Forced Oscillation of Second-Order Half-Linear Dynamic Equations on Time Scales

Abstract

We will establish a new interval oscillation criterion for second-order half-linear dynamic equation (𝑟(𝑡)[𝑥^Δ(𝑡)]^𝛼)^Δ+𝑝(𝑡)𝑥^𝛼(𝜎(𝑡))=𝑓(𝑡) on a time scale 𝕋 which is unbounded, which is a extension of the oscillation result for second order linear dynamic equation established by Erbe et al. (2008). As an application, we obtain a sufficient condition of oscillation of the second-order half-linear differential equation ([𝑥^′(𝑡)]^𝛼)^′+𝑐sin𝑡𝑥^𝛼(𝑡)=cos𝑡, where 𝛼=𝑝/𝑞, 𝑝, 𝑞 are odd positive integers.

1. Introduction

The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. thesis [1] in order to unify continuous and discrete analysis. Not only can this theory of so-called “dynamic equations” unify the theories of differential equations and of difference equations, but also it is able to extend these classical cases to cases “in between,” for example, to so-called q-difference equations. A time scale 𝕋 is an arbitrary 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. Many other interesting time scales exist, and they give rise to plenty of applications, among them the study of population dynamic models (see [2]). A book on the subject of time scale by Bohner and Peterson [2] summarizes and organizes much of the time scale calculus (see also [3]). For the notions used below, we refer to [2] and to the next section, where we recall some of the main tools used in the subsequent sections of this paper.

In the last years, there has been much research activity concerning the oscillation and nonoscillation of solutions of some dynamic equations on time scales, and we refer the reader to the paper [4–6]. Following this trend, in this paper we will provide some sufficient conditions for oscillation of second-order half-linear dynamic equation.

Consider the second-order half-linear dynamic equation(𝑟(𝑡)[𝑥^Δ(𝑡)]^𝛼)^Δ+𝑝(𝑡)𝑥^𝛼(𝜎(𝑡))=𝑓(𝑡) (1.1) on a time scale 𝕋 which is unbounded above, 𝑟(𝑡), 𝑝(𝑡), and 𝑓(𝑡) are rd-continuous functions. 𝛼 is a quotient of odd positive integer. When 𝛼=1, (1.1) is the second-order linear dynamic equation(𝑟(𝑡)(𝑥^Δ(𝑡)))^Δ+𝑝(𝑡)𝑥(𝜎(𝑡))=𝑓(𝑡). (1.2)

In [7], by using the Riccati substitution the authors established a interval oscillation criterion, that is, a criterion given by the behavior of 𝑝(𝑡) and 𝑞(𝑡) on a sequence of subintervals of [𝑎,∞)_𝕋. In this paper, we extend the result of [7] to the second-order half-linear dynamic (1.1). As a application, we prove the equation([𝑥^′(𝑡)]^𝛼)^′+𝑐sin𝑡𝑥^𝛼(𝑡)=cos𝑡 (1.3) is oscillatory, if 𝑐≥𝑐₀, where 𝛼=𝑝/𝑞, 𝑝,𝑞 are odd positive integers and 𝑐₀ is defined in Example 3.1.

For completeness, (see [2, 3] for elementary results for the time scale calculus), we recall some basic results for dynamic equations and the calculus on time scales. Let 𝕋 be a time scale (i.e., a closed nonempty subset of ℝ) with sup𝕋=∞. The forward jump operator is defined by 𝜎(𝑡)=inf{𝑠∈𝕋∶𝑠>𝑡} (1.4) and the backward jump operator is defined by 𝜌(𝑡)=sup{𝑠∈𝕋∶𝑠<𝑡}, (1.5) where sup∅=inf𝕋, where ∅ denotes the empty set. If 𝜎(𝑡)>𝑡, we say 𝑡 is right-scattered, while if 𝜌(𝑡)<𝑡, we say 𝑡 is left-scattered. If 𝜎(𝑡)=𝑡, we say 𝑡 is right-dense, while if 𝜌(𝑡)=𝑡 and 𝑡≠inf𝕋, we say 𝑡 is left-dense. Given a time scale interval [𝑐,𝑑]_𝕋∶={𝑡∈𝕋∶𝑐≤𝑡≤𝑑} in 𝕋 the notation [𝑐,𝑑]^𝜅_𝕋 denotes the interval [𝑐,𝑑]_𝕋 in case 𝜌(𝑑)=𝑑 and denotes the interval [𝑐,𝑑)_𝕋 in case 𝜌(𝑑)<𝑑. The graininess function 𝜇 for a time scale 𝕋 is defined by 𝜇(𝑡)=𝜎(𝑡)−𝑡, and for any function 𝑓∶𝕋→ℝ the notation 𝑓^𝜎(𝑡) denotes 𝑓(𝜎(𝑡)). We say that 𝑥∶𝕋→ℝ is differentiable at 𝑡∈𝕋 provided that𝑥^Δ(𝑡)∶=lim_𝑠→𝑡𝑥(𝑡)−𝑥(𝑠)𝑡−𝑠 (1.6) exists, when 𝜎(𝑡)=𝑡 (here by 𝑠→𝑡 it is understood that 𝑠 approaches 𝑡 in the time scale) and when 𝑥 is continuous at 𝑡 and 𝜎(𝑡)>𝑡, 𝑥^Δ(𝑡)∶=𝑥(𝜎(𝑡))−𝑥(𝑡).𝜇(𝑡) (1.7) Note that if 𝕋=ℝ, then the delta derivative is just the standard derivative, and when 𝕋=ℤ the delta derivative is just the forward difference operator. Hence, our results contain the discrete and continuous cases as special cases and generalize these results to arbitrary time scales (e.g., the time scale 𝑞^ℕ0∶={1,𝑞,𝑞²,…} which is very important in quantum theory [8]).

2. Main Theorem

Theorem 2.1. Assume that given any 𝑇∈[𝑎,∞)_𝕋, there exists points 𝑇≤𝑠₁<𝑡₁≤𝑠₂<𝑡₂ such that 𝑓(𝑡)≤0,𝑡∈[𝑠₁,𝑡₁]_𝕋,≥0,𝑡∈[𝑠₂,𝑡₂]_𝕋. (2.1) Further assume that there exists a function 𝑢∈𝐶¹_𝑟𝑑 such that for 𝑖=1,2, one has 𝑄_𝑖[𝑢]∶=^𝑡𝑖_𝑠𝑖[𝑟(𝑡)(𝑢^Δ(𝑡))^𝛼+1−𝑝(𝑡)(𝑢^𝜎(𝑡))^𝛼+1]Δ𝑡 (2.2) satisfies 𝑄_𝑖[𝑢]≤0, and 𝑢(𝑡)≢0 on [𝑠_𝑖,𝑡_𝑖]_𝕋, with 𝑢(𝑠_𝑖)=0=𝑢(𝑡_𝑖). Then the dynamic (1.1) is oscillatory on [𝑎,∞)_𝕋.

Remark 2.2. When 𝛼=1, the above theorem becomes [7, Theorem 2.1].

Proof. Assume that (1.1) is nonoscillatory. Then there is a solution 𝑥(𝑡) of (1.1) and a 𝑇∈[𝑎,∞)_𝕋 such that 𝑥(𝑡) is of one sign on [𝑇,∞)_𝕋. We consider the case 𝑥(𝑡)>0 on [𝑇,∞)_𝕋. Make the substitution 𝑥𝜔(𝑡)=𝑟(𝑡)[^Δ(𝑡)]𝑥(𝑡)^𝛼𝑢^𝛼+1(𝑡),𝑡∈[𝕋,∞)_𝕋. (2.3) Then, for 𝑡∈[𝑠₁,𝑡₁]_𝕋 (note that 𝑓(𝑡)≤0 on [𝑠₁,𝑡₁]_𝕋), 𝜔^Δ𝑥(𝑡)=[𝑟(𝑡)(^Δ(𝑡))𝑥(𝑡)^𝛼]^Δ𝑢^𝛼+1𝑥(𝜎(𝑡))+𝑟(𝑡)[^Δ(𝑡)]𝑥(𝑡)^𝛼(𝑢^𝛼+1(𝑡))^Δ=−𝑝(𝑡)𝑢^𝛼+1(𝜎(𝑡))+𝑟(𝑡)(𝑢^Δ(𝑡))^𝛼+1+𝑓(𝑡)𝑥^𝛼𝑢(𝜎(𝑡))^𝛼+1(𝜎(𝑡))−𝑟(𝑡)[(𝑢^Δ(𝑡))^𝛼+1𝑥−(^Δ(𝑡))𝑥(𝑡)^𝛼(𝑢^𝛼+1(𝑡))^Δ+(𝑥^Δ(𝑡))^𝛼(𝑥^𝛼(𝑡))^Δ𝑥^𝛼(𝑡)𝑥^𝛼𝑢(𝜎(𝑡))^𝛼+1(𝜎(𝑡))]≤−𝑝(𝑡)𝑢^𝛼+1(𝜎(𝑡))+𝑟(𝑡)(𝑢^Δ(𝑡))^𝛼+1−𝑟(𝑡)[(𝑢^Δ(𝑡))^𝛼+1𝑥−(^Δ(𝑡))𝑥(𝑡)^𝛼(𝑢^𝛼+1(𝑡))^Δ+(𝑥^Δ(𝑡))^𝛼(𝑥^𝛼(𝑡))^Δ𝑥^𝛼(𝑡)𝑥^𝛼𝑢(𝜎(𝑡))^𝛼+1(𝜎(𝑡))]. (2.4) If we define 𝐹(𝑡)∶=𝑟(𝑡)[(𝑢^Δ(𝑡))^𝛼+1𝑥−(^Δ(𝑡))𝑥(𝑡)^𝛼(𝑢^𝛼+1(𝑡))^Δ+(𝑥^Δ(𝑡))^𝛼(𝑥^𝛼(𝑡))^Δ𝑥^𝛼(𝑡)𝑥^𝛼𝑢(𝜎(𝑡))^𝛼+1(𝜎(𝑡))], (2.5) then we have 𝜔^Δ(𝑡)≤−𝑝(𝑡)𝑢^𝛼+1(𝜎(𝑡))+𝑟(𝑡)(𝑢^Δ(𝑡))^𝛼+1−𝐹(𝑡),𝑡∈[𝑠₁,𝑡₁]_𝕋. (2.6)
(i) Suppose that 𝑡∈[𝑠₁,𝑡₁]_𝕋 is right-dense. Then, (𝑢^𝛼+1(𝑡))^Δ=(𝛼+1)𝑢^𝛼(𝑡)𝑢^Δ(𝑡), (2.7) so we have 𝐹(𝑡)=(𝛼+1)𝑟(𝑡)[(𝑢^Δ(𝑡))^𝛼+1𝛼+1−𝑢^Δ𝑥(𝑡)[^Δ(𝑡)𝑢(𝑡)]𝑥(𝑡)^𝛼+[(𝑥^Δ(𝑡)𝑢(𝑡)/𝑥(𝑡))^𝛼]^{(𝛼+1)/𝛼}(𝛼+1)/𝛼]. (2.8) We use Young's inequality [9], which says that |𝐴|^𝑝𝑝−𝐴𝐵+|𝐵|^𝑞𝑞1≥0,𝑝>1,𝑞>1,𝑝+1𝑞=1, (2.9) with equality if and only if 𝐵=𝐴^𝛼,𝛼∶=𝑝/𝑞.
So if we let 𝐴=𝑢^Δ𝑥(𝑡),𝐵=[^Δ(𝑡)𝑢(𝑡)]𝑥(𝑡)^𝛼,𝑝=𝛼+1,𝑞=𝛼+1𝛼, (2.10) then we have 𝐹(𝑡)≥0,𝑡∈[𝑠₁,𝑡₁]_𝕋, and 𝑥𝐹(𝑡)=0,iff^Δ(𝑡)𝑢(𝑡)𝑥(𝑡)=𝑢^Δ(𝑡). (2.11)
(ii) Suppose next that 𝑡∈[𝑠₁,𝑡₁)_𝕋 is right-scattered and 𝑢(𝑡)≠0. Then, 𝑥^Δ(𝑡)=𝑥(𝜎(𝑡))−𝑥(𝑡)𝜇(𝑡),(𝑥^𝛼(𝑡))^Δ=𝑥^𝛼(𝜎(𝑡))−𝑥^𝛼(𝑡),𝑢𝜇(𝑡)_Δ(𝑡)=𝑢(𝜎(𝑡))−𝑢(𝑡)𝜇(𝑡),(𝑢^𝛼+1(𝑡))^Δ=𝑢^𝛼+1(𝜎(𝑡))−𝑢^𝛼+1(𝑡).𝜇(𝑡) (2.12)
Let us put 𝑎∶=𝑢(𝜎(𝑡))/𝑢(𝑡), 𝑏∶=𝑥(𝜎(𝑡))/𝑥(𝑡). Then we have 𝐹(𝑡)=𝑟(𝑡)𝑢^𝛼+1(𝑡)𝜇^𝛼+1(𝑡)𝑓(𝑎,𝑏), (2.13) where 𝑓(𝑎,𝑏)∶=(1−𝑎⁻¹)^𝛼+1−(𝑏−1)^𝛼(1−𝑎^−(𝛼+1))+(𝑏−1)^𝛼(1−𝑏^−𝛼).
Note that 𝑓(𝑏,𝑏)=0 and 𝜕𝑓𝜕𝑎(𝑎,𝑏)=(𝛼+1)𝑎⁻²𝑎^𝛼[(𝑎−1)^𝛼−(𝑏−1)^𝛼]. (2.14) It follows that if 𝑎>𝑏, then 𝜕𝑓/𝜕𝑎>0, and so 𝑓(𝑎,𝑏)>0. Likewise, if 𝑎<𝑏, then 𝜕𝑓/𝜕𝑎<0, and so 𝑓(𝑎,𝑏)>0.
In other words, 𝑓(𝑎,𝑏)≥0 and 𝑓(𝑎,𝑏)=0⇔𝑎=𝑏⇔𝑢(𝜎(𝑡))=𝑢(𝑡)𝑥(𝜎(𝑡))⇔𝑥𝑥(𝑡)^Δ(𝑡)=𝑢𝑥(𝑡)^Δ(𝑡).𝑢(𝑡) (2.15)
(iii) Suppose next that 𝑡∈[𝑠₁,𝑡₁)_𝕋 is right-scattered but 𝑢(𝑡)=0. It is easy to get that 𝐹(𝑡)=𝑟(𝑡)𝑢^𝛼+1(𝜎(𝑡))𝜇^𝛼+1(𝑡)[1−(1−𝑥(𝑡))𝑥(𝜎(𝑡))^𝛼]. (2.16) So 𝐹(𝑡)≥0 and 𝑥𝐹(𝑡)=0⇔𝑢(𝜎(𝑡))=0⇔^Δ(𝑡)𝑢(𝑡)𝑥(𝑡)=𝑢^Δ(𝑡). (2.17)
From (i), (ii), and (iii), we get that 𝐹(𝑡)≥0,𝑡∈[𝑠₁,𝑡₁)_𝕋 and 𝑥𝐹(𝑡)=0,iff^Δ(𝑡)𝑢(𝑡)𝑥(𝑡)=𝑢^Δ(𝑡). (2.18)
Integrating (2.6) from 𝑠₁ to 𝑡₁ (using 𝑢(𝑠₁)=𝑢(𝑡₁)=0), we get that 0≤^𝑡1_𝑠1[𝑟(𝑡)(𝑢^Δ(𝑡))^𝛼+1−𝑝(𝑡)𝑢^𝛼+1(𝜎(𝑡))]Δ𝑡−^𝑡1_𝑠1𝐹(𝑡)Δ𝑡. (2.19) Since 𝑄₁[𝑢]≤0, we obtain that ^𝑡1_𝑠1𝐹(𝑡)Δ𝑡≤0. (2.20) From (2.18) and 𝐹(𝑡)≥0, 𝑡∈[𝑠₁,𝑡₁]_𝕋, we get that 𝐹(𝑡)≡0, 𝑡∈[𝑠₁,𝑡₁)_𝕋. That is, 𝑥^Δ(𝑡)𝑢(𝑡)𝑥(𝑡)=𝑢^Δ(𝑡). (2.21) So, (𝑢(𝑡))𝑥(𝑡)^Δ=0. (2.22) Hence 𝑢(𝑡)/𝑥(𝑡)=𝐶. From 𝑢(𝑠₁)=0, we get that 𝑢(𝑡)≡0,𝑡∈[𝑠₁,𝑡₁]_𝕋, which is a contradiction.

3. Example

Example 3.1. Consider the second-order half-linear differential equation ([𝑥^′(𝑡)]^𝛼)^′+𝑐sin𝑡𝑥^𝛼(𝑡)=cos𝑡, (3.1) where 𝛼=𝑝/𝑞, 𝑝,𝑞 are odd positive integers. Let 𝑢(𝑡)=sin^𝑞𝜋2𝑡,2𝑘𝜋+2≤𝑡≤2𝑘𝜋+𝜋,−sin^𝑞𝜋2𝑡,2(𝑘+1)𝜋≤𝑡≤2(𝑘+1)𝜋+2,𝑘∈ℕ₀, (3.2) and [𝑠₁,𝑡₁]=[2𝑘𝜋+𝜋/2,2𝑘𝜋+𝜋]. We have 𝑄₁[𝑢]∶=^{2𝑘𝜋+𝜋}_{2𝑘𝜋+𝜋/2}[(𝑢^′(𝑡))^{(𝑝+𝑞)/𝑞}−𝑐sin𝑡𝑢^{(𝑝+𝑞)/𝑞}(𝑡)]𝑑𝑡=^𝜋_𝜋/2{[2𝑞sin^𝑞−12𝑡cos2𝑡]^{(𝑝+𝑞)/𝑞}−𝑐sin𝑡sin^𝑝+𝑞2𝑡}𝑑𝑡=₀^𝜋/2{[2𝑞sin^𝑞−12𝑠cos2𝑠]^{(𝑝+𝑞)/𝑞}−𝑐cos𝑠sin^𝑝+𝑞=12𝑠}𝑑𝑠2^𝜋₀[2𝑞sin^𝑞−1𝑡cos𝑡]^{(𝑝+𝑞)/𝑞}𝑑𝑡−2^𝑝+𝑞𝑐₀^𝜋/2sin^𝑝+𝑞𝑡cos^𝑝+𝑞+1=1𝑡𝑑𝑡2₀^𝜋/2[2𝑞sin^𝑞−1𝑡cos𝑡]^{(𝑝+𝑞)/𝑞}1𝑑𝑡+2₀^𝜋/2[2𝑞cos^𝑞−1𝑡sin𝑡]^{(𝑝+𝑞)/𝑞}𝑑𝑡−2^𝑝+𝑞𝑐Γ((𝑝+𝑞+1)/2)Γ((𝑝+𝑞+2)/2)=12Γ(𝑝+𝑞+3/2)2(2𝑞)^{(𝑝+𝑞)/𝑞}Γ((𝑝+𝑞)/2−𝑝/2𝑞)Γ(1+𝑝/2𝑞)Γ((𝑝+𝑞)/2+1)−2^𝑝+𝑞𝑐Γ((𝑝+𝑞+1)/2)Γ((𝑝+𝑞+2)/2).2Γ(𝑝+𝑞+3/2) (3.3) Noticing that 𝑝+𝑞 is even number and Γ(𝑝+𝑞2+𝑝2𝑞)=(𝑝+𝑞2𝑝−1+2𝑞)(𝑝+𝑞2𝑝−2+𝑝2𝑞)⋯(1+𝑝2𝑞)Γ(1+2𝑞), (3.4) using the following formula (see [10]) Γ(𝑛+𝑧)Γ(𝑛−𝑧)=𝜋𝑧sin𝜋𝑧[(𝑛−1)!]^2𝑛−1_𝑘=1𝑧(1−²𝑘²1),Γ(𝑛+2)=(2𝑛−1)!!2^𝑛√𝜋, (3.5) we get that 𝑄₁1[𝑢]=2(2𝑞)^{(𝑝+𝑞)/𝑞}Γ((𝑝+𝑞)/2−𝑝/2𝑞)Γ((𝑝+𝑞)/2+𝑝/2𝑞)((𝑝+𝑞)/2−1+𝑝/2𝑞)((𝑝+𝑞)/2−2+𝑝/2𝑞)⋯(1+𝑝/2𝑞)((𝑝+𝑞)/2)!−2^𝑝+𝑞𝑐((𝑝+𝑞−1)!!/2^{(𝑝+𝑞)/2})√𝜋⋅((𝑝+𝑞)/2)!((2(𝑝+𝑞+1)−1)!!/2^𝑝+𝑞)√𝜋=12(2𝑞)^{(𝑝+𝑞)/𝑞}((𝑝𝜋/2𝑞)/sin(𝑝𝜋/2𝑞))[((𝑝+𝑞)/2−1)!]²∏^{(𝑝+𝑞)/2−1}_𝑘=1[1−𝑝²/4𝑘²𝑞²]((𝑝+𝑞)/2−1+𝑝/2𝑞)((𝑝+𝑞)/2−2+𝑝/2𝑞)⋯(1+𝑝/2𝑞)((𝑝+𝑞)/2)!−2^{3(𝑝+𝑞)/2}𝑐(𝑝+𝑞−1)!!((2(𝑝+𝑞)+1)!!𝑝+𝑞2)! (3.6) It is easy to see that 𝑄₂[𝑢]=𝑄₁[𝑢], so from (3.6), we obtain that when 𝑐≥𝑝𝜋(2𝑞)^𝔄/𝑞2^−3𝔄/2(2𝔄+1)!![(𝔄/2−1)!]²∏^𝔄/2−1_𝑘=1[1−𝑝²/4𝑘²𝑞²]4𝑞sin(𝑝𝜋/2𝑞)(𝑝+𝑞−1)!!(𝔄/2−1+𝑝/2𝑞)(𝔄/2−2+𝑝/2𝑞)⋯(1+𝑝/2𝑞)[(𝔄/2)!]², (3.7)where 𝔄 denotes (𝑝+𝑞)., equation (3.1) is oscillatory.
In particular, take 𝑝=1,𝑞=3. From (3.7), we get that when 𝑐≥525𝜋√³6,512 (3.8) the second-order half-linear equation ([𝑥^′(𝑡)]^1/3)^′+𝑐sin𝑡𝑥^1/3(𝑡)=cos𝑡 (3.9) is oscillatory.

Example 3.2. Consider the second-order half-linear difference equation Δ([Δ𝑥(𝑛)]^𝛼)+𝑝(𝑛)𝑥^𝛼(𝑛+1)=𝑛^𝛽𝑓(𝑛), (3.10) where 𝑝(𝑛)=𝑐(−1)^𝑛, 𝛼 is a quotient of odd positive integers, 𝛽∈ℝ,𝑓(𝑛)=0,𝑛=4𝑘or4𝑘+2,1,𝑛=4𝑘+1,−1,𝑛=4𝑘+3,𝑘∈ℕ₀, (3.11) Let 𝑢(𝑛)=𝑓(𝑛),𝑡∈𝑁₀, and note that 𝑓(𝑛)=≥0,𝑛=4𝑘,4𝑘+1,4𝑘+2𝑘∈ℕ₀,≤0,𝑛=4𝑘+2,4𝑘+3,4𝑘+4∈ℕ₀. (3.12)
Furthermore, we have (note that (−1)^𝛼+1=1) 𝑄₁[𝑢]∶=^2𝑘+2_2𝑘[(𝑢^Δ(𝑛))^𝛼+1−𝑝(𝑛)𝑢^𝛼+1(𝑛+1)]Δ𝑛=[𝑢(2𝑘+1)−𝑢(2𝑘)]^𝛼+1−𝑝(2𝑘)𝑢^𝛼+1(2𝑘+1)+[𝑢(2𝑘+2)−𝑢(2𝑘+1)]^𝛼+1−𝑝(2𝑘+1)𝑢^𝛼+1(2𝑘+2)=2−𝑐. (3.13) Therefore, if 𝑐≥2, then (3.10) is oscillatory.

Example 3.3. Consider the second-order half-linear q-difference equation ([𝑥^Δ(𝑡)]^𝛼)^Δ+𝑝(𝑡)𝑥^𝛼(𝑞𝑡)=𝑡^𝛽𝑓(𝑡), (3.14) where 𝑝(𝑡)=𝑐(−1)^𝑛, 𝑡=𝑞^𝑛∈𝕋=𝑞^ℕ0,𝑞>1,𝑐>0,𝛽∈𝑅. 𝛼 is a quotient of odd positive integers. 𝑓(𝑡)=0,𝑡=𝑞^𝑛,𝑛=4𝑘or4𝑘+2,1,𝑡=𝑞^𝑛,𝑛=4𝑘+1,−1,𝑡=𝑞^𝑛,𝑛=4𝑘+3,𝑘∈ℕ₀, (3.15)
Let 𝑢(𝑡)=𝑓(𝑡),𝑡=𝑞^𝑛∈𝕋. We have 𝑄₁[𝑢]∶=^𝑞4𝑘+2_𝑞4𝑘[(𝑢^Δ(𝑡))^𝛼+1−𝑝(𝑡)𝑢^𝛼+1(𝑞𝑡)]Δ𝑡={[𝑢(𝑞^4𝑘+1)−𝑢(𝑞^4𝑘)𝑞^4𝑘](𝑞−1)^𝛼+1−𝑐𝑢^𝛼+1(𝑞^4𝑘+1)}𝑞^4𝑘(𝑞−1)+{[𝑢(𝑞^4𝑘+2)−𝑢(𝑞^4𝑘+1)𝑞^4𝑘+1](𝑞−1)^𝛼+1+𝑐𝑢^𝛼+1(𝑞^4𝑘+2)}𝑞^4𝑘+11(𝑞−1)=[𝑞^{4𝑘(𝛼+1)}(𝑞−1)^𝛼+1𝑞−𝑐+𝑞^{(4𝑘+1)(𝛼+1)}(𝑞−1)^𝛼+1]𝑞^4𝑘(𝑞−1). (3.16) Similarly, we have 𝑄₂[𝑢]∶=^𝑞4𝑘+4_𝑞4𝑘+2[(𝑢^Δ(𝑡))^𝛼+1−𝑝(𝑡)𝑢^𝛼+11(𝑞𝑡)]Δ𝑡=[𝑞^{(4𝑘+2)(𝛼+1)}(𝑞−1)^𝛼+1𝑞−𝑐+𝑞^{(4𝑘+3)(𝛼+1)}(𝑞−1)^𝛼+1]𝑞^4𝑘+2(𝑞−1). (3.17) Therefore, if 𝑐>0, then 𝑄₁[𝑢]≤0,𝑄₂[𝑢]≤0, for large 𝑘. So (3.14) is oscillatory.

Many other interesting examples can be similarly given.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (no.10971232) and by NSF of Maoming University (no. LK201002).