Discrete Dynamics in Nature and Society

Volume 2011 (2011), Article ID 428976, 8 pages

http://dx.doi.org/10.1155/2011/428976

## A New Oscillation Criterion for Forced Second-Order Quasilinear Differential Equations

Department of Mathematics, Jining University, Qufu 273155, China

Received 17 February 2011; Accepted 23 March 2011

Academic Editor: M. De la Sen

Copyright © 2011 Shao Jing. 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

By using the generalized variational principle and Riccati technique, a new oscillation criterion is established for second-order quasilinear differential equation with an oscillatory forcing term, which improves and generalizes some of new results in the literature.

#### 1. Introduction

In this paper, we are concerned with oscillation for the second order forced quasilinear differential equation where with and being constants.

By a solution of (1.1), we mean a function , where depends on the particular solution, which has the property and satisfies (1.1). We restrict our attention to the nontrivial solutions of (1.1) only, that is, to solutions such that for all . 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 nontrivial solutions are oscillatory.

We note that when , (1.1) turns into the half-linear differential equation while , (1.1) is a super-half-linear differential equation.

Oscillation for (1.2) and its special case (without a forcing term) are widely studied by many authors (see [1–6] and references cited therein) since the foundation work of Elbert [7]. Manojlović [8], Ayanlar and Tiryaki [9] obtained some Kamenev type oscillation criteria for (1.3) by using integral averaging techniques; a more detailed description of oscillation criteria can be found in the book in [10].

The half-linear differential equation (1.3) has similar properties to linear differential equation; for example, Sturm's comparison theorem and separation theorem (see Elbert [7], Jaros and Kusano [5], Li and Yeh [11] for details) are still true for (1.3). In particular, the Riccati type equation (related to (1.3) by the substitution ) and the -degree functional play the same role as the classical Riccati equation and quadratic functional in the linear oscillation theory. (The proof of the relationship between (1.3), (1.4), and (1.5) is based on the recently established Picone's identity, see [5].)

When , (1.2) is reduced to the forced linear differential equation

In paper [12], based on the oscillatory behavior of the forcing term, Wong proved the following result for (1.6).

Theorem 1.1. *Suppose that, for any , there exist such that for and for . Let for . If there exists such that
**
then (1.6) is oscillatory.*

This result involves the “oscillatory interval” of and Leighton's variational principle for (1.6). Recently, using a similar method, Wong's result was extended to (1.2) by Li and Cheng [1] with a positive and nondecreasing function .

Theorem 1.2. *Suppose that, for any , there exist such that for and for . Let for . If there exist and a positive, nondecreasing function such that
**
for , where , then (1.2) is oscillatory.*

However, inequality (1.8) has no relation to the -degree functional (1.5), and Theorem 1.2 cannot be applied to oscillation when , since is the denominator of the fraction in the right-side integral of (1.8), and .

Zheng and Meng [4] improved the paper [1] and gave an oscillation criterion without any restriction on the signs of , , and , which is closely related to the -degree functional (1.5). Here we list the main result of Zheng and Meng (Jaros et al. [6] obtained a similar result using the generalized Pocone's formulae with ).

Theorem 1.3. *Suppose that, for any , there exist such that for and for . Let for . If there exist and a positive function such that
**
for , then (1.1) is oscillatory, where is the same as (2.4).*

Meanwhile, among the oscillation criteria, Komkov [13] gave a generalized Leighton's variational principle, which also can be applied to oscillation for linear equation.

Theorem 1.4. *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 the equation must vanish on .*

As far as we know, there is no oscillation criterion which relates to the generalized Leighton's variational principle for (1.1). The question then arises as to whether a more general result can be established to this equation. In this paper, we give an affirmative answer for this question and give a new oscillation criterion for the quasilinear differential equation (1.1); this oscillation criterion is closely related to the generalized Leighton's variational principle, which generalizes and improves the results mention above.

#### 2. Main Results

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

Lemma 2.1 (see [14]). *Supposing 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 , then (1.1) is oscillatory, where
**
with the convention that .*

*Proof. *Suppose on the contrary that there is a nontrivial nonoscillatory solution. We assume that on for some . Set
Then for all , we obtain the Riccati type equation
By the assumption, we can choose so that on the interval . For given , setting , we have , , where . So obtains its minimum on , and
So (2.6) and (2.7) 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 (1.2), we have
which contradicts (2.3) with .

When is a negative solution for , we define the Riccati transformation as (2.5), and we obtain (2.8) also. In this case, we choose so that on the interval . For given , setting , we have . We use and on to reach a similar contradiction. The proof is complete.

Corollary 2.3. *If in Theorem 2.2 and (2.3) is replaced by
**
for , then (1.1) is oscillatory.*

*Remark 2.4. *Corollary 2.3 is closely related to the generalized variational formulae similar to (1.10), so Theorem 2.2, Corollary 2.3 are generalizations of Theorems 1.1 and 1.3 (if we choose in Corollary 2.3, then we obtain Corollary 2.3 of paper [4]), and when , we have , so Corollary 2.3 improves the main result of paper [1], since the positive constant in Theorem 2.2, Corollary 2.3 can be selected as any number lying in .

*Remark 2.5. *Hypothesis (2.2) in Theorem 2.2, Corollary 2.3 can be replaced by the following condition
The conclusion is still true for this case.

Now we give an example to illustrate the efficiency of our result.

*Example 2.6. *Consider the following forced quasilinear differential equation:
where are constants, for , and for , is an integer. Here , in Theorem 2.2. Since , Theorems 1.1 and 1.2 cannot be applied to this case. However, we can obtain oscillation for (2.14) using Theorem 2.2. In fact, 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 ), and we can show that the integral inequality for . So (2.14) is oscillatory for by Theorem 2.2. We note further that Theorem 1.3 cannot be applied to (2.14) with .

#### Acknowledgments

Thanks are due to the referees for giving the author some useful comments which improve the main result. This paper was partially supported by the NSF of China (Grant 10801089), NSF of Shandong (Grant ZR2009AM011, ZR2009AQ010), and NSF of Jining University (Grant 2010QNKJ09).

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