About this Journal Submit a Manuscript Table of Contents
Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 539213, 10 pages
http://dx.doi.org/10.1155/2012/539213
Research Article

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

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: ||𝑦𝑟(𝑡)||(𝑡)𝛼1𝑦(𝑡)||||+𝑝(𝑡)𝑦(𝑡)𝛼1𝑦(𝑡)+𝑚𝑗=1𝑞𝑗||||(𝑡)𝑦(𝑡)𝛽𝑗1𝑦(𝑡)=𝑒(𝑡),𝑡𝑡0,(1.1) where 𝑟,𝑝,𝑞𝑗(1𝑗𝑚),𝑒𝐶([𝑡0,),) with 𝑟(𝑡)>0 and 0<𝛼<𝛽1<𝛽2<<𝛽𝑚 are real numbers, 𝑝,𝑞𝑗 (1𝑗𝑚), and 𝑒 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 𝑦𝐶1[𝑇𝑦,)(𝑇𝑦𝑡0, where 𝑇𝑦𝑡0 depends on the particular solution) which has the property 𝑝(𝑡)|𝑦(𝑡)|𝛼1𝑦(𝑡)𝐶1[𝑇𝑦,) 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 𝑚=0, we have the following second-order half-linear differential equation without or with forcing term: ||𝑦𝑟(𝑡)||(𝑡)𝛼1𝑦(𝑡)||||+𝑞(𝑡)𝑦(𝑡)𝛼1𝑦(𝑡)=0,𝑡𝑡0,||𝑦(1.2)𝑟(𝑡)(||𝑡)𝛼1𝑦(𝑡)||||+𝑞(𝑡)𝑦(𝑡)𝛼1𝑦(𝑡)=𝑒(𝑡),𝑡𝑡0.(1.3) There are a lot of papers involved oscillation (see [16]) 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 𝑇𝑡0, there exist 𝑇𝑠1<𝑡1𝑠2<𝑡2 such that 𝑒(𝑡)0 for 𝑡[𝑠1,𝑡1] and 𝑒(𝑡)0 for 𝑡[𝑠2,𝑡2]. Let 𝐷(𝑠𝑖,𝑡𝑖)={𝑢𝐶1[𝑠𝑖,𝑡𝑖]𝑢(𝑡)0,𝑢(𝑠𝑖)=𝑢(𝑡𝑖)=0} for 𝑖=1,2. If there exist 𝐻𝐷(𝑠𝑖,𝑡𝑖) and a positive, nondecreasing function 𝜌𝐶1([𝑡0,),) such that 𝑡𝑖𝑠𝑖𝐻21(𝑡)𝜌(𝑡)𝑞(𝑡)𝑑𝑡>𝛼+1𝛼+1𝑡𝑖𝑠𝑖𝑟(𝑡)𝜌(𝑡)||𝐻||(𝑡)𝛼12||𝐻||+||||(𝑡)𝐻(𝑡)𝜌𝜌𝛼+1𝑑𝑡(1.4) for 𝑖=1,2. Then, (1.3) is oscillatory.

Unfortunately, Theorem 1.1 cannot be applied to the case where 𝛼>1, since for 𝜌(𝑡)1, the term |𝐻(𝑡)|𝛼1 will appear as a denominator in (1.4) so that the requirement 𝐻(𝑠𝑖)=𝐻(𝑡𝑖)=0 will cause trouble. This certainly calls for investigation of oscillation criteria that can handle with such cases.

When 𝛼=1, (1.2) and (1.3) are reduced to the linear differential equation: 𝑟(𝑡)𝑦(𝑡)+𝑞(𝑡)𝑦(𝑡)=0,𝑡𝑡0,𝑟(1.5)(𝑡)𝑦(𝑡)+𝑞(𝑡)𝑦(𝑡)=𝑒(𝑡),𝑡𝑡0.(1.6) In paper [7], Wong proved the following result for (1.6).

Theorem 1.2. Suppose that for any 𝑇𝑡0, there exist 𝑇𝑠1<𝑡1𝑠2<𝑡2 such that 𝑒(𝑡)0 for 𝑡[𝑠1,𝑡1] and 𝑒(𝑡)0 for 𝑡[𝑠2,𝑡2]. Let 𝐷(𝑠𝑖,𝑡𝑖)={𝑢𝐶1[𝑠𝑖,𝑡𝑖]𝑢(𝑡)0,𝑢(𝑠𝑖)=𝑢(𝑡𝑖)=0} for 𝑖=1,2. If there exists 𝑢𝐷(𝑠𝑖,𝑡𝑖) such that 𝑄𝑖(𝑢)=𝑡𝑖𝑠𝑖𝑞(𝑡)𝑢2(𝑢𝑡)𝑟(𝑡)(𝑡)2𝑑𝑡>0,𝑖=1,2,(1.7) 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 𝐶1 function 𝑢(𝑡) defined on [𝑠1,𝑡1] and a function 𝐺(𝑢) such that 𝐺(𝑢(𝑡)) is not constant on [𝑠1,𝑡1], 𝐺(𝑢(𝑠1))=𝐺(𝑢(𝑡1))=0, 𝑔(𝑢)=𝐺(𝑢) is continuous, 𝑡1𝑠1𝑢𝑞(𝑡)𝐺(𝑢(𝑡))𝑟(𝑡)(𝑡)2𝑑𝑡>0,(1.8) and (𝑔(𝑢(𝑡)))24𝐺(𝑢(𝑡)) for 𝑡[𝑠1,𝑡1]. Then, every solution of (1.5) must vanish on [𝑠1,𝑡1].

We note that when 𝐺(𝑢)𝑢2, the left-hand side of (1.8) is the energy functional related to (1.5).

When 𝑝(𝑡)0, 𝑚=1, (1.1) turns into the quasilinear differential equation: ||𝑦𝑟(𝑡)||(𝑡)𝛼1𝑦(𝑡)||||+𝑞(𝑡)𝑦(𝑡)𝛽1𝑦(𝑡)=𝑒(𝑡),𝑡𝑡0,(1.9) where 𝑝,𝑞,𝑒𝐶([𝑡0,),) with 𝑝(𝑡)>0 and 0<𝛼𝛽 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 𝑇𝑡0, there exist 𝑇𝑠1<𝑡1𝑠2<𝑡2 such that 𝑠𝑒(𝑡)0,𝑡1,𝑡1,𝑠0,𝑡2,𝑡2.(1.10) Let 𝑢𝐶1[𝑠𝑖,𝑡𝑖] and nonnegative functions 𝐺1,𝐺2 satisfying 𝐺𝑖(𝑢(𝑠𝑖))=𝐺𝑖(𝑢(𝑡𝑖))=0, 𝑔𝑖(𝑢)=𝐺𝑖(𝑢) are continuous and (𝑔𝑖(𝑢(𝑡)))𝛼+1(𝛼+1)𝛼+1𝐺𝛼𝑖(𝑢(𝑡)) for 𝑡[𝑠𝑖,𝑡𝑖], 𝑖=1,2. If there exists a positive function 𝜙𝐶1([𝑡0,),) such that 𝑄𝜙𝑖(𝑢)=𝑡𝑖𝑠𝑖𝑄𝜙(𝑡)𝑒(𝑡)𝐺𝑖||𝑢(𝑢(𝑡))𝑟(𝑡)||+𝐺(𝑡)𝑖1/(𝛼+1)||𝜙(𝑢(𝑡))||(𝑡)(𝛼+1)𝜙(𝑡)𝛼+1𝑑𝑡>0(1.11) for 𝑖=1,2. Then (1.9) is oscillatory, where 𝑄𝑒(𝑡)=𝛼𝛼/𝛽𝛽(𝛽𝛼)(𝛼𝛽)/𝛽[]𝑞(𝑡)𝛼/𝛽||||𝑒(𝑡)(𝛽𝛼)/𝛽,(1.12) with the convention that 00=1.

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 𝑇𝑡0, there exist 𝑇𝑠1<𝑡1𝑠2<𝑡2 such that 𝑞𝑗(𝑡)0(1𝑗𝑚) for 𝑡[𝑠1,𝑡1][𝑠2,𝑡2] and 𝑠𝑒(𝑡)0,𝑡1,𝑡1,𝑠0,𝑡2,𝑡2.(1.13) Let 𝐷(𝑠𝑖,𝑡𝑖)={𝑢𝐶1[𝑠𝑖,𝑡𝑖]𝑢𝛼+1(𝑡)>0,𝑡(𝑠𝑖,𝑡𝑖),𝑢(𝑠𝑖)=𝑢(𝑡𝑖)=0} for 𝑖=1,2. If there exist 𝐻𝐷(𝑠𝑖,𝑡𝑖) and a positive function 𝜙𝐶1([𝑡0,),) such that 𝑡𝑖𝑠𝑖𝜙(𝑡)𝑝(𝑡)+𝑚𝑗=1𝑄𝑗𝐻(𝑡)𝛼+1||𝐻(𝑡)𝑟(𝑡)||+||(𝑡)𝐻(𝑡)𝜙||(𝑡)(𝛼+1)𝜙(𝑡)𝛼+1𝑑𝑡>0(1.14) for 𝑖=1,2. Then (1.1) is oscillatory, where 𝑄𝑗(𝑡)=𝛼𝛼/𝛽𝑗𝛽𝑗𝑚𝛽𝑗𝛼(𝛼𝛽𝑗)/𝛽𝑗𝑞𝑗(𝑡)𝛼/𝛽𝑗||||𝑒(𝑡)(𝛽𝑗𝛼)/𝛽𝑗,1𝑗𝑚,(1.15) with the convention that 00=1.

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, 𝐻𝐷(𝑠𝑖,𝑡𝑖) is admissible, and the functional is 𝑡𝑖𝑠𝑖1𝛼+1𝛼+1𝑝(𝑡)𝜌(𝑡)||𝐻||(𝑡)𝛼12||𝐻||+||||𝜌(𝑡)𝐻(𝑡)(𝑡)𝜌(𝑡)𝛼+1𝐻2(𝑡)𝜌(𝑡)𝑞(𝑡)𝑑𝑡.(1.16) Our emphasis will be directed towards oscillation criteria that are closely related to the generalized energy functional (the generalization of (𝛼+1)-degree energy functional) for half-linear equations (see [4, 1113] 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 𝑋 and 𝑌 are nonnegative, then 𝛾𝑋𝑌𝛾1𝑋𝛾(𝛾1)𝑌𝛾,𝛾>1,(2.1) where equality holds if and only if 𝑋=𝑌.

Now, we will give our main results.

Theorem 2.2. Assume that for any 𝑇𝑡0, there exist 𝑇𝑠1<𝑡1𝑠2<𝑡2 such that 𝑠𝑒(𝑡)0,𝑡1,𝑡1,𝑠0,𝑡2,𝑡2.(2.2) Let 𝑢𝐶1[𝑠𝑖,𝑡𝑖] and nonnegative functions 𝐺1,𝐺2 satisfying 𝐺𝑖(𝑢(𝑠𝑖))=𝐺𝑖(𝑢(𝑡𝑖))=0, 𝑔𝑖(𝑢)=𝐺𝑖(𝑢) are continuous and (𝑔𝑖(𝑢(𝑡)))𝛼+1(𝛼+1)𝛼+1𝐺𝛼𝑖(𝑢(𝑡)) for 𝑡[𝑠𝑖,𝑡𝑖],𝑖=1,2. If there exists a positive function 𝜙𝐶1([𝑡0,),) such that 𝑄𝜙𝑖(𝑢)=𝑡𝑖𝑠𝑖𝐺𝜙(𝑡)𝑖(𝑢(𝑡))𝑝(𝑡)+𝑚𝑗=1𝑄𝑗||𝑢(𝑡)(2.3)𝑟(𝑡)||+𝐺(𝑡)𝑖1/(𝛼+1)(||𝜙𝑢(𝑡))(||𝑡)(𝛼+1)𝜙(𝑡)𝛼+1𝑑𝑡>0(2.4) for 𝑖=1,2, where 𝑄𝑗(𝑡) is defined as (1.15) with the convention that 00=1. Then, (1.1) is oscillatory.

Proof. Suppose to the contrary that there is a nontrivial nonoscillatory solution 𝑦=𝑦(𝑡). We assume that 𝑦(𝑡)0 on [𝑇0,) for some 𝑇0𝑡0. Set ||𝑦𝑤(𝑡)=𝜙(𝑡)𝑟(𝑡)||(𝑡)𝛼1𝑦(𝑡)||||𝑦(𝑡)𝛼1𝑦(𝑡),𝑡𝑇0.(2.5) Then differentiating (2.5) and making use of (1.1), it follows that for all 𝑡𝑇0, 𝑤𝜙(𝑡)=(𝑡)𝜙(𝑡)𝑤(𝑡)𝜙(𝑡)𝑝(𝑡)+𝜙(𝑡)𝑒(𝑡)||||𝑦(𝑡)𝛼1||||𝑦(𝑡)𝛼𝑤(𝑡)(𝛼+1)/𝛼(𝑟(𝑡)𝜙(𝑡))1/𝛼𝜙(𝑡)𝑚𝑗=1𝑞𝑗||𝑦||(𝑡)𝛽𝑗𝛼.(2.6) By the assumptions, we can choose 𝑠𝑖,𝑡𝑖𝑇0 for 𝑖=1,2 so that 𝑒(𝑡)0 on the interval 𝐼1=[𝑠1,𝑡1], with 𝑠1<𝑡1 and 𝑦(𝑡)0, or 𝑒(𝑡)0 on the interval 𝐼2=[𝑠2,𝑡2], with 𝑠2<𝑡2 and 𝑦(𝑡)0. For given 𝑡𝐼1 or 𝑡𝐼2, set 𝐹𝑗(𝑥)=𝑞𝑗(𝑡)𝑥𝛽𝑗𝛼𝑒(𝑡)/𝑚𝑥𝛼, 1𝑗𝑚, we have 𝐹𝑗(𝑥𝑗)=0, 𝐹𝑗(𝑥𝑗)>0, where 𝑥𝑗=[𝛼𝑒(𝑡)/𝑚(𝛽𝑗𝛼)𝑞𝑗(𝑡)]1/𝛽𝑗. So, 𝐹𝑗(𝑥) obtains it minimum on 𝑥𝑗 and 𝐹𝑗(𝑥)𝐹𝑗𝑥𝑗=𝑄𝑗(𝑡).(2.7) So on the interval 𝐼1 or 𝐼2, (2.6) and (2.2) imply that 𝑤(𝑡) satisfies 𝜙(𝑡)𝑝(𝑡)+𝑚𝑗=1𝑄𝑗(𝑡)𝑤𝜙(𝑡)+(𝑡)||||𝜙(𝑡)𝑤(𝑡)𝛼𝑤(𝑡)(𝛼+1)/𝛼(𝑟(𝑡)𝜙(𝑡))1/𝛼.(2.8) Multiplying 𝐺𝑖(𝑢(𝑡)) through (2.8) and integrating (2.8) from 𝑠𝑖 to 𝑡𝑖, using the fact that 𝐺𝑖(𝑢(𝑠1))=𝐺𝑖(𝑢(𝑡1))=0, we obtain 𝑡𝑖𝑠𝑖𝜙(𝑡)𝑝(𝑡)+𝑚𝑗=1𝑄𝑗𝐺(𝑡)𝑖(𝑢(𝑡))𝑑𝑡𝑡𝑖𝑠𝑖𝐺𝑖(𝑢(𝑡))𝑤𝜙(𝑡)+(𝑡)𝑤||||𝜙(𝑡)(𝑡)𝛼𝑤(𝑡)(𝛼+1)/𝛼(𝑟(𝑡)𝜙(𝑡))1/𝛼𝑑𝑡=𝐺𝑖(||𝑢(𝑡))𝑤(𝑡)𝑡𝑖𝑠𝑖+𝑡𝑖𝑠𝑖𝑔𝑖(𝑢(𝑡))𝑢(+𝑡)𝑤(𝑡)𝑑𝑡𝑡𝑖𝑠𝑖𝐺𝑖𝜙(𝑢(𝑡))(𝑡)||||𝜙(𝑡)𝑤(𝑡)𝛼𝑤(𝑡)(𝛼+1)/𝛼(𝑟(𝑡)𝜙(𝑡))1/𝛼=𝑑𝑡𝑡𝑖𝑠𝑖𝑔𝑖(𝑢(𝑡))𝑢(𝑡)+𝐺𝑖𝜙(𝑢(𝑡))(𝑡)𝜙(𝑡)𝑤(𝑡)𝑑𝑡𝛼𝑡𝑖𝑠𝑖𝐺𝑖||𝑤||(𝑢(𝑡))(𝑡)(𝛼+1)/𝛼(𝑟(𝑡)𝜙(𝑡))1/𝛼𝑑𝑡𝑡𝑖𝑠𝑖||𝑔𝑖(||||𝑢𝑢(𝑡))(||𝑡)+𝐺𝑖(||𝜙𝑢(𝑡))||(𝑡)||||𝜙(𝑡)𝑤(𝑡)𝑑𝑡𝛼𝑡𝑖𝑠𝑖𝐺𝑖||𝑤||(𝑢(𝑡))(𝑡)(𝛼+1)/𝛼(𝑟(𝑡)𝜙(𝑡))1/𝛼𝑑𝑡(𝛼+1)𝑡𝑖𝑠𝑖𝐺𝑖𝛼/(𝛼+1)||𝑢(𝑢(𝑡))||(𝑡)+𝐺𝑖||𝜙(𝑢(𝑡))||(𝑡)||||(𝛼+1)𝜙(𝑡)𝑤(𝑡)𝑑𝑡𝛼𝑡𝑖𝑠𝑖𝐺𝑖||||(𝑢(𝑡))𝑤(𝑡)(𝛼+1)/𝛼(𝑟(𝑡)𝜙(𝑡))1/𝛼𝑑𝑡.(2.9) Let 𝛼𝑋=(𝑟(𝑡)𝜙(𝑡))1/𝛼𝛼/(𝛼+1)𝐺𝑖𝛼/(𝛼+1)||||1𝑤(𝑡),𝛾=1+𝛼,𝑌=(𝛼𝜙(𝑡)𝑟(𝑡))𝛼/(𝛼+1)||𝑢||+𝐺(𝑡)𝑖1/(𝛼+1)||𝜙||(𝑡)(𝛼+1)𝜙(𝑡)𝛼,(2.10) by Lemma 2.1 and (2.9), we have 𝑡𝑖𝑠𝑖𝜙(𝑡)𝑝(𝑡)+𝑚𝑗=1𝑄𝑗𝐺(𝑡)𝑖(𝑢(𝑡))𝑑𝑡𝑡𝑖𝑠𝑖||𝑢𝜙(𝑡)𝑟(𝑡)||+𝐺(𝑡)𝑖1/(𝛼+1)||𝜙(𝑢(𝑡))||(𝑡)(𝛼+1)𝜙(𝑡)𝛼+1𝑑𝑡,(2.11) which contradicts with (2.3). This completes the proof of Theorem 2.2.

Corollary 2.3. If 𝜙(𝑡)1 in Theorem 2.2, and (2.3) is replaced by 𝑄𝑖(𝑢)=𝑡𝑖𝑠𝑖𝑝(𝑡)+𝑚𝑗=1𝑄𝑗𝐺(𝑡)𝑖||𝑢(𝑢(𝑡))𝑟(𝑡)||(𝑡)𝛼+1𝑑𝑡>0,(2.12) for 𝑖=1,2. Then, (1.1) is oscillatory.

If we choose 𝐺1(𝑢)=𝐺2(𝑢)=𝑢𝛼+1 in Corollary 2.3, then we have the following corollary.

Corollary 2.4. Suppose that for any 𝑇𝑡0, there exist 𝑇𝑠1<𝑡1𝑠2<𝑡2 such that (2.2) is true. Let 𝐷(𝑠𝑖,𝑡𝑖)={𝑢𝐶1[𝑠𝑖,𝑡𝑖]𝑢(𝑡)0,𝑢(𝑠𝑖)=𝑢(𝑡𝑖)=0} for 𝑖=1,2. If there exist 𝑢𝐷(𝑠𝑖,𝑡𝑖) such that 𝑄𝑖(𝑢)=𝑡𝑖𝑠𝑖𝑝(𝑡)+𝑚𝑗=1𝑄𝑗||||(𝑡)𝑢(𝑡)𝛼+1||𝑢𝑟(𝑡)||(𝑡)𝛼+1𝑑𝑡>0,(2.13) for 𝑖=1,2. Then, (1.3) is oscillatory.

Remark 2.5. Corollary 2.4 is closely related to the (𝛼+1)-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 𝛼 in Theorem 2.2 and Corollary 2.3 can be selected as any number lying in (0,). We note further that in most cases, oscillation criteria are obtained using the same auxiliary function on [𝑠1,𝑡1] and [𝑠2,𝑡2], we note that such functions can be selected differently.

Remark 2.6. If 𝐺(𝑢)𝑢𝛼+1, then Theorem 2.2 reduces to Theorem 1.5, and if 𝑝(𝑡)0, 𝑗=1, 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: 𝑠𝑒(𝑡)0,𝑡1,𝑡1,𝑠0,𝑡2,𝑡2.(2.14) The conclusion is still true for these cases.

Example 2.8. Consider the following forced mixed nonlinearities differential equation: 𝛾𝑡𝜆/3𝑦(𝑡)||||+𝑝(𝑡)𝑦(𝑡)+𝑞(𝑡)𝑦(𝑡)2𝑦(𝑡)=sin3𝑡,𝑡2𝜋,(2.15) where 𝛾,𝜆>0 are constants, 𝑞(𝑡)=𝑡𝜆exp(3sin𝑡), 𝑝(𝑡)=𝑡𝜆/3exp(sin𝑡), for 𝑡[2𝑛𝜋,(2𝑛+1)𝜋), and 𝑞(𝑡)=𝑡𝜆exp(3sin𝑡), 𝑝(𝑡)=𝑡𝜆/3exp(sin𝑡), for 𝑡[(2𝑛+1)𝜋,(2𝑛+2)𝜋), 𝑛>0 is an integer, Shao [9] obtain oscillation for (2.15) when 𝐾(𝑡)0. Using Theorem 2.2, we can easily verify that 𝑄1(𝑡)=(3/2)32𝑡𝜆/3exp(sin𝑡)sin2𝑡 for 𝑡[2𝑛𝜋,(2𝑛+1)𝜋), and 𝑄1(𝑡)=(3/2)32𝑡𝜆/3exp(sin𝑡)sin2𝑡 for 𝑡[(2𝑛+1)𝜋,(2𝑛+2)𝜋). For any 𝑇1, we choose 𝑛 sufficiently large so that 𝑛𝜋=2𝑘𝜋𝑇 and 𝑠1=2𝑘𝜋 and 𝑡1=(2𝑘+1)𝜋, we select 𝑢(𝑡)=sin𝑡0, 𝐺1(𝑢)=𝑢2exp(𝑢) (we note that (𝐺1(𝑢))24𝐺1(𝑢) for 𝑢0), 𝜙(𝑡)=𝑡𝜆/3, then we have 𝑡1𝑠1𝜙(𝑡)𝑝(𝑡)+𝑄1(𝐺𝑡)1(𝑢(𝑡))𝑑𝑡=𝜋0sin23𝑡𝑑𝑡+232𝜋0sin4𝜋𝑡𝑑𝑡=2+9832,𝑡1𝑠1||𝑢𝜙(𝑡)𝑝(𝑡)||+𝐺(𝑡)11/(𝛼+1)||𝜙(𝑢(𝑡))||(𝑡)(𝛼+1)𝜙(𝑡)𝛼+1𝑑𝑡=𝛾(2𝑘+1)𝜋2𝑘𝜋𝜆|||||cos𝑡|+sin𝑡exp(3sin𝑡/2)2𝑡2𝑑𝑡<𝛾(2𝑘+1)𝜋2𝑘𝜋1+𝜆𝑒3/222𝑑𝑡=𝛾1+𝜆𝑒3/222𝜋.(2.16) So we have 𝑄𝜙1(𝑢)>0 provided, 0<𝛾<(4𝜋+932)/2(2+𝜆𝑒3/2)2𝜋. Similarly, for 𝑠2=(2𝑘+1)𝜋 and 𝑡2=(2𝑘+2)𝜋, we select 𝑢(𝑡)=sin𝑡0, 𝐺2(𝑢)=𝑢2exp(𝑢) (we note that (𝐺2(𝑢))24𝐺2(𝑢) for 𝑢0), we can show that the integral inequality 𝑄𝜙2(𝑢)>0 for 0<𝛾<(4𝜋+932)/2(2+𝜆𝑒3/2)2𝜋. So (2.15) is oscillatory for 0<𝛾<(4𝜋+932)/2(2+𝜆𝑒3/2)2𝜋 by Theorem 2.2.

Example 2.9. Consider the following forced mixed nonlinearities differential equation: 𝑡𝜆||𝑦||(𝑡)𝛼1𝑦(𝑡)||||+𝑝(𝑡)𝑦(𝑡)𝛼1𝑦(𝑡)+𝑞(𝑡)𝑦3(𝑡)=sin1/3𝑡,(2.17) for 𝑡2𝜋, where 𝑝(𝑡)=𝐾𝑡𝜆exp(sin𝑡), 𝑞(𝑡)=𝑡9𝜆/5exp(9sin𝑡/5), for 𝑡[2𝑛𝜋,(2𝑛+1)𝜋), and 𝑝(𝑡)=𝐾𝑡𝜆exp(sin𝑡), 𝑞(𝑡)=𝑡9𝜆/5exp(9sin𝑡/5), for 𝑡[(2𝑛+1)𝜋,(2𝑛+2)𝜋), 𝑛>0 is an integer, 𝐾,𝜆>0 are constants and 𝛼=5/3>1, 𝛽=3. Obviously, Theorem 1.1 cannot be applied to this case. However, we conclude that (2.17) is oscillatory for 𝐾>(3/4)(1+3𝜆𝑒/8)8/3𝜋9/55/944/9. Since the zeros of the forcing term sin1/3𝑡 are 𝑛𝜋, let 𝑢(𝑡)=sin𝑡 and 𝜙(𝑡)=𝑡𝜆. Using Theorem 2.2, we can easily verify that 𝑄(𝑡)=(9/55/944/9)𝑡𝜆exp(sin𝑡)sin4/27𝑡 for 𝑡[2𝑛𝜋,(2𝑛+1)𝜋), and 𝑄(𝑡)=(9/55/944/9)𝑡𝜆exp(sin𝑡)sin4/27𝑡 for 𝑡[(2𝑛+1)𝜋,(2𝑛+2)𝜋). For any 𝑇1, choose 𝑛 sufficiently large so that 𝑛𝜋=2𝑘𝜋𝑇 and 𝑠1=2𝑘𝜋 and 𝑡1=(2𝑘+1)𝜋. For 𝑡[𝑠1,𝑡1], we select 𝐺1(𝑢)=𝑢8/3exp(𝑢) (we note that (𝐺1(𝑢))8/3(8/3)8/3(𝐺1(𝑢))5/3 for 𝑢0). It is easy to verify the following estimations: 𝑡1𝑠1𝜙(𝑡)(𝑝(𝑡)+𝑄(𝑡))𝐺1(=𝑢(𝑡))𝑑𝑡(2𝑘+1)𝜋2𝑘𝜋sin8/3𝑡9𝐾+55/944/9sin4/27𝑡>9𝑑𝑡𝐾+55/944/9(2𝑘+1)𝜋2𝑘𝜋sin34𝑡𝑑𝑡=39𝐾+55/944/9,𝑡1𝑠1||𝑢𝜙(𝑡)𝑟(𝑡)||+𝐺(𝑡)11/(𝛼+1)||𝜙(𝑢(𝑡))||(𝑡)(𝛼+1)𝜙(𝑡)𝛼+1=𝑑𝑡(2𝑘+1)𝜋2𝑘𝜋|cos𝑡|+3𝜆𝑒3sin𝑡/8||||sin𝑡8𝑡8/3<𝑑𝑡(2𝑘+1)𝜋2𝑘𝜋1+3𝜆𝑒88/3𝑑𝑡=1+3𝜆𝑒88/3𝜋.(2.18) So we have 𝑄𝜙1(𝑢)>0. Similarly, for 𝑠2=(2𝑘+1)𝜋 and 𝑡2=(2𝑘+2)𝜋, we select 𝑢(𝑡)=sin𝑡<0, 𝐺2(𝑢)=𝑢8/3exp(𝑢) (we note that (𝐺2(𝑢))8/3(8/3)8/3(𝐺2(𝑢))5/3 for 𝑢0), we can show that the integral inequality 𝑄𝜙2(𝑢)>0. So (2.17) is oscillatory for 𝐾>(3/4)(1+3𝜆𝑒/8)8/3𝜋9/55/944/9 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).

References

  1. W.-T. Li and S. S. Cheng, “An oscillation criterion for nonhomogeneous half-linear differential equations,” Applied Mathematics Letters, vol. 15, no. 3, pp. 259–263, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. Á. Elbert, “A half-linear second order differential equation,” in Qualitative Theory of Differential Equations, Szeged, Ed., vol. 30 of Colloquia Mathematica Societatis János Bolyai, pp. 153–180, North-Holland, Amsterdam, The Netherlands, 1979. View at Google Scholar · View at Zentralblatt MATH
  3. W. Leighton, “Comparison theorems for linear differential equations of second order,” Proceedings of the American Mathematical Society, vol. 13, pp. 603–610, 1962. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. J. Jaroš and T. Kusano, “A Picone type identity for second order half-linear differential equations,” Acta Mathematica Universitatis Comenianae, vol. 68, no. 1, pp. 137–151, 1999. View at Google Scholar · View at Zentralblatt MATH
  5. H. J. Li and C. C. Yeh, “Sturmian comparison theorem for half-linear second-order differential equations,” Proceedings of the Royal Society of Edinburgh A, vol. 125, no. 6, pp. 1193–1204, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. J. V. Manojlović, “Oscillation criteria for second-order half-linear differential equations,” Mathematical and Computer Modelling, vol. 30, no. 5-6, pp. 109–119, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. J. S. W. Wong, “Oscillation criteria for a forced second-order linear differential equation,” Journal of Mathematical Analysis and Applications, vol. 231, no. 1, pp. 235–240, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. V. Komkov, “A generalization of Leighton's variational theorem,” Applicable Analysis, vol. 2, pp. 377–383, 1972. View at Google Scholar
  9. J. Shao, “A new oscillation criterion for forced second-order quasilinear differential equations,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 428976, 8 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. Z. Zheng, X. Wang, and H. Han, “Oscillation criteria for forced second order differential equations with mixed nonlinearities,” Applied Mathematics Letters, vol. 22, no. 7, pp. 1096–1101, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. J. Jaroš, T. Kusano, and N. Yoshida, “Forced superlinear oscillations via Picone's identity,” Acta Mathematica Universitatis Comenianae, vol. 69, no. 1, pp. 107–113, 2000. View at Google Scholar · View at Zentralblatt MATH
  12. J. Jaroš, T. Kusano, and N. Yoshida, “Generalized Picone's formula and forced oscillations in quasilinear differential equations of the second order,” Archivum Mathematicum, vol. 38, no. 1, pp. 53–59, 2002. View at Google Scholar · View at Zentralblatt MATH
  13. O. Došlý, Half-Linear Differential Equations, vol. 202 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 2005.
  14. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK, 2nd edition, 1988.