Oscillation and Nonoscillation Criteria for Nonlinear Dynamic Systems on Time Scales

Abstract

We consider the nonlinear dynamic system 𝑥^Δ(𝑡)=𝑎(𝑡)𝑔(𝑦(𝑡)),𝑦^Δ(𝑡)=−𝑓(𝑡,𝑥^𝜎(𝑡)). We establish some necessary and sufficient conditions for the existence of oscillatory and nonoscillatory solutions with special asymptotic properties for the system. We generalize the known results in the literature. Some examples are included to illustrate the results.

1. Introduction

In this paper we investigate the nonlinear two-dimensional dynamic system: 𝑥^Δ(𝑡)=𝑎(𝑡)𝑔𝑦(𝑡),𝑦^Δ(𝑡)=−𝑓𝑡,𝑥^𝜎(𝑡),𝑡∈𝑡₀,∞_𝕋, (1.1) where 𝑎(𝑡) is a nonnegative, rd-continuous function which is defined for 𝑡∈[𝑡₀,∞)_𝕋=[𝑡₀,∞)∩𝕋. Here, 𝕋 is a time scale unbounded from above. We assume throughout that 𝑔∶ℝ→ℝ is a continuous function with 𝑢𝑔(𝑢)>0 for 𝑢≠0, and 𝑓(𝑡,𝑢)∶[𝑡₀,∞)_𝕋×ℝ→ℝ is continuous as a function of 𝑢∈ℝ with sign property 𝑢𝑓(𝑡,𝑢)>0 for 𝑢≠0 and 𝑡∈[𝑡₀,∞)_𝕋.

By the solution of system (1.1), we mean a pair of nontrivial real-valued functions (𝑥(𝑡),𝑦(𝑡)) which has property 𝑥,𝑦∈𝐶¹_rd([𝑡₀,∞)_𝕋,ℝ) and satisfies system (1.1) for 𝑡∈[𝑡₀,∞)_𝕋. Our attention is restricted to those solutions (𝑥(𝑡),𝑦(𝑡)) of system (1.1) which exist on some half-line [𝑡_𝑥,∞)_𝕋 and satisfy sup{|𝑥(𝑡)|+|𝑦(𝑡)|∶𝑡≥𝑡_𝑥}>0 for any 𝑡_𝑥≥𝑡₀. As usual, a continuous real-valued function defined on [𝑇₀,∞) is said to be oscillatory if it has arbitrarily large zeros, otherwise it is said to be nonoscillatory. A solution (𝑥(𝑡),𝑦(𝑡)) of system (1.1) is called oscillatory if both 𝑥(𝑡) and 𝑦(𝑡) are oscillatory functions, and otherwise it will be called nonoscillatory. System (1.1) is called oscillatory if its solutions are oscillatory.

The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. thesis in 1990 in order to unify continuous and discrete analysis (see [1]). Not only can this theory of the so-called “dynamic equations” unify the theories of differential equations and difference equations, but also extend these classical cases to cases “in between”, for example, to the so-called 𝑞-difference equations and can be applied on other different types of time scales. Since Hilger formed the definition of derivatives and integral on time scales, several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. [2] and references cited therein. A book on the subject of time scales (see [3]) summarizes and organizes much of time scale calculus. The reader is referred to Chapter 1 in [3] for the necessary time scale definitions and notations used throughout this paper.

The system (1.1) includes two-dimensional linear/nonlinear differential and difference systems, which were investigated in the literature, see for example [4–9] and the references therein.

On the other hand, the system (1.1) reduces to some important second-order dynamic equations in the particular case, for example 𝑥^Δ(𝑡)𝑎(𝑡)^Δ+𝑏(𝑡)𝑓𝑥^𝜎(𝑡)𝑥=0,_ΔΔ(𝑡)+𝑏(𝑡)𝑥^𝜎(𝑡)^𝜆−1𝑥^𝜎(𝑡)=0,𝜆>0, (1.2) where 𝑏(𝑡) is rd-continuous on [𝑡₀,∞)_𝕋. In recent years there has been much research activity concerning the oscillation and nonoscillation of solutions of dynamic equation (1.2) on time scales. We refer the reader to the recent papers [10–13] and the references therein. However, most of previous studies for the system (1.1) have been restricted to the case where 𝑓(𝑡,𝑢)=𝑏(𝑡)𝑓(𝑢), for example [4–8, 14–18] and the references therein. Erbe and Mert [14, 17] obtained some oscillation results for the system (1.1). Fu and Lin [15] obtained some oscillation and nonoscillation criteria for the linear dynamic system (1.1).

Since there are few works about oscillation and nonoscillation of dynamic systems on time scales (see [15]), motivated by [9, 14, 15], in this paper we investigate oscillatory and nonoscillatory properties for the system (1.1) in the case of general 𝑓(𝑡,𝑢) in which 𝑡 and 𝑢 are not necessarily separable. In the next section, by means of appropriate hypotheses on 𝑓(𝑡,𝑢) and fixed point theorem, we establish some new sufficient and necessary conditions for the existence of nonoscillatory solutions with special asymptotic properties for the system (1.1). In Section 3, we obtain sufficient and necessary conditions for all solutions of the system (1.1) to be oscillatory via the results in Section 2 and some inequality techniques without using Riccati transformation. Our results not only unify the known results of differential and difference systems but also extend and improve the existing results of dynamic systems on time scales in the literature.

2. Nonoscillation Results

In this section, we generalize and improve some results of [7–9, 15, 18]. Some necessary and sufficient conditions are given for the system (1.1) to admit the existence of nonoscillatory solutions with special asymptotic properties. These results will be used for the next section. Additional hypotheses on 𝑔(𝑢) and 𝑓(𝑡,𝑢) are needed for this purpose.(H₁) For any positive constant 𝑙 and 𝐿 with 𝑙<𝐿, there exist positive constants ℎ and 𝐻, depending possibly on 𝑙 and 𝐿, such that 𝑙≤|𝑢|≤𝐿 implies ℎ𝑓(𝑡,𝑙)≤𝑓(𝑡,𝑢)≤𝐻𝑓(𝑡,𝐿),𝑡∈𝑡₀,∞_𝕋. (2.1)(H₂) There exists a positive constant 𝑘 such that 𝑔(𝑢𝑣)≥𝑘𝑔(𝑢)𝑔(𝑣) for 𝑢𝑣>0.(H₃) For any positive constant 𝑙 and 𝐿 with 𝑙<𝐿, there exist positive constants ℎ and 𝐻, depending possibly on 𝑙 and 𝐿, such that 𝑙≤|𝑢|≤𝐿 implies ℎ𝑓(𝑡,𝑙𝜃(𝑡))≤𝑓(𝑡,𝑢𝜃(𝑡))≤𝐻𝑓(𝑡,𝐿𝜃(𝑡)),𝑡∈𝑡₀,∞_𝕋, (2.2) where 𝜃(𝑡) is a positive nondecreasing function.

For convenience, we will employ the following notation: 𝐴(𝑠,𝑡)=^𝑡_𝑠𝑎(𝜏)Δ𝜏,𝑠,𝑡∈𝑡₀,∞_𝕋. (2.3)

Theorem 2.1. Assume that 𝑔 is nondecreasing and that (H₁) holds. Then system (1.1) has a nonoscillatory solution (𝑥(𝑡),𝑦(𝑡)) such that lim_𝑡→∞𝑥(𝑡)=𝛼≠0 and lim_𝑡→∞𝑦(𝑡)=0 if and only if for any positive constant 𝑑^∞_𝑡0𝑎(𝑡)𝑔𝑑^∞_𝑡𝑓(𝑠,𝑐)Δ𝑠Δ𝑡<∞𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑐≠0. (2.4)

Proof. Suppose that (𝑥(𝑡),𝑦(𝑡)) is a nonoscillatory solution of (1.1) such that lim_𝑡→∞𝑥(𝑡)=𝛼≠0 and lim_𝑡→∞𝑦(𝑡)=0. Without loss of generality, we assume that 𝛼>0. Then there exist 𝑡₁∈[𝑡₀,∞)_𝕋 and positive constants 𝑙 and 𝐿 such that 𝑙≤𝑥(𝑡)≤𝐿 for 𝑡∈[𝑡₁,∞)_𝕋. Condition (H₁) implies that 𝑓𝑡,𝑥^𝜎(𝑡)≥ℎ𝑓(𝑡,𝑙) (2.5) for 𝑡∈[𝑡₁,∞)_𝕋 and some constant ℎ>0. It follows from the second equation in (1.1) that 𝑦(𝑠)−𝑦(𝑡)=−^𝑠_𝑡𝑓𝜏,𝑥^𝜎(𝜏)Δ𝜏. (2.6) Let 𝑠→∞ and noting that lim_𝑠→∞𝑦(𝑠)=0, we have 𝑦(𝑡)=^∞_𝑡𝑓𝜏,𝑥^𝜎(𝜏)Δ𝜏,𝑡∈𝑡₁,∞_𝕋. (2.7) Thus, from (2.5), (2.7) and the first equation in (1.1), we obtain that ∞>lim_𝑡→∞𝑥(𝑡)−𝑥𝑡₁=^∞_𝑡1𝑎(𝑠)𝑔𝑦(𝑠)=Δ𝑠_∞𝑡1𝑎(𝑠)𝑔^∞_𝑠𝑓𝜏,𝑥^𝜎(𝜏)≥Δ𝜏Δ𝑠_∞𝑡1𝑎(𝑠)𝑔ℎ^∞_𝑠𝑓(𝜏,𝑙)Δ𝜏Δ𝑠, (2.8) which implies that (2.4) holds.
Conversely, suppose that (2.4) holds, we may assume that 𝑐>0. In view of (H₁), there is a constant 𝐻>0 such that 𝑐/2≤𝑥(𝑡)≤𝑐 implies 𝑓𝑡,𝑥^𝜎(𝑡)≤𝐻𝑓(𝑡,𝑐),𝑡∈𝑡₀,∞_𝕋. (2.9) Since (2.4) holds, we can choose 𝑡₁∈[𝑡₀,∞)_𝕋 large enough such that ^∞_𝑡1𝑎(𝑠)𝑔𝐻^∞_𝑠𝑓()𝑐𝜏,𝑐Δ𝜏Δ𝑠≤2. (2.10) Let 𝐵𝐶[𝑡₀,∞)_𝕋 be the Banach space of all real-valued rd-continuous functions on [𝑡₀,∞)_𝕋 endowed with the norm ‖𝑥‖=sup_{𝑡∈[𝑡0,∞)𝕋}|𝑥(𝑡)|<∞. We defined a bounded, convex, and closed subset Ω of 𝐵𝐶[𝑡₀,∞)_𝕋 as Ω=𝑥∈𝐵𝐶𝑡₀,∞_𝕋∶𝑐2≤𝑥(𝑡)≤𝑐. (2.11) Define an operator Γ∶Ω→𝐵𝐶[𝑡₀,∞)_𝕋 as follows: (Γ𝑥)(𝑡)=𝑐−^∞_𝑡𝑎(𝑠)𝑔^∞_𝑠𝑓𝜏,𝑥^𝜎(𝜏)Δ𝜏Δ𝑠,𝑡∈𝑡₁,∞_𝕋,𝑐−^∞_𝑡1𝑎(𝑠)𝑔^∞_𝑠𝑓𝜏,𝑥^𝜎(𝜏)Δ𝜏Δ𝑠,𝑡∈𝑡₀,𝑡_1𝕋. (2.12) Now we show that Γ satisfies the assumptions of Schauder's fixed-point theorem (see [19, Corollary 6]).(i) We will show that Γ𝑥∈Ω for any 𝑥∈Ω. In fact, for any 𝑥∈Ω and 𝑡∈[𝑡₁,∞)_𝕋, in view of (2.10), we get 𝑐≥(Γ𝑥)(𝑡)=𝑐−^∞_𝑡𝑎(𝑠)𝑔^∞_𝑠𝑓𝜏,𝑥^𝜎(𝜏)Δ𝜏Δ𝑠≥𝑐−^∞_𝑡1𝑎(𝑠)𝑔𝐻^∞_𝑠𝑓()𝑐𝜏,𝑐Δ𝜏Δ𝑠≥𝑐−2=𝑐2. (2.13) Similarly, we can prove that 𝑐/2≤(Γ𝑥)(𝑡)≤𝑐 for any 𝑥∈Ω and 𝑡∈[𝑡₀,𝑡₁)_𝕋. Hence, Γ𝑥∈Ω for any 𝑥∈Ω.(ii) We prove that Γ is a completely continuous mapping. First, we consider the continuity of Γ. Let 𝑥_𝑛∈Ω and ‖𝑥_𝑛−𝑥‖→0 as 𝑛→∞, then 𝑥∈Ω and |𝑥_𝑛(𝑡)−𝑥(𝑡)|→0 as 𝑛→∞ for any 𝑡∈[𝑡₀,∞)_𝕋. Consequently, by the continuity of 𝑔 and 𝑓, for any 𝑡∈[𝑡₁,∞)_𝕋, we have lim_𝑛→∞𝑎(𝑡)𝑔^∞_𝑡𝑓𝜏,𝑥^𝜎_𝑛(𝜏)Δ𝜏−𝑔^∞_𝑡𝑓𝜏,𝑥^𝜎(𝜏)Δ𝜏=0. (2.14)From (2.9), we obtain that 𝑎(𝑡)𝑔^∞_𝑡𝑓𝜏,𝑥^𝜎_𝑛(𝜏)Δ𝜏−𝑔^∞_𝑡𝑓𝜏,𝑥^𝜎(𝜏)Δ𝜏≤2𝑎(𝑡)𝑔𝐻^∞_𝑡𝑓(𝜏,𝑐)Δ𝜏. (2.15) On the other hand, from (2.12) we have Γ𝑥_𝑛(𝑡)−(Γ𝑥)(𝑡)≤^∞_𝑡1𝑎(𝑠)𝑔^∞_𝑠𝑓𝜏,𝑥^𝜎_𝑛(𝜏)Δ𝜏−𝑔^∞_𝑠𝑓𝜏,𝑥^𝜎(𝜏)Δ𝜏Δ𝑠 (2.16) for 𝑡∈[𝑡₀,𝑡₁]_𝕋 and Γ𝑥_𝑛(𝑡)−(Γ𝑥)(𝑡)≤^∞_𝑡𝑎(𝑠)𝑔^∞_𝑠𝑓𝜏,𝑥^𝜎_𝑛(𝜏)Δ𝜏−𝑔^∞_𝑠𝑓𝜏,𝑥^𝜎(𝜏)Δ𝜏Δ𝑠 (2.17) for 𝑡∈[𝑡₁,∞]_𝕋. Therefore, from (2.16) and (2.17), we have Γ𝑥_𝑛−Γ𝑥≤^∞_𝑡1𝑎(𝑠)𝑔^∞_𝑠𝑓𝜏,𝑥^𝜎_𝑛(𝜏)Δ𝜏−𝑔^∞_𝑠𝑓𝜏,𝑥^𝜎(𝜏)Δ𝜏Δ𝑠. (2.18) Referring to Chapter 5 in [20], we see that the Lebesgue dominated convergence theorem holds for the integral on time scales. Then, from (2.14) and (2.15), (2.18) yields lim_𝑛→∞‖Γ𝑥_𝑛−Γ𝑥‖=0, which implies that Γ is continuous on Ω.
Next, we show that ΓΩ is uniformly cauchy. In fact, for any 𝜀>0, take 𝑡₂∈[𝑡₁,∞)_𝕋 and 𝑡₂>𝑡₁ such that ^∞_𝑡2𝑎(𝑠)𝑔𝐻^∞_𝑠𝑓(𝜏,𝑐)Δ𝜏Δ𝑠≤𝜀. (2.19) Then for any 𝑥∈Ω and 𝑡,𝑟∈[𝑡₂,∞)_𝕋, we have |(Γ𝑥)(𝑡)−(Γ𝑥)(𝑟)|≤^∞_𝑡𝑎(𝑠)𝑔𝐻^∞_𝑠𝑓()+𝜏,𝑐Δ𝜏Δ𝑠_∞𝑟𝑎(𝑠)𝑔𝐻^∞_𝑠𝑓(𝜏,𝑐)Δ𝜏Δ𝑠≤2^∞_𝑡2𝑎(𝑠)𝑔𝐻^∞_𝑠𝑓(𝜏,𝑐)Δ𝜏Δ𝑠≤2𝜀. (2.20) This means that ΓΩ is uniformly cauchy.
Finally, we prove that ΓΩ is equicontinuous on [𝑡₀,𝑡₂]_𝕋 for any 𝑡₂∈[𝑡₀,∞)_𝕋. Without loss of generality, we set 𝑡₂>𝑡₁. For any 𝑥∈Ω, we have |(Γ𝑥)(𝑡)−(Γ𝑥)(𝑟)|≡0 for 𝑡,𝑟∈[𝑡₀,𝑡₁]_𝕋 and |(Γ𝑥)(𝑡)−(Γ𝑥)(𝑟)|=^∞_𝑡𝑎(𝑠)𝑔^∞_𝑠𝑓𝜏,𝑥^𝜎(𝜏)Δ𝜏Δ𝑠−^∞_𝑟𝑎(𝑠)𝑔^∞_𝑠𝑓𝜏,𝑥^𝜎(𝜏)≤Δ𝜏Δ𝑠_𝑟𝑡𝑎(𝑠)𝑔𝐻^∞_𝑠𝑓(𝜏,𝑐)Δ𝜏Δ𝑠 (2.21) for 𝑡,𝑟∈[𝑡₁,𝑡₂]_𝕋.
Now, we see that for any 𝜀>0, there exists 𝛿>0 such that when 𝑡,𝑟∈[𝑡₀,𝑡₂]_𝕋 with |𝑡−𝑟|<𝛿, |(Γ𝑥)(𝑡)−(Γ𝑥)(𝑟)|<𝜀 for any 𝑥∈Ω. This means that ΓΩ is equicontinuous on [𝑡₀,𝑡₂]_𝕋 for any 𝑡₂∈[𝑡₀,∞)_𝕋. By Arzela-Ascoli theorem (see [19, Lemma 4]), ΓΩ is relatively compact. From the above, we have proved that Γ is a completely continuous mapping.
By Schauder’s fixed point theorem, there exists 𝑥∈Ω such that Γ𝑥=𝑥. Therefore, we have 𝑥(𝑡)=𝑐−^∞_𝑡𝑎(𝑠)𝑔^∞_𝑠𝑓𝜏,𝑥^𝜎(𝜏)Δ𝜏Δ𝑠,𝑡∈𝑡₁,∞_𝕋. (2.22) Set 𝑦(𝑡)=^∞_𝑡𝑓𝜏,𝑥^𝜎(𝜏)Δ𝜏,𝑡∈𝑡₁,∞_𝕋. (2.23) Then lim_𝑡→∞𝑦(𝑡)=0 and 𝑦^Δ(𝑡)=−𝑓(𝑡,𝑥^𝜎(𝑡)). On the other hand, 𝑥(𝑡)=𝑐−^∞_𝑡𝑎(𝑠)𝑔𝑦(𝑠)Δ𝑠, (2.24) which implies that lim_𝑡→∞𝑥(𝑡)=𝑐 and 𝑥^Δ(𝑡)=𝑎(𝑡)𝑔(𝑦(𝑡)). The proof is complete.

Corollary 2.2. Suppose that 𝑔 is nondecreasing and that (H₁) and (H₂) hold. Then system (1.1) has a nonoscillatory solution (𝑥(𝑡),𝑦(𝑡)) such that lim_𝑡→∞𝑥(𝑡)=𝛼≠0 and lim_𝑡→∞𝑦(𝑡)=0 if and only if for some 𝑐≠0^∞_𝑡0𝑎(𝑡)𝑔^∞_𝑡𝑓(𝑠,𝑐)Δ𝑠Δ𝑡<∞. (2.25)

Theorem 2.3. Suppose that lim_𝑡→∞𝐴(𝑡₀,𝑡)=∞ and 𝑔 is nondecreasing. Suppose further that (H₃) holds. Then system (1.1) has a nonoscillatory solution (𝑥(𝑡),𝑦(𝑡)) such that lim_𝑡→∞(𝑥(𝑡)/𝐴(𝑡₀,𝑡))=𝛼≠0 and lim_𝑡→∞𝑦(𝑡)=𝛽≠0 if and only if for some 𝑐≠0^∞_𝑡0𝑓𝑡,𝑐𝐴𝑡₀,𝜎(𝑡)Δ𝑡<∞. (2.26)

Proof. Suppose that (𝑥(𝑡),𝑦(𝑡)) is a nonoscillatory solution of (1.1) such that lim_𝑡→∞(𝑥(𝑡)/𝐴(𝑡₀,𝑡))=𝛼≠0 and lim_𝑡→∞𝑦(𝑡)=𝛽≠0. We may assume that 𝛼>0. Hence, there exist 𝑡₁∈[𝑡₀,∞)_𝕋 and positive constant 𝑙,𝐿 such that 𝑙𝐴(𝑡₀,𝑡)≤𝑥(𝑡)≤𝐿𝐴(𝑡₀,𝑡) and 𝑦^Δ(𝑡)<0,𝑦(𝑡)>𝛽 for 𝑡∈[𝑡₁,∞)_𝕋. By condition (H₃), there exists a constant ℎ>0 such that 𝑓(𝑡,𝑥^𝜎(𝑡))≥ℎ𝑓(𝑡,𝑙𝐴(𝑡₀,𝜎(𝑡))) for 𝑡∈[𝑡₁,∞)_𝕋. According to the second equation in (1.1), we have ∞>𝑦𝑡₁−𝛽=^∞_𝑡1𝑓𝑡,𝑥^𝜎(𝑡)Δ𝑡≥ℎ^∞_𝑡1𝑓𝑡,𝑙𝐴𝑡₀,𝜎(𝑡)Δ𝑡, (2.27) which implies that (2.26) holds with 𝑐=𝑙.
Conversely, Let (2.26) holds for some 𝑐=2𝑝, where 𝑝>0. By (H₃), there exists a constant 𝐻>0 such that 𝑝≤𝑢≤2𝑝 implies 𝑓(𝑡,𝑢𝐴(𝑡₀,𝜎(𝑡)))≤𝐻𝑓(𝑡,2𝑝𝐴(𝑡₀,𝜎(𝑡))) for 𝑡∈[𝑡₀,∞)_𝕋. Take 𝑡₁∈[𝑡₀,∞)_𝕋 so large that 𝐻^∞_𝑡1𝑓𝑡,𝑐𝐴𝑡₀,𝜎(𝑡)Δ𝑡≤𝑑, (2.28) where 𝑑=𝑔⁻¹(𝑐)/2. We introduce 𝐵𝐶[𝑡₁,∞)_𝕋 be the partially ordered Banach space of all real-valued and rd-continuous functions 𝑥(𝑡) with the norm ‖𝑥‖=sup_{𝑡∈[𝑡1,∞)𝕋}(|𝑥(𝑡)|/𝐴(𝑡₁,𝑡)), and the usual pointwise ordering ≤.
Define Ω=𝑥∈𝐵𝐶𝑡₁,∞_𝕋∶𝑔(𝑑)𝐴𝑡₁,𝑡≤𝑥(𝑡)≤𝑔(2𝑑)𝐴𝑡₁,𝑡. (2.29) It is easy to see that Ω is a bounded, convex, and closed subset of 𝐵𝐶[𝑡₁,∞)_𝕋. Let us further define an operator Γ∶Ω→𝐵𝐶[𝑡₁,∞)_𝕋 as follows: (Γ𝑥)(𝑡)=^𝑡_𝑡1𝑎(𝑠)𝑔𝑑+^∞_𝑠𝑓𝜏,𝑥^𝜎(𝜏)Δ𝜏Δ𝑠,𝑡∈𝑡₁,∞_𝕋. (2.30)
Since it can be shown that Γ is continuous and sends Ω into a relatively compact subset of Ω, the Schauder’s fixed point theorem ensures that the existence of an 𝑥∈Ω such that 𝑥=Γ𝑥, this is 𝑥(𝑡)=^𝑡_𝑡1𝑎(𝑠)𝑔𝑑+^∞_𝑠𝑓𝜏,𝑥^𝜎(𝜏)Δ𝜏Δ𝑠,𝑡∈𝑡₁,∞_𝕋. (2.31) Set 𝑦(𝑡)=𝑑+^∞_𝑡𝑓𝜏,𝑥^𝜎(𝜏)Δ𝜏,𝑡∈𝑡₁,∞_𝕋. (2.32) Then lim_𝑡→∞𝑦(𝑡)=𝑑 and 𝑦^Δ(𝑡)=−𝑓(𝑡,𝑥^𝜎(𝑡)). On the other hand, by L'Hôpital's Rule (see [15, Lemma 2.11]), we have lim_𝑡→∞𝑥(𝑡)𝐴𝑡₀,𝑡=lim_𝑡→∞𝑎(𝑡)𝑔𝑑+∫^∞_𝑡𝑓𝜏,𝑥^𝜎(𝜏)Δ𝜏𝑎(𝑡)=lim_𝑡→∞𝑔𝑑+^∞_𝑡𝑓𝜏,𝑥^𝜎(𝜏)Δ𝜏=𝑔(𝑑)≠0, (2.33) and 𝑥^Δ(𝑡)=𝑎(𝑡)𝑔(𝑦(𝑡)). The proof is complete.

Remark 2.4. Theorems 2.1 and 2.3 extend and improve essentially the known results of [7–9, 15, 18].

3. Oscillation Results

In this section, we need some additional conditions to guarantee that the system (1.1) has oscillatory solutions.(H₄) There exists a continuous nondecreasing function 𝜑∶ℝ→ℝ such that sgn𝜑(𝑢)=sgn𝑢,^±∞𝑑𝑢𝑔𝜑(𝑢)<∞, (3.1) and |𝑓(𝑡,𝑢)|≥|𝑓(𝑡,𝑙)||𝜑(𝑢)|,𝑡∈[𝑡₀,∞)_𝕋,|𝑢|≥𝑢₀ for some constants 𝑢₀>0 and 𝑙≠0 with sgn𝑙=sgn𝑢.(H₅) There exists a continuous nondecreasing function 𝜑∶[−𝑀,𝑀]→ℝ,𝑀>0 being a constant, such that sgn𝜑(𝑣)=sgn𝑣,₀^±𝑀𝑑𝑣𝜑𝑔(𝑣)<∞, (3.2) and |𝑓(𝑡,𝑢𝑣)|≥𝑘|𝑓(𝑡,𝑢)||𝜑(𝑣)|,𝑡∈[𝑡₀,∞)_𝕋,𝑢≠0,0<|𝑣|<𝑣₀ for some positive constant 𝑘>0 and 𝑣₀>0.

Theorem 3.1. Suppose that lim_𝑡→∞𝐴(𝑡₀,𝑡)=∞ and 𝑔 is nondecreasing. Suppose further that (H₁), (H₂) and (H₄) hold. Then system (1.1) is oscillatory if and only if for all 𝑐≠0^∞_𝑡0𝑎(𝑡)𝑔^∞_𝑡𝑓(𝑠,𝑐)Δ𝑠Δ𝑡=∞. (3.3)

Proof. If (3.3) does not hold, by Theorem 2.1, system (1.1) has a nonoscillatory solution (𝑥(𝑡),𝑦(𝑡)) such that lim_𝑡→∞𝑥(𝑡)=𝛼≠0 and lim_𝑡→∞𝑦(𝑡)=0.
Conversely, suppose that (3.3) holds and that (1.1) has a nonoscillatory solution (𝑥(𝑡),𝑦(𝑡)) for 𝑡∈[𝑡₀,∞)_𝕋. We may assume that 𝑥(𝑡)>0 for 𝑡∈[𝑡₁,∞)_𝕋, where 𝑡₁∈[𝑡₀,∞)_𝕋. Since lim_𝑡→∞𝐴(𝑡₀,𝑡)=∞, it is easy to show that 𝑦(𝑡)>0,𝑡∈[𝑡₁,∞)_𝕋. From the second equation in (1.1), we have 𝑦^Δ(𝑡)<0,𝑡∈[𝑡₁,∞)_𝕋. Hence, lim_𝑡→∞𝑦(𝑡)≥0. It follows from the first equation in (1.1) that 𝑥^Δ(𝑡)>0,𝑡∈[𝑡₁,∞)_𝕋, and lim_𝑡→∞𝑥(𝑡)=∞ by Theorem 2.1. Integrating the second equation in (1.1) from 𝑡 to ∞ yields that 𝑦(𝑡)≥^∞_𝑡𝑓𝑠,𝑥^𝜎(𝑠)Δ𝑠,𝑡∈𝑡₁,∞_𝕋. (3.4) By (3.4), (H₂) and in view of nondecreasing 𝜑, it follows that 𝑥^Δ(𝑡)𝑔𝜑(𝑥^𝜎(𝑡=𝑎(𝑡)𝑔𝑦(𝑡)))𝑔𝜑(𝑥^𝜎(𝑡≥𝑎(𝑡)𝑔∫))^∞𝑡𝑓𝑠,𝑥^𝜎(𝑠)Δ𝑠𝑔𝜑(𝑥^𝜎(𝑡(𝑡)𝑔))≥𝑘𝑎^∞_𝑡𝑓𝑠,𝑥^𝜎(𝑠)𝜑(𝑥^𝜎(𝑠))Δ𝑠 (3.5) for 𝑡∈[𝑡₁,∞)_𝕋.
Since (H₄) holds and lim_𝑡→∞𝑥(𝑡)=∞, there is 𝑡₂∈[𝑡₁,∞)_𝕋 and 𝑙>0 such that 𝑓𝑡,𝑥^𝜎(𝑡)𝜑(𝑥^𝜎(𝑡()))≥𝑓𝑡,𝑙 (3.6) for 𝑡∈[𝑡₂,∞)_𝕋. From (3.5) and (3.6), we get 𝑥^Δ(𝑡)𝑔𝜑(𝑥^𝜎(𝑡(𝑡)𝑔))≥𝑘𝑎^∞_𝑡𝑓(𝑠,𝑙)Δ𝑠. (3.7) Integrating (3.7) from 𝑡₂ to 𝑡, we have ^𝑡_𝑡2𝑥^Δ(𝑠)𝑔𝜑(𝑥^𝜎(𝑠))Δ𝑠≥𝑘^𝑡_𝑡2𝑎(𝑠)𝑔^∞_𝑠𝑓(𝜏,𝑙)Δ𝜏Δ𝑠. (3.8) Since 𝑔,𝜑, and 𝑥 are nondecreasing, we obtain ^𝑡_𝑡2𝑥^Δ(𝑠)𝑔𝜑(𝑥^𝜎(𝑠))Δ𝑠≤^𝑡_𝑡2𝑥^Δ(𝑠)𝑔𝜑(𝑥(𝑠))Δ𝑠. (3.9) By (H₄), (3.8) and (3.9), we get 𝑘^𝑡_𝑡2𝑎(𝑠)𝑔^∞_𝑠𝑓(𝜏,𝑙)Δ𝜏Δ𝑠≤_𝑥𝑡^𝑥(𝑡)₂𝑑𝑢𝑔𝜑(𝑢)<∞, (3.10) which contradicts (3.3) when 𝑡→∞. The proof is complete.

Theorem 3.2. Suppose that lim_𝑡→∞𝐴(𝑡₀,𝑡)=∞ and (H₅) holds. Suppose further that 𝑓(𝑡,𝑢) is nondecreasing in 𝑢 for each fixed 𝑡∈[𝑡₀,∞)_𝕋 and 𝑔 is nondecreasing. Then system (1.1) is oscillatory if and only if for all 𝑐≠0^∞_𝑡0𝑓𝑡,𝑐𝐴𝑡₀,𝜎(𝑡)Δ𝑡=∞. (3.11)

Proof. If (3.11) does not hold, by Theorem 2.3, system (1.1) has a nonoscillatory solution (𝑥(𝑡),𝑦(𝑡)) such that lim_𝑡→∞(𝑥(𝑡)/𝐴(𝑡₀,𝑡))=𝛼≠0 and lim_𝑡→∞𝑦(𝑡)=𝛽≠0.
Conversely, suppose that (3.11) holds and that system (1.1) has a nonoscillatory solution (𝑥(𝑡),𝑦(𝑡)) for 𝑡∈[𝑡₀,∞)_𝕋. We assume that 𝑥(𝑡)>0 for 𝑡∈[𝑡₁,∞)_𝕋, where 𝑡₁∈[𝑡₀,∞)_𝕋. Then by same argument in the proof of Theorem 3.1, we have 𝑥^Δ(𝑡)>0,𝑦^Δ(𝑡)<0,𝑦(𝑡)>0 eventually. We claim that (3.11) implies lim_𝑡→∞𝑦(𝑡)=0. In fact, if lim_𝑡→∞𝑦(𝑡)=𝛽>0, then 𝑦(𝑡)≥𝛽 for 𝑡∈[𝑡₁,∞)_𝕋. According to the first equation in (1.1), we get 𝑥^𝜎(𝑡)=𝑥𝑡₁+_𝑡^𝜎(𝑡)₁𝑎(𝑠)𝑔𝑦(𝑠)Δ𝑠≥^𝑡_𝑡1𝑎(𝑠)𝑔𝑦(𝑠)Δ𝑠+_𝑡^𝜎(𝑡)𝑎(𝑠)𝑔𝑦(𝑠)𝑦(𝑡)𝐴𝑡Δ𝑠≥𝑔₁,𝑡+𝜇(𝑡)𝑎(𝑡)=𝑔𝑦(𝑡)𝐴𝑡₁,𝜎(𝑡)≥𝑔𝛽𝐴𝑡₁,𝜎(𝑡). (3.12) Integrating the second equation in (1.1) from 𝑡₁ to ∞, we have 𝛽−𝑦𝑡₁=−^∞_𝑡1𝑓𝑠,𝑥^𝜎(𝑠)Δ𝑠≤−^∞_𝑡1𝑓𝑠,𝑔𝛽𝐴𝑡₁,𝜎(𝑠)Δ𝑠=−∞, (3.13) which is a contradiction. Hence, lim_𝑡→∞𝑦(𝑡)=0.
By (H₅), we have 𝑘𝑓𝑡,𝐴𝑡₁(𝑡)𝑥,𝜎≤𝑘𝑓𝑡,^𝜎(𝑡)𝑔𝑦(𝑡)≤𝑓𝑡,𝑥^𝜎(𝑡)𝜑𝑔𝑦(𝑡)=−𝑦^Δ(𝑡)𝜑𝑔𝑦(𝑡). (3.14) From (3.14), it follows 𝑘𝑓𝑠,𝐴𝑡^𝑡_𝑡11,𝜎(𝑠)Δ𝑠≤−^𝑡_𝑡1𝑦^Δ(𝑠)𝜑𝑔𝑦(𝑠)Δ𝑠. (3.15) Hence, 𝑘^𝑡_𝑡1𝑓𝑠,𝐴𝑡₁,𝜎(𝑠)Δ𝑠≤−_𝑦𝑡^𝑦(𝑡)₁𝑑𝑢𝜑𝑔(𝑢). (3.16) In view of (H₅) and (3.11), this is a contradiction. The proof is complete.

Remark 3.3. Theorems 3.1 and 3.2 improve the existing results of [15, 18].

Example 3.4. Consider the system: 𝑥^Δ(𝑡)=𝑦(𝑡)^1/𝛼−1𝑦(𝑡),𝑦^Δ(𝑡)𝑡=−^𝑣𝑥^𝜎(𝑡)^𝛾−1𝑥^𝜎(𝑡)1+𝑡^𝑢|𝑥^𝜎(𝑡)|^𝑚, (3.17) where 𝕋=𝑎ℕ={𝑎𝑛∣𝑛∈ℕ},𝑎,𝑚,𝛾,𝑢,𝛼>0 and 𝑣 are constants as well as 𝛾>𝑚.
Let 𝑎(𝑡)=1,𝑔𝑦=𝑦^(1/𝛼)−1()=𝑡𝑦,𝑓𝑡,𝑥^𝑣|𝑥|^𝛾−1𝑥1+𝑡^𝑢|𝑥|^𝑚. (3.18) It is easy to see that 𝑔(𝑦) is increasing and for 0<𝑙≤𝑥≤𝐿,𝛾≥𝑚, 𝑓()()(),𝑓()()().𝑡,𝑙≤𝑓𝑡,𝑥≤𝑓𝑡,𝐿𝑡,𝑙𝑡≤𝑓𝑡,𝑥𝑡≤𝑓𝑡,𝐿𝑡 (3.19) For 𝑢>𝑣+2𝛼, we have ^∞_𝑡0𝑎(𝑡)𝑔^∞_𝑡𝑓(𝑠,𝑐)Δ𝑠Δ𝑡=^∞_𝑡0^∞_𝑡𝑠^𝑣|𝑐|^𝛾1+𝑠^𝑢|𝑐|^𝑚Δ𝑠^1/𝛼≤|𝑐|Δ𝑡_{(𝛾−𝑚)/𝛼∞𝑡0∞𝑡}𝑠^𝑣−𝑢Δ𝑠^1/𝛼≤|𝑐|Δ𝑡_{(𝛾−𝑚)/𝛼}𝑎^{∞(𝑣−𝑢+𝛼+1)/𝛼}_𝑛=𝑛0^∞_𝑟=𝑛𝑟^{(𝑣−𝑢)/𝛼}=|𝑐|^{(𝛾−𝑚)/𝛼}𝑎^{∞(𝑣−𝑢+𝛼+1)/𝛼}_𝑛=𝑛0𝑛^{(𝑣−𝑢+𝛼)/𝛼}<∞, (3.20) that is, (2.25) holds. By Corollary 2.2, system (3.17) has a nonoscillatory solution (𝑥(𝑡),𝑦(𝑡)) such that lim_𝑡→∞𝑥(𝑡)≠0 and lim_𝑡→∞𝑦(𝑡)=0.
On the other hand, For 𝑢+𝑚>𝑣+𝛾+1, we obtain ^∞_𝑎𝑓(𝑡,𝑐𝐴(𝑎,𝜎(𝑡)))Δ𝑡=^∞_𝑎𝑡^𝑣+𝛾|𝑐|^𝛾1+𝑡^𝑢+𝑚|𝑐|^𝑚≤|𝑐|Δ𝑡_{𝛾−𝑚∞𝑎}𝑡^{𝑣+𝛾−𝑢−𝑚}=|𝑐|Δ𝑡_𝛾−𝑚𝑎^{∞𝑣+𝛾−𝑢−𝑚+1}_𝑛=1𝑛^{𝑣+𝛾−𝑢−𝑚}<∞. (3.21) Hence, (2.26) holds. By Theorem 2.3, system (3.17) has a nonoscillatory solution (𝑥(𝑡),𝑦(𝑡)) such that lim_𝑡→∞(𝑥(𝑡)/𝑡)≠0 and lim_𝑡→∞𝑦(𝑡)≠0.

Example 3.5. Consider the system: 𝑥^Δ(𝑡)=1𝑡𝑦⁵(𝑡),𝑦^Δ(𝑡)𝑡=−³𝑥^𝜎(𝑡)^4/3𝑥^𝜎(𝑡)1+𝑡³(𝑥^𝜎(𝑡))², (3.22) where 𝕋=𝑎^ℕ0,ℕ₀={0}∪ℕ and 𝑎>1.
Let 𝑎(𝑡)=1𝑡,𝑔𝑦=𝑦⁵()=𝑡,𝑓𝑡,𝑥³|𝑥|^4/3𝑥1+𝑡³𝑥². (3.23) Obviously, 𝑓(𝑡,𝑥) is increasing in 𝑥 for fixed 𝑡, and taking 𝜑(𝑢)=|𝑢|^−2/3𝑢, we have 𝑓(𝑡,𝑥)≥𝑓𝑡,sgn𝑥|𝑥|^1/3,|𝑥|≥1,_±∞𝑑𝑢𝑔𝜑(𝑢)=^±∞𝑑𝑢𝑢^5/3<∞. (3.24) On the other hand, we obtain ^∞₁𝑎(𝑡)𝑔^∞_𝑡𝑓(𝑠,𝑐)Δ𝑠Δ𝑡=^∞₁1𝑡^∞_𝑡𝑠³|𝑐|^7/31+𝑠³|𝑐|²Δ𝑠⁵≤()Δ𝑡𝑎−1⁶|𝑐|^∞35/3_𝑛=0^∞_𝑟=𝑛𝑎^4𝑟1+𝑎^3𝑟|𝑐|²⁵=∞, (3.25) that is, (3.3) holds. Hence, system (3.22) is oscillatory by Theorem 3.1.

Example 3.6. Consider the system: 𝑥^Δ(𝑡)=𝑦(𝑡),𝑦^Δ(𝑡)=−𝑏(𝑡)𝑥^𝜎(𝑡)^𝜆sgn𝑥^𝜎(𝑡) (3.26) on a time scale 𝕋 which contains only isolated points and is unbounded above. Here, 𝑎(𝑡)=1,𝑔(𝑦)=𝑦,0<𝜆<1,𝑓(𝑡,𝑥)=𝑏(𝑡)|𝑥|^𝜆sgn𝑥, 𝑏(𝑡) is a nonnegative rd-continuous function on [𝑡₀,∞)_𝕋.
We take 𝜑(𝑥)=|𝑥|^𝜆−1𝑥,0<|𝑥|≤1, then all conditions of Theorem 3.2 are satisfied. Hence, system (3.26) is oscillatory if and only if ^∞_𝑡0𝜎^𝜆(𝑡)𝑏(𝑡)Δ𝑡=∞. (3.27) On the other hand, system (3.26) can be written in the Emden-Fowler equation: 𝑥^ΔΔ(𝑡)+𝑏(𝑡)𝑥^𝜎(𝑡)^𝜆sgn𝑥^𝜎(𝑡)=0. (3.28) Since we do not assume that 𝜆 is a quotient of odd positive integers, (3.28) includes the equation studied in [21]. Theorem 3.2 generalizes and improves Theorem 7 of [21].

Acknowledgment

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have lead to the present improved version of the original paper.