Abstract

By using a generalized arithmetic-geometric mean inequality on time scales, we study the forced oscillation of second-order dynamic equations with nonlinearities given by Riemann-Stieltjes integrals of the form [𝑝(𝑡)𝜙𝛼(𝑥Δ(𝑡))]Δ+𝑞(𝑡)𝜙𝛼(𝑥(𝜏(𝑡)))+𝑎𝜎(𝑏)𝑟(𝑡,𝑠)𝜙𝛾(𝑠)(𝑥(𝑔(𝑡,𝑠)))Δ𝜉(𝑠)=𝑒(𝑡), where 𝑡[𝑡0,)𝕋=[𝑡0𝕋,), 𝕋 is a time scale which is unbounded from above; 𝜙(𝑢)=|𝑢|sgn𝑢; 𝛾[𝑎,𝑏]𝕋1 is a strictly increasing right-dense continuous function; 𝑝,𝑞,𝑒[𝑡0,)𝕋, 𝑟[𝑡0,)𝕋×[𝑎,𝑏]𝕋1, 𝜏[𝑡0,)𝕋[𝑡0,)𝕋, and 𝑔[𝑡0,)𝕋×[𝑎,𝑏]𝕋1[𝑡0,)𝕋 are right-dense continuous functions; 𝜉[𝑎,𝑏]𝕋1 is strictly increasing. Some interval oscillation criteria are established in both the cases of delayed and advanced arguments. As a special case, the work in this paper unifies and improves many existing results in the literature for equations with a finite number of nonlinear terms.

1. Introduction

Following Hilger’s landmark paper [1], there have been plenty of references focused on the theory of time scales in order to unify continuous and discrete analysis, where a time scale is an arbitrary nonempty closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations. The oscillation theory has been developed very rapidly since the discovery of time scale calculus with this understanding. Throughout this paper, a knowledge and understanding of time scale calculus is assumed. For an introduction to time scale calculus and dynamic equations, we refer to the seminal books by Bohner and Peterson [2, 3].

In this paper, we consider the following second-order dynamic equation with the nonlinearity given by a Riemann-Stieltjes integral of the form 𝑝(𝑡)𝜙𝛼𝑥Δ(𝑡)Δ+𝑞(𝑡)𝜙𝛼(𝑥(𝜏(𝑡)))+𝑎𝜎(𝑏)𝑟(𝑡,𝑠)𝜙𝛾(𝑠)(𝑥(𝑔(𝑡,𝑠)))Δ𝜉(𝑠)=𝑒(𝑡),(1.1) where 𝑡[𝑡0,)𝕋=[𝑡0,)𝕋, 𝑡0𝕋, 𝕋 is a time scale (a closed nonempty subset of real numbers) which is unbounded from above; 𝜙(𝑢)=|𝑢|sgn𝑢; 𝑎,𝑏𝕋1, 𝑏>𝑎, 𝕋1 is another time scale; 𝛾[𝑎,𝑏]𝕋1 is a strictly increasing right-dense continuous function satisfying 0<𝛾(𝑎)<𝛼<𝛾(𝑏);𝑝, 𝑞, 𝑒[𝑡0,)𝕋 are right-dense continuous with 𝑝>0; 𝑟[𝑡0,)𝕋×[𝑎,𝑏]𝕋1 is right-dense continuous; 𝜏[𝑡0,)𝕋[𝑡0,)𝕋, 𝑔[𝑡0,)𝕋×[𝑎,𝑏]𝕋1[𝑡0,)𝕋 are right-dense continuous functions satisfying lim𝑡𝜏(𝑡)=lim𝑡𝑔(𝑡,𝑠)=; 𝜉[𝑎,𝑏]𝕋1 is strictly increasing. Here 𝑎𝜎(𝑏)𝑓(𝑠)Δ𝜉(𝑠) denotes the Riemann-Stieltjes integral of the function 𝑓 on [𝑎,𝜎(𝑏)]𝕋1 with respect to 𝜉, 𝜎[𝑡0,)𝕋[𝑡0,)𝕋 is the forward jump operator.

We restrict our attention to those solutions of (1.1) which exist on the time scale half-line [𝑇𝑥,)𝕋, where 𝑇𝑥𝑡0 may depend on the particular solution, a nontrivial function in any neighborhood of infinity. As usual, such a solution of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Equation (1.1) is said to be oscillatory if every proper solution is oscillatory.

Recently, people have been interested in the combined effects of linear, superlinear, sublinear terms, and a forced term in oscillation. For instance, Sun and Wong [4] investigated the following forced differential equation with mixed nonlinearities 𝑝(𝑡)𝑥(𝑡)+𝑞(𝑡)𝑥(𝑡)+𝑛𝑗=1𝑞𝑗(𝑡)𝜙𝛼𝑗(𝑥(𝑡))=𝑒(𝑡),(1.2) where 𝑝,𝑞,𝑞𝑗,𝑒𝐶[𝑡0,), and 0<𝛼1<<𝛼𝑚<1<𝛼𝑚+1<<𝛼𝑛. The authors obtained interval oscillation criteria for (1.2) by using an arithmetic-geometric inequality and employing arguments developed earlier in [59]. Sun and Meng [10] studied (1.2) again by making use of some of the arguments developed by Kong [11]. In [12], Agarwal and Zafer extended the results in [4] to dynamic equations on time scales of the form 𝑝(𝑡)𝜙𝛼𝑥Δ(𝑡)Δ+𝑞(𝑡)𝜙𝛼(𝑥(𝑡))+𝑛𝑗=1𝑞𝑗(𝑡)𝜙𝛽𝑗(𝑥(𝑡))=𝑒(𝑡),(1.3) where 𝑡[𝑡0,)𝕋, 𝛽1>𝛽2>>𝛽𝑚>𝛼>𝛽𝑚+1>>𝛽𝑛.(1.4)

Very recently, Agarwal et al. [13] further to extend the results in [12] to the case of several delays of the form𝑝(𝑡)𝜙𝛼𝑥Δ(𝑡)+𝑞(𝑡)𝜙𝛼(𝑥(𝜏(𝑡)))+𝑛𝑗=1𝑞𝑗(𝑡)𝜙𝛽𝑗𝑥𝜏𝑗(𝑡)=𝑒(𝑡),(1.5) where 𝜏(𝑡) and 𝜏𝑗(𝑡) are right-dense continuous functions satisfying lim𝑡𝜏(𝑡)=lim𝑡𝜏𝑗(𝑡)= for 𝑗=1,2,,𝑛. Sun and Kong [14] studied the oscillation of the second-order forced differential equation with the nonlinearity given by a Riemann-Stieltjes integral of the form 𝑝(𝑡)𝑥(𝑡)+𝑞(𝑡)𝑥(𝑡)+𝑏0𝑟(𝑡,𝑠)𝜙𝛾(𝑠)(𝑥(𝜏(𝑠)))𝑑𝜉(𝑠)=𝑒(𝑡),(1.6) where 𝑡0,𝛾𝐶[0,𝑏) is a strictly increasing function satisfying 0𝛾(0)<1<𝛾(𝑏). Some interval oscillation criteria of the El-Sayed type and the Kong type are established which unify many existing results in the literature.

It is obvious that (1.2), (1.3), (1.5), and (1.6) are special cases of (1.1). Some other particular cased of (1.1) can be found in [1520]. In this paper, we will establish interval oscillation criteria for the more general (1.1). Clearly, our work is of significance because (1.1) allows an infinite number of nonlinear terms and even a continuum of nonlinearities determined by the function 𝜉. Moreover, even for the special cases of (1.2), (1.3), (1.5), and (1.6), our results generalize many existing oscillation criteria in the literature.

This paper is organized as follows. We present some lemmas in Section 2 which play a key role in the proof of the main results. The main results are given in Section 3. Two examples are given to illustrate the main results in Section 4.

2. Preliminaries

We here present four lemmas which play a key role in the proof of the main results in the next section. In the sequel, we denote by 𝐿𝜉[𝑎,𝑏]𝕋1 the set of Riemann-Stieltjes integrable functions on [𝑎,𝜎(𝑏))𝕋1 with respect to 𝜉. Assume that 𝛾, 𝛾1𝐿𝜉[𝑎,𝑏]𝕋1. Let =sup{𝑠(𝑎,𝑏)𝕋1𝛾(𝑠)𝛼}.

We first present the following two Lemmas 2.1, and 2.2, which generalize Lemma  2.1 and Lemma  3.1 in [14].

Lemma 2.1. Let 𝑚1=𝛼𝜎(𝑏)𝜎()𝛾1(𝑠)Δ𝜉(𝑠)𝜎(𝑏)𝜎()Δ𝜉(𝑠)1,𝑚2=𝛼𝑎𝜎()𝛾1(𝑠)Δ𝜉(𝑠)𝑎𝜎()Δ𝜉(𝑠)1.(2.1) Then for any 𝛿(𝑚1,𝑚2), there exists 𝜂𝐿𝜉([𝑎,𝑏]𝕋1 such that 𝜂(𝑠)>0 on [𝑎,𝑏]𝕋1, 𝑎𝜎(𝑏)𝛾(𝑠)𝜂(𝑠)Δ𝜉(𝑠)=𝛼,(2.2)𝑎𝜎(𝑏)𝜂(𝑠)Δ𝜉(𝑠)=𝛿.(2.3)

Proof. By the choice of and the definitions of 𝑚1 and 𝑚2, we have that 0<𝑚1<1<𝑚2. Set 𝜂1(𝑠)=𝛼𝛾1(𝑠)𝜎(𝑏)𝜎()Δ𝜉(𝑠)1[],𝑠𝜎(),𝑏𝕋,[0,𝑠𝑎,𝜎())𝕋,𝜂2[](𝑠)=0,𝑠𝜎(),𝑏𝕋,𝛼𝛾1(𝑠)𝑎𝜎()Δ𝜉(𝑠)1[,𝑠𝑎,𝜎())𝕋.(2.4) It is easy to see that 𝜂𝑖𝐿𝜉[𝑎,𝑏]𝕋1 and 𝑎𝜎(𝑏)𝛾(𝑠)𝜂𝑖(𝑠)Δ𝜉(𝑠)=𝛼,𝑖=1,2.(2.5) Moreover, 𝑎𝜎(𝑏)𝜂1(𝑠)Δ𝜉(𝑠)=𝑚1,𝑎𝜎(𝑏)𝜂2(𝑠)Δ𝜉(𝑠)=𝑚2.(2.6) Let 𝜂(𝑠,𝑙)=(1𝑙)𝜂1(𝑠)+𝑙𝜂2[](𝑠),𝑠𝑎,𝑏𝕋1[],𝑙0,1.(2.7) Then we have that 𝑎𝜎(𝑏)𝜂(𝑠,0)Δ𝜉(𝑠)=𝑎𝜎(𝑏)𝜂1(𝑠)Δ𝜉(𝑠)=𝑚1<1,𝑎𝜎(𝑏)𝜂(𝑠,1)Δ𝜉(𝑠)=𝑎𝜎(𝑏)𝜂2(𝑠)Δ𝜉(𝑠)=𝑚2>1.(2.8) By the continuous dependence of 𝜂(𝑠,𝑙) on 𝑙, there exists 𝑙(0,1) such that 𝜂(𝑠)=𝜂(𝑠,𝑙) satisfies 𝑎𝜎(𝑏)𝜂(𝑠)Δ𝜉(𝑠)=𝛿(𝑚1,𝑚2). Note that 𝜂(𝑠)>0 on [𝑎,𝑏]𝕋1 and 𝑎𝜎(𝑏)𝛾(𝑠)𝜂(𝑠)Δ𝜉(𝑠)=𝛼. This completes the proof of Lemma 2.1.

The next lemma is a generalized arithmetic-geometric mean inequality on time scales.

Lemma 2.2. Assume that 𝑢[𝑎,𝑏]𝕋1𝑅 is right-dense continuous, 𝜂𝐿𝜉[𝑎,𝑏]𝕋, 𝑢>0, 𝜂>0 on [𝑎,𝑏]𝕋1 and 𝑎𝜎(𝑏)𝜂(𝑠)Δ𝜉(𝑠)=1. Then 𝑎𝜎(𝑏)𝜂(𝑠)𝑢(𝑠)Δ𝜉(𝑠)exp𝑎𝜎(𝑏)𝜂(𝑠)ln𝑢(𝑠)Δ𝜉(𝑠).(2.9)

Proof. Define an operator 𝐿 as follows: 𝐿(𝑓)=𝑎𝜎(𝑏)𝜂(𝑠)𝑓(𝑠)Δ𝜉(𝑠).(2.10) It is obvious that 𝐿 is a linear operator satisfying that 𝐿(1)=1 and 𝐿(𝑢)>0. To derive inequality (2.9) it suffices to show that 𝐿(𝑢)exp(𝐿(ln𝑢)).(2.11) Note that ln𝑡𝑡1 for 𝑡>0. Thus, for any 𝑠[𝑎,𝑏]𝕋1 we have ln𝑢(𝑠)𝐿(𝑢)𝑢(𝑠)𝐿(𝑢)1,(2.12) which follows that ln𝑢(𝑠)ln𝐿(𝑢)𝑢(𝑠)𝐿(𝑢)1.(2.13) Taking the operator 𝐿 on both sides of (2.13), we get 𝑢𝐿(ln𝑢)ln𝐿(𝑢)=𝐿(ln𝑢ln𝐿(𝑢))𝐿𝐿(𝑢)𝐿(1)=11=0,(2.14) which implies (2.11). This completes the proof.

The following two lemmas generalize Lemma  2.4 and Lemma  6.1 in [13].

Lemma 2.3. Let 𝜏[𝑡0,]𝕋[𝑡0,]𝕋 be a right-dense continuous function satisfying 0𝜏(𝑡)𝑡, 𝑐,𝑑[𝑡0,)𝕋 with 𝑐<𝑑, and 𝜏𝑐𝑑=min{𝜏(𝑡)𝑡[𝑐,𝑑]𝕋}. Assume 𝑥[𝜏𝑐𝑑,𝑑]𝕋 is a positive right-dense continuous function such that 𝑝(𝑡)𝜙𝛼(𝑥Δ(𝑡)) is nonincreasing on [𝜏𝑐𝑑,𝑑]𝕋. Then 𝑥(𝜏(𝑡))𝑃𝜏𝑥(𝜎(𝑡))(𝑡),𝜏𝑐𝑑𝑃𝜎(𝑡),𝜏𝑐𝑑[,𝑡𝑐,𝑑),(2.15) where 𝑃(𝑡,𝑠)=𝑡𝑠𝑝1/𝛼(𝑠)Δ𝑠.

Proof. Set 𝑧(𝑡)=𝑝1/𝛼(𝑡)𝑥Δ(𝑡). It is not difficult to verify that 𝑧(𝑡) is nonincreasing on [𝜏𝑐𝑑,𝑑]𝕋 since 𝑝(𝑡)𝜙𝛼(𝑥Δ(𝑡)) is nonincreasing on [𝜏𝑐𝑑,𝑑]𝕋. Then we have 𝜏𝑥(𝑡)=𝑥𝑐𝑑+𝑡𝜏𝑐𝑑𝑥Δ(𝜏𝑠)Δ𝑠=𝑥𝑐𝑑+𝑡𝜏𝑐𝑑𝑝1/𝛼(𝑠)𝑧(𝑠)Δ𝑠𝑧(𝑡)𝑡𝜏𝑐𝑑𝑝1/𝛼(𝑠)Δ𝑠=𝑝1/𝛼(𝑡)𝑃𝑡,𝜏𝑐𝑑𝑥Δ𝜏(𝑡),𝑡𝑐𝑑,𝑏𝕋.(2.16) Next, for 𝑠[𝜏(𝑡),𝜎(𝑡)]𝕋, and 𝑡[𝑐,𝑑)𝕋 we define 𝜖(𝑠)=𝑥(𝑠)𝑝1/𝛼(𝑠)𝑃𝑠,𝜏𝑐𝑑𝑥Δ(𝑠).(2.17) Then (2.16) yields that 𝜖(𝑠)0 for 𝑠[𝜏(𝑡),𝜎(𝑡)]𝕋 and 𝑡[𝑐,𝑑)𝕋. Consequently, for 𝑡[𝑐,𝑑)𝕋, we have 0𝜎(𝑡)𝜏(𝑡)𝑝1/𝛼(𝑠)𝜖(𝑠)𝑥(𝑠)𝑥𝜎(𝑠)Δ𝑠=𝜎(𝑡)𝜏(𝑡)𝑃𝑠,𝜏𝑐𝑑𝑥(𝑠)Δ=𝑃𝜎Δ𝑠(𝑡),𝜏𝑐𝑑𝑥𝑃𝜏(𝜎(𝑡))(𝑡),𝜏𝑐𝑑𝑥.(𝜏(𝑡))(2.18) This implies (2.15). The proof of Lemma 2.3 is complete.

Similar to the proof of Lemma 2.3, we can get the following result.

Lemma 2.4. Let 𝜏[𝑡0,]𝕋[𝑡0,]𝕋 be a right-dense continuous function satisfying 𝜏(𝑡)>𝑡, 𝑐,𝑑[𝑡0,)𝕋 with 𝑐<𝑑, and 𝜏𝑐𝑑=max{𝜏(𝑡)𝑡[𝑐,𝑑]𝕋}. If 𝑥[𝑐,𝜏𝑐𝑑]𝕋 is a positive right-dense continuous function for which 𝑝(𝑡)𝜙𝛼(𝑥Δ(𝑡)) is nonincreasing on [𝑐,𝜏𝑐𝑑]𝕋, then 𝑥(𝜏(𝑡))𝑃𝜏𝑥(𝜎(𝑡))𝑐𝑑,𝜏(𝑡)𝑃𝜏𝑐𝑑[),𝜎(𝑡),𝑡𝑐,𝑑𝕋,(2.19) where 𝑃(𝑡,𝑠) is defined as in Lemma 2.3.

3. Main Results

We note from the definition of 𝑚1 and 𝑚2 that 0<𝑚1<1<𝑚2. In the following we will use the values of 𝛿 in the interval (𝑚1,1] to establish interval criteria for oscillation of (1.1). For 𝑐,𝑑[𝑡0,)𝕋 with 𝑐<𝑑, we define the function class 𝒰(𝑐,𝑑)={𝑢𝐶1rd[𝑐,𝑑]𝕋𝑢(𝑐)=0=𝑢(𝑑),𝑢0}, where 𝐶1rd[𝑐,𝑑]𝕋 denotes the set of right-dense continuously Δ—differentiable functions on [𝑐,𝑑]𝕋. In the following, let 𝜏𝑐𝑑 and 𝜏𝑐𝑑 be defined as in Lemmas 2.3 and 2.4. Set 𝑔𝑐𝑑=min[](𝑡,𝑠)𝑐,𝑑𝕋×[]𝑎,𝑏𝕋1𝑔(𝑡,𝑠),𝑔𝑐𝑑=max[](𝑡,𝑠)𝑐,𝑑𝕋×[]𝑎,𝑏𝕋1𝑔(𝑡,𝑠).(3.1)

Theorem 3.1. Assume that 𝜏(𝑡),𝑔(𝑡,𝑠)𝑡 for 𝑡[𝑡0,)𝕋 and 𝑠[𝑎,𝑏]𝕋1. Suppose also that for any 𝑇𝑡0, there exist subintervals [𝑐𝑖,𝑑𝑖]𝕋 of [𝑇,), 𝑖=1,2, such that 𝑐𝑖,𝑑𝑖𝕋, 𝑑𝑖>𝑐𝑖, and 𝑟(𝑡,𝑠)0,(𝑡,𝑠)𝑖,𝑑𝑖𝕋×[]𝑎,𝑏𝕋1,(1)𝑖𝑒(𝑡)0,𝑡𝑖,𝑑𝑖𝕋,(3.2) where 𝑖=min{𝜏𝑐𝑖𝑑𝑖,𝑔𝑐𝑖𝑑𝑖}. For each 𝛿(𝑚1,1], let 𝜂𝐿𝜉[𝑎,𝑏]𝕋1 be defined as in Lemma 2.1. If there exists 𝑢𝑖𝒰(𝑐𝑖,𝑑𝑖) for 𝑖=1,2 such that sup𝑚𝛿1,1𝑑𝑖𝑐𝑖𝑄𝑖||𝑢(𝑡)𝑖||(𝜎(𝑡))𝛼+1||𝑢𝑝(𝑡)Δ𝑖||(𝑡)𝛼+1Δ𝑡0,(3.3) where 𝑄𝑖𝑃(𝑡)=𝑞(𝑡)𝜏(𝑡),𝜏𝑐𝑖𝑑𝑖𝑃𝜎(𝑡),𝜏𝑐𝑖𝑑𝑖𝛼+||||𝑒(𝑡)1𝛿1𝛿×exp𝑎𝜎(𝑏)𝜂(𝑠)ln𝑟(𝑡,𝑠)𝜂(𝑠)𝑃(𝑔(𝑡,𝑠),𝑔𝑐𝑖𝑑𝑖)𝑃𝜎(𝑡),𝑔𝑐𝑖𝑑𝑖𝛾(𝑠).Δ𝜉(𝑠)(3.4) Here we use the convention that ln0=, 𝑒=0, and 01𝛿=0 and (1𝛿)1𝛿=1 for 𝛿=1 due to the fact that lim𝑡0𝑡𝑡=1. Then (1.1) is oscillatory.

Proof. We prove this result by the contradiction method. Assume the contrary. Then (1.1) has an extendible solution 𝑥(𝑡) which is eventually positive or negative. Without loss of generality, we may assume that 𝑥(𝑡)>0 for all 𝑡[𝑡0,)𝕋. When 𝑥(𝑡) is eventually negative, the proof is in the same way except that the interval [𝑐2,𝑑2]𝕋, instead of [𝑐1,𝑑1]𝕋, is used. Define 𝑤(𝑡)=𝑝(𝑡)𝜙𝛼𝑥Δ(𝑡)𝜙𝛼𝑐(𝑥(𝑡)),𝑡1,𝑑1𝕋.(3.5) It follows that 𝑤Δ𝜙(𝑡)=𝑞(𝑡)𝛼(𝑥(𝜏(𝑡)))𝜙𝛼+(𝑥(𝜎(𝑡)))𝑎𝜎(𝑏)𝜙𝑟(𝑡,𝑠)𝛾(𝑠)(𝑥(𝑔(𝑡,𝑠)))𝜙𝛼𝑒(𝑥(𝜎(𝑡)))Δ𝜉(𝑠)(𝑡)𝜙𝛼𝜙(𝑥(𝜎(𝑡)))+𝑝(𝑡)𝛼𝑥Δ𝜙(𝑡)𝛼(𝑥(𝑡))Δ𝜙𝛼(𝑥(𝑡))𝜙𝛼.(𝑥(𝜎(𝑡)))(3.6) It is obvious that the conditions in Lemma 2.3 are satisfied with 𝜏 replaced by 𝑔(𝑡,𝑠). By (2.15) we have for 𝑡[𝑐1,𝑑1)𝕋𝑥(𝜏(𝑡))𝑥𝑃𝜏(𝜎(𝑡))(𝑡),𝜏𝑐1𝑑1𝑃𝜎(𝑡),𝜏𝑐1𝑑1,𝑥(𝑔(𝑡,𝑠))𝑥𝑃𝑔(𝜎(𝑡))(𝑡,𝑠),𝑔𝑐1𝑑1𝑃𝜎(𝑡),𝑔𝑐1𝑑1.(3.7) By (3.2), (3.6), and (3.7), and the fact that 𝜙 is increasing, we get 𝑤Δ𝑃(𝑡)𝑞(𝑡)𝜏(𝑡),𝜏𝑐1𝑑1𝑃𝜎(𝑡),𝜏𝑐1𝑑1𝛼𝑒(𝑡)𝜙𝛼+(𝑥(𝜎(𝑡)))𝑎𝜎(𝑏)𝑃𝑟(𝑡,𝑠)𝑔(𝑡,𝑠),𝑔𝑐1𝑑1𝑃𝜎(𝑡),𝑔𝑐1𝑑1𝛾(𝑠)[]𝑥(𝜎(𝑡))𝛾(𝑠)𝛼𝜙Δ𝜉(𝑠)+𝑝(𝑡)𝛼𝑥Δ(𝜙𝑡)𝛼(𝑥(𝑡))Δ𝜙𝛼(𝑥(𝑡))𝜙𝛼(.𝑥(𝜎(𝑡)))(3.8)
(I) We first consider the case where the supremum in (3.3) is assumed at 𝛿=1. From (3.2) and (3.8) we have that for 𝑡[𝑐1,𝑑1)𝕋𝑤Δ𝑃(𝑡)𝑞(𝑡)𝜏(𝑡),𝜏𝑐1𝑑1𝑃𝜎(𝑡),𝜏𝑐1𝑑1𝛼𝜙+𝑝(𝑡)𝛼𝑥Δ𝜙(𝑡)𝛼(𝑥(𝑡))Δ𝜙𝛼(𝑥(𝑡))𝜙𝛼+(𝑥(𝜎(𝑡)))𝑎𝜎(𝑏)𝑃𝑟(𝑡,𝑠)𝑔(𝑡,𝑠),𝑔𝑐1𝑑1𝑃𝜎(𝑡),𝑔𝑐1𝑑1𝛾(𝑠)[]𝑥(𝜎(𝑡))𝛾(𝑠)𝛼Δ𝜉(𝑠).(3.9) Let 𝜂𝐿𝜉[𝑎,𝑏]𝕋1 be defined as in Lemma 2.1 with 𝛿=1. Then 𝜂 satisfies (2.2) and (2.3) with 𝛿=1. This follows that 𝑎𝜎(𝑏)[]𝜂(𝑠)𝛾(𝑠)𝛼Δ𝜉(𝑠)=0.(3.10) Therefore, by Lemma 2.2 we have that for 𝑡[𝑐1,𝑑1]𝕋𝑎𝜎(𝑏)𝑃𝑔𝑟(𝑡,𝑠)(𝑡,𝑠),𝑔𝑐1𝑑1𝑃𝜎(𝑡),𝑔𝑐1𝑑1𝛾(𝑠)[]𝑥(𝜎(𝑡))𝛾(𝑠)𝛼=Δ𝜉(𝑠)𝑎𝜎(𝑏)𝜂(𝑠)𝜂1𝑃(𝑠)𝑟(𝑡,𝑠)𝑔(𝑡,𝑠),𝑔𝑐1𝑑1𝑃𝜎(𝑡),𝑔𝑐1𝑑1𝛾(𝑠)[]𝑥(𝜎(𝑡))𝛾(𝑠)𝛼Δ𝜉(𝑠)exp𝑎𝜎(𝑏)𝜂(𝑠)ln𝑟(𝑡,𝑠)𝑃𝜂(𝑠)𝑔(𝑡,𝑠),𝑔𝑐1𝑑1𝑃𝜎(𝑡),𝑔𝑐1𝑑1𝛾(𝑠)[]𝑥(𝜎(𝑡))𝛾(𝑠)𝛼Δ𝜉(𝑠)=exp𝑎𝜎(𝑏)𝜂(𝑠)ln𝑟(𝑡,𝑠)𝑃𝜂(𝑠)𝑔(𝑡,𝑠),𝑔𝑐1𝑑1𝑃𝜎(𝑡),𝑔𝑐1𝑑1𝛾(𝑠)Δ𝜉(𝑠)×expln𝑥(𝜎(𝑡))𝑎𝜎(𝑏)𝜂[𝛾](𝑠)(𝑠)𝛼Δ𝜉(𝑠)=exp𝑎𝜎(𝑏)𝜂(𝑠)ln𝑟(𝑡,𝑠)𝑃𝜂(𝑠)𝑔(𝑡,𝑠),𝑔𝑐1𝑑1𝑃𝜎(𝑡),𝑔𝑐1𝑑1𝛾(𝑠).Δ𝜉(𝑠)(3.11) Substituting (3.11) into (3.9) we obtain 𝑤Δ(𝑡)𝑄1𝜙(𝑡)+𝑝(𝑡)𝛼𝑥Δ𝜙(𝑡)𝛼(𝑥(𝑡))Δ𝜙𝛼(𝑥(𝑡))𝜙𝛼𝑐(𝑥(𝜎(𝑡))),𝑡1,𝑑1𝕋,(3.12) where 𝑄1(𝑡) is defined by (3.4) with 𝑖=1 and 𝛿=1. Multiplying both sides of the above inequality by |𝑢1(𝜎(𝑡))|𝛼+1 and proceeding as in the proof of Theorem 3.1 in [13], we can get a contradiction with (3.3).
(II) Now we consider the case where the supremum in (3.3) is assumed at 𝛿(𝑚1,1). Let ̃𝜂(𝑠)=𝛿1𝜂(𝑠). Then from (2.2) and (2.3), we get 𝑎𝜎(𝑏)̃𝜂(𝑠)Δ𝜉(𝑠)=1,𝑎𝜎(𝑏)[]̃𝜂(𝑠)𝛿𝛾(𝑠)𝛼Δ𝜉(𝑠)=0.(3.13) Hence for 𝑡[𝑐1,𝑑1]𝕋𝑒(𝑡)𝜙𝛼+(𝑥(𝜎(𝑡)))𝑎𝜎(𝑏)𝑃𝑔𝑟(𝑡,𝑠)(𝑡,𝑠),𝑔𝑐1𝑑1𝑃𝜎(𝑡),𝑔𝑐1𝑑1𝛾(𝑠)[]𝑥(𝜎(𝑡))𝛾(𝑠)𝛼Δ𝜉(𝑠)=𝑎𝜎(𝑏)̃𝜂(𝑠)Ω(𝑡,𝑠)Δ𝜉(𝑠),(3.14) where Ω(𝑡,𝑠)=𝛿𝑟(𝑡,𝑠)𝜂𝑃𝑔(𝑠)(𝑡,𝑠),𝑔𝑐1𝑑1𝑃𝜎(𝑡),𝑔𝑐1𝑑1𝛾(𝑠)[]𝑥(𝜎(𝑡))𝛾(𝑠)𝛼+||||𝑒(𝑡)𝑥𝛼.(𝜎(𝑡))(3.15) On the other hand, by the basic arithmetic-geometric mean inequality, we have that Ω(𝑡,𝑠)𝑟(𝑡,𝑠)𝜂(𝑠)𝛿𝑃𝑔(𝑡,𝑠),𝑔𝑐1𝑑1𝑃𝜎(𝑡),𝑔𝑐1𝑑1𝛿𝛾(𝑠)||||𝑒(𝑡)1𝛿1𝛿[]𝑥(𝜎(𝑡))𝛿𝛾(𝑠)𝛼.(3.16) Substituting (3.16) into (3.14), using Lemma 2.2 and similar to the computation in (I), for 𝑡[𝑐1,𝑑1]𝕋 we can get 𝑒(𝑡)+𝜙𝛼(𝑥(𝜎(𝑡)))𝑎𝜎(𝑏)𝑃𝑔𝑟(𝑡,𝑠)(𝑡,𝑠),𝑔𝑐1𝑑1𝑃𝜎(𝑡)𝑔𝑐1𝑑1𝛾(𝑠)𝑥𝛾(𝑠)𝛼(𝜎(𝑡))Δ𝜉(𝑠)𝑎𝜎(𝑏)̃𝜂(𝑠)𝑟(𝑡,𝑠)𝜂(𝑠)𝛿𝑃𝑔(𝑡,𝑠),𝑔𝑐1𝑑1𝑃𝜎(𝑡),𝑔𝑐1𝑑1𝛿𝛾(𝑠)||||𝑒(𝑡)1𝛿1𝛿[]𝑥(𝜎(𝑡))𝛿𝛾(𝑠)𝛼||||Δ𝜉(𝑠)𝑒(𝑡)1𝛿1𝛿exp𝑎𝜎(𝑏)𝜂(𝑠)ln𝑟(𝑡,𝑠)𝑃𝜂(𝑠)𝑔(𝑡,𝑠),𝑔𝑐1𝑑1𝑃𝜎(𝑡),𝑔𝑐1𝑑1𝛾(𝑠),Δ𝜉(𝑠)(3.17) which also implies (3.12) for 𝑡[𝑐1,𝑑1]𝕋. The rest of the proof is similar to Part (I) and hence is omitted. This completes the proof of Theorem 3.1.

For the case when 𝜏(𝑡),𝑔(𝑡,𝑠)>𝑡 for 𝑡[𝑡0,)𝕋 and 𝑠[𝑎,𝑏]𝕋1, using Lemma 2.4 and following the proof of Theorem 3.1, we have the following oscillation result for (1.1) immediately.

Theorem 3.2. Assume that 𝜏(𝑡),𝑔(𝑡,𝑠)>𝑡 for 𝑡[𝑡0,)𝕋 and 𝑠[𝑎,𝑏]𝕋1. Suppose also that for any 𝑇𝑡0, there exist subintervals [𝑐𝑖,𝑑𝑖]𝕋 of [𝑇,), 𝑖=1,2, such that 𝑐𝑖,𝑑𝑖𝕋, 𝑑𝑖>𝑐𝑖, and (3.2) holds for 𝑡[𝑐𝑖,𝑖)𝕋 and 𝑠[𝑎,𝑏]𝕋, where 𝑖=max{𝜏𝑐𝑖𝑑𝑖,𝑔𝑐𝑖𝑑𝑖}. For each 𝛿(𝑚1,1], let 𝜂𝐿𝜉[𝑎,𝑏]𝕋1 be defined as in Lemma 2.1. If there exists 𝑢𝑖𝒰(𝑐𝑖,𝑑𝑖) for 𝑖=1,2 such that (3.4) holds, where 𝑄𝑖𝑃𝜏(𝑡)=𝑞(𝑡)𝑐1𝑑1,𝜏(𝑡)𝑃𝜏𝑐1𝑑1,𝜎(𝑡)𝛼+||||𝑒(𝑡)1𝛿1𝛿×exp𝑎𝜎(𝑏)𝜂(𝑠)ln𝑟(𝑡,𝑠)𝑃𝑔𝜂(𝑠)𝑐1𝑑1,𝑔(𝑡,𝑠)𝑃𝑔𝑐1𝑑1,𝜎(𝑡)𝛾(𝑠).Δ𝜉(𝑠)(3.18) Then (1.1) is oscillatory.

Remark 3.3. We see from the proof of Lemma 2.1 in Section 2 that for each 𝛿(𝑚1,1], the function 𝜂 can be constructed explicitly for any nondecreasing function 𝜉, and hence the functions 𝑄𝑖 in Theorems 3.1 and 3.2 are explicitly given.

Remark 3.4. We observe that in Theorems 3.1 and 3.2, if the supremum in (3.3) is assumed at 𝛿=1, the effect of 𝑒(𝑡) is neglected in some extent. This implies that the magnitude of 𝑒(𝑡) in [𝑐𝑖,𝑑𝑖]𝕋 cannot be large. For otherwise, the supremum would have been taken at some 𝛿(𝑚1,1).
Now, we interpret the results for (1.1) to the special case of (1.5). Set 𝕋1=, 𝑎=1, 𝑏=𝑛+1 for 𝑛, and 𝛾𝜉(𝑠)=𝑠,𝑠=1,2,,𝑛+1,(𝑠)=𝛽𝑠satisfying(1.4),𝑠=1,2,,𝑛,𝑟(𝑡,𝑠)=𝑞𝑠(𝑡),𝑠=1,2,,𝑛,𝑔(𝑡,𝑠)=𝜏𝑠(𝑡),𝑠=1,2,,𝑛.(3.19) Then (1.1) reduces to (1.5). By a straightforward computation, we have that 𝑚1=𝛼𝑛𝑚𝑛𝑗=𝑚+1𝛽𝑗1,𝑚2=𝛼𝑚𝑚𝑗=1𝛽𝑗1.(3.20) Then Lemma 2.1 can be restated as the following: for any 𝛿(𝑚1,𝑚2), there exists a positive 𝑛-tuple (𝜂1,,𝜂𝑛) satisfying 𝑛𝑗=1𝛼𝑗𝜂𝑗=𝛼,𝑛𝑗=1𝜂𝑗=𝛿.(3.21) Therefore, by Theorems 3.1 and 3.2, we obtain the following oscillation results for (1.5) which generalize the results in [13].

Corollary 3.5. Assume that 𝜏(𝑡),𝜏𝑗(𝑡)𝑡 for 𝑡[𝑡0,)𝕋 and 𝑗=1,,𝑛. Suppose also that for any 𝑇𝑡0, there exist subintervals [𝑐𝑖,𝑑𝑖]𝕋 of [𝑇,), 𝑖=1,2, such that 𝑐𝑖,𝑑𝑖𝕋, 𝑑𝑖>𝑐𝑖, and 𝑞𝑗(𝑡)0,(1)𝑖𝑒𝜃(𝑡)0,𝑡𝑖,𝑑𝑖𝕋,(3.22) where 𝜃𝑖=min{𝜔𝑖,𝜔𝑖𝑗𝑗=1,2,,𝑛}, 𝜔𝑖=min{𝜏(𝑡)𝑡[𝑐𝑖,𝑑𝑖]𝕋} and 𝜔𝑖𝑗=min{𝜏𝑗(𝑡)𝑡[𝑐𝑖,𝑑𝑖]𝕋}. For each 𝛿(𝑚1,1], let (𝜂1,,𝜂𝑛) be defined by (3.21). We further assume that there exists a function 𝑢𝑖𝒰(𝑎𝑖,𝑏𝑖) such that (3.4) holds, where 𝑄𝑖𝑃𝜏(𝑡)=𝑞(𝑡)(𝑡),𝜔𝑖𝑃𝜎(𝑡),𝜔𝑖𝛼+||||𝑒(𝑡)1𝛿𝑛1𝛿𝑗=1𝑞𝑗(𝑡)𝜂𝑗𝜂𝑗𝑃𝜏𝑗(𝑡),𝜔𝑖𝑗𝑃𝜎(𝑡),𝜔𝑖𝑗𝛽𝑗𝜂𝑗.(3.23) Then (1.5) is oscillatory.

Corollary 3.6. Assume that 𝜏(𝑡),𝜏𝑗(𝑡)>𝑡 for 𝑡[𝑡0,)𝕋 and 𝑗=1,,𝑛. Suppose also that for any 𝑇𝑡0, there exist subintervals [𝑐𝑖,𝑑𝑖]𝕋 of [𝑇,), 𝑖=1,2, such that 𝑐𝑖,𝑑𝑖𝕋, 𝑑𝑖>𝑐𝑖, and 𝑞𝑗(𝑡)0,(1)𝑖𝑐𝑒(𝑡)0,𝑡𝑖,𝜃𝑖𝕋,(3.24) where 𝜃𝑖=max{𝜔𝑖,𝜔𝑖𝑗𝑗=1,2,,𝑛}, 𝜔𝑖=max{𝜏(𝑡)𝑡[𝑐𝑖,𝑑𝑖]𝕋} and 𝜔𝑖𝑗=max{𝜏𝑗(𝑡)𝑡[𝑐𝑖,𝑑𝑖]𝕋}. For each 𝛿(𝑚1,1], let (𝜂1,,𝜂𝑛) be defined by (3.21). We further assume there exists a function 𝑢𝑖𝒰(𝑎𝑖,𝑏𝑖) such that (3.4) holds, where 𝑄𝑖𝑃(𝑡)=𝑞(𝑡)𝜔𝑖,𝜏(𝑡)𝑃𝜔𝑖,𝜎(𝑡)𝛼+||||𝑒(𝑡)1𝛿𝑛1𝛿𝑗=1𝑞𝑗(𝑡)𝜂𝑗𝜂𝑗𝑃𝜔𝑖𝑗,𝜏𝑗(𝑡)𝑃𝜔𝑖𝑗,𝜎(𝑡)𝛽𝑗𝜂𝑗.(3.25) Then (1.5) is oscillatory.

Remark 3.7. Corollaries 3.5 and 3.6 generalize those results in [13] since the sufficient condition for oscillation of (1.5) is given here in the form of sup𝛿(𝑚1,1]𝑑𝑖𝑐𝑖[]Δ𝑡0.

4. Examples

We will give two examples to illustrate Theorems 3.1 and 3.2 in the case when 𝜉(𝑠)=𝑠, 𝕋1=, and 𝕋=().

Example 4.1. Consider on 𝕋= the following differential equation 𝜙3/2𝑥(𝑡)+𝑘1sin𝑡𝜙3/2𝑥𝜋𝑡4+𝑘2sin𝑡21𝜙𝑠(𝑥(𝑡))𝑑𝑠=𝑓(𝑡)cos𝑡,(4.1) where 𝑡0, 𝑘1,𝑘2>0 are constants, 𝑓𝐶[0,) is an arbitrary nonnegative function. Here we have 𝑝(𝑡)=1, 𝑞(𝑡)=𝑘1sin𝑡, 𝑟(𝑡,𝑠)=𝑘2sin𝑡, 𝑒(𝑡)=𝑓(𝑡)cos𝑡, 𝑎=1, 𝑏=2, 𝛾(𝑠)=𝑠, 𝜏(𝑡)=𝑡2𝜋, and 𝑔(𝑡,𝑠)=𝑡. For any 𝑇0, we choose 𝑘 large enough so that 2𝑘𝜋𝑇 and let 𝑐1=2𝑘𝜋+𝜋/4, 𝑑1=2𝑘𝜏+𝜋/2, 𝑐2=2𝑘𝜋+3𝜋/4 and 𝑑2=2𝑘𝜋+𝜋. Then it is easy to verify that (3.2) holds, and 𝑄𝑖(𝑡)=𝑘1sin𝑡𝑡𝑐𝑖𝑡𝑐𝑖+(𝜋/4)3/2+𝑘2sin𝑡,𝑖=1,2.(4.2)
A straightforward computation yields that 𝑚1=3ln(4/3)=0.863 and 𝑚2=3ln(3/2)=1.2164. By Lemma 2.1, for any 𝛿(𝑚1,𝑚2), there exists a positive Riemann integrable function on [1,2] such that (2.2) and (2.3) hold. Particularly, we can choose 𝛿=1 and hence 𝜂(𝑠)=1. If we choose 𝑢𝑖(𝑡)=(𝑡𝑐𝑖)(𝑑𝑖𝑡) for 𝑖=1,2, then we have 𝑑𝑖𝑐𝑖𝑄𝑖||𝑢(𝑡)𝑖||(𝑡)5/2=𝑑𝑡𝜋/2𝜋/4𝑘sin𝑡1(𝑡(𝜋/4))4((𝜋/2)𝑡)5/2𝑡3/2+𝑘2𝜋𝑡4𝜋2𝑡5/2𝑑𝑡,𝑑𝑖𝑐𝑖||𝑢𝑖||(𝑡)5/2𝑑𝑡=𝜋/2𝜋/4|||3𝜋4|||2𝑡5/22𝑑𝑡=7𝜋47/2.(4.3) If there exist positive constants 𝑘1 and 𝑘2 such that 𝑑𝑖𝑐𝑖𝑄𝑖||𝑢(𝑡)𝑖||(𝑡)5/22𝑑𝑡7𝜋47/2,(4.4) then condition (3.3) holds. By Theorem 3.1, we have that (4.1) is oscillatory.

Example 4.2. Consider on 𝕋= the following difference equation 𝜙3/2𝑥Δ(𝑡)Δ+𝑞(𝑡)𝜙3/2(𝑥(𝑡+1))+𝑟(𝑡)21𝜙𝑠(𝑥(𝑡+1))𝑑𝑠=𝑒(𝑡),(4.5) where 𝑡, 𝑘𝑞(𝑡)=1𝑓,𝑡=8𝑗,8𝑗+1,8𝑗+2,8𝑗+3,8𝑗+4,8𝑗+5,1(𝑘𝑡),𝑡=8𝑗+6,8𝑗+7,𝑟(𝑡)=2𝑓,𝑡=8𝑗,8𝑗+1,8𝑗+2,8𝑗+3,8𝑗+4,8𝑗+5,2𝑓(𝑡),𝑡=8𝑗+6,8𝑗+7,𝑒(𝑡)=3(𝑓𝑡),𝑡=8𝑗,8𝑗+1,8𝑗+2,4𝑓(𝑡),𝑡=8𝑗+3,8𝑗+4,8𝑗+5,5(𝑡),𝑡=8𝑗+6,8𝑗+7,(4.6) for 𝑗,𝑘1 and 𝑘2 are positive constants, 𝑓𝑖(𝑡) (𝑖=1,2,3,4,5) are arbitrary real-valued functions with 𝑓3(𝑡)0 and 𝑓4(𝑡)0. For any 𝑁, we can choose 𝑗 large enough so that 8𝑗𝑁. Let 𝑐1=8𝑗+1, 𝑑1=8𝑗+3, 𝑐2=8𝑗+4, and 𝑑2=8𝑗+6. Then (3.2) is valid. Choose 𝜂(𝑠)=1 and 𝑢𝑖(𝑡)=(𝑡𝑐𝑖)(𝑑𝑖𝑡) for 𝑖=1,2. Then (2.2) and (2.3) hold with 𝛿=1. By the straightforward computation, we get 𝑑𝑖𝑐𝑖𝑄𝑖||𝑢(𝑡)𝑖||(𝜎(𝑡))5/2||𝑢Δ𝑖||(𝑡)5/2Δ𝑡=𝑘1+𝑘21.(4.7) By Theorem 3.2, (4.5) is oscillatory if 𝑘1+𝑘21.

Acknowledgments

The author thanks the reviewer for his/her valuable comments on this paper. This paper was supported by the national natural science foundations of China (60704039, 61174217) and the natural science foundations of Shandong Province (ZR2010AL002, JQ201119).