations of the papers by Zheng et al. [10] and Shao [9].

Remark 2.7. The hypothesis (2.2) in Theorem 2.2 and Corollary 2.3 can be replaced by the following condition:

The conclusion is still true for these cases.

Example 2.8. Consider the following forced mixed nonlinearities differential equation:

where are constants, , , for , and , , for , is an integer, Shao [9] obtain oscillation for (2.15) when . Using Theorem 2.2, we can easily verify that for , and for . For any , we choose sufficiently large so that and and , we select , (we note that for ), , then we have

So we have provided, . Similarly, for and , we select , (we note that for ), we can show that the integral inequality for . So (2.15) is oscillatory for by Theorem 2.2.

Example 2.9. Consider the following forced mixed nonlinearities differential equation:

for , where , , for , and , , for , is an integer, are constants and , . Obviously, Theorem 1.1 cannot be applied to this case. However, we conclude that (2.17) is oscillatory for . Since the zeros of the forcing term are , let and . Using Theorem 2.2, we can easily verify that for , and for . For any , choose sufficiently large so that and and . For , we select (we note that for ). It is easy to verify the following estimations:

So we have . Similarly, for and , we select , (we note that for ), we can show that the integral inequality . So (2.17) is oscillatory for by Theorem 2.2.

Acknowledgment

This research was partially supported by the NSF of China (Grants nos. 11171178 and 11271225) and Science and Technology Project of High Schools of Shandong Province (Grant no. J12LI52).

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