We consider the nonlinear dynamic system xΔ(t)=a(t)g(y(t)), yΔ(t)=-f(t,xσ(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:
(1.1)xΔ(t)=a(t)g(y(t)),yΔ(t)=-f(t,xσ(t)),t∈[t0,∞)T,
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∈Crd1([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 [tx,∞)𝕋 and satisfy sup{|x(t)|+|y(t)|∶t≥tx}>0 for any tx≥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
(1.2)[xΔ(t)a(t)]Δ+b(t)f(xσ(t))=0,xΔΔ(t)+b(t)|xσ(t)|λ-1xσ(t)=0,λ>0,
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.

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
(2.1)hf(t,l)≤|f(t,u)|≤Hf(t,L),t∈[t0,∞)T.

There exists a positive constant k such that g(uv)≥kg(u)g(v) for uv>0.

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
(2.2)hf(t,lθ(t))≤|f(t,uθ(t))|≤Hf(t,Lθ(t)),t∈[t0,∞)T,

where θ(t) is a positive nondecreasing function.

For convenience, we will employ the following notation:
(2.3)A(s,t)=∫sta(τ)Δτ,s,t∈[t0,∞)T.

Theorem 2.1.

Assume that g is nondecreasing and that (H_{1}) holds. Then system (1.1) has a nonoscillatory solution (x(t),y(t)) such that limt→∞x(t)=α≠0 and limt→∞y(t)=0 if and only if for any positive constant d(2.4)∫t0∞a(t)g(d∫t∞|f(s,c)|Δs)Δt<∞forsomec≠0.

Proof.

Suppose that (x(t),y(t)) is a nonoscillatory solution of (1.1) such that limt→∞x(t)=α≠0 and limt→∞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 (H_{1}) implies that
(2.5)f(t,xσ(t))≥hf(t,l)
for t∈[t1,∞)𝕋 and some constant h>0. It follows from the second equation in (1.1) that
(2.6)y(s)-y(t)=-∫tsf(τ,xσ(τ))Δτ.
Let s→∞ and noting that lims→∞y(s)=0, we have
(2.7)y(t)=∫t∞f(τ,xσ(τ))Δτ,t∈[t1,∞)T.
Thus, from (2.5), (2.7) and the first equation in (1.1), we obtain that
(2.8)∞>limt→∞x(t)-x(t1)=∫t1∞a(s)g(y(s))Δs=∫t1∞a(s)g(∫s∞f(τ,xσ(τ))Δτ)Δs≥∫t1∞a(s)g(h∫s∞f(τ,l)Δτ)Δs,
which implies that (2.4) holds.

Conversely, suppose that (2.4) holds, we may assume that c>0. In view of (H_{1}), there is a constant H>0 such that c/2≤x(t)≤c implies
(2.9)f(t,xσ(t))≤Hf(t,c),t∈[t0,∞)T.
Since (2.4) holds, we can choose t1∈[t0,∞)𝕋 large enough such that
(2.10)∫t1∞a(s)g(H∫s∞f(τ,c)Δτ)Δs≤c2.
Let BC[t0,∞)𝕋 be the Banach space of all real-valued rd-continuous functions on [t0,∞)𝕋 endowed with the norm ∥x∥=supt∈[t0,∞)𝕋|x(t)|<∞. We defined a bounded, convex, and closed subset Ω of BC[t0,∞)𝕋 as
(2.11)Ω={x∈BC[t0,∞)T:c2≤x(t)≤c}.
Define an operator Γ:Ω→BC[t0,∞)𝕋 as follows:
(2.12)(Γx)(t)={c-∫t∞a(s)g(∫s∞f(τ,xσ(τ))Δτ)Δs,t∈[t1,∞)T,c-∫t1∞a(s)g(∫s∞f(τ,xσ(τ))Δτ)Δs,t∈[t0,t1]T.
Now we show that Γ satisfies the assumptions of Schauder's fixed-point theorem (see [19, Corollary 6]).

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
(2.13)c≥(Γx)(t)=c-∫t∞a(s)g(∫s∞f(τ,xσ(τ))Δτ)Δs≥c-∫t1∞a(s)g(H∫s∞f(τ,c)Δτ)Δs≥c-c2=c2.

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.

We prove that Γ is a completely continuous mapping. First, we consider the continuity of Γ. Let xn∈Ω and ∥xn-x∥→0 as n→∞, then x∈Ω and |xn(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
(2.14)limn→∞|a(t)[g(∫t∞f(τ,xnσ(τ))Δτ)-g(∫t∞f(τ,xσ(τ))Δτ)]|=0.

From (2.9), we obtain that
(2.15)a(t)|g(∫t∞f(τ,xnσ(τ))Δτ)-g(∫t∞f(τ,xσ(τ))Δτ)|≤2a(t)g(H∫t∞f(τ,c)Δτ).
On the other hand, from (2.12) we have
(2.16)|(Γxn)(t)-(Γx)(t)|≤∫t1∞a(s)|g(∫s∞f(τ,xnσ(τ))Δτ)-g(∫s∞f(τ,xσ(τ))Δτ)|Δs
for t∈[t0,t1]𝕋 and
(2.17)|(Γxn)(t)-(Γx)(t)|≤∫t∞a(s)|g(∫s∞f(τ,xnσ(τ))Δτ)-g(∫s∞f(τ,xσ(τ))Δτ)|Δs
for t∈[t1,∞]𝕋. Therefore, from (2.16) and (2.17), we have
(2.18)‖Γxn-Γx‖≤∫t1∞a(s)|g(∫s∞f(τ,xnσ(τ))Δτ)-g(∫s∞f(τ,xσ(τ))Δτ)|Δs.
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 limn→∞∥Γxn-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
(2.19)∫t2∞a(s)g(H∫s∞f(τ,c)Δτ)Δs≤ε.
Then for any x∈Ω and t,r∈[t2,∞)𝕋, we have
(2.20)|(Γx)(t)-(Γx)(r)|≤|∫t∞a(s)g(H∫s∞f(τ,c)Δτ)Δs|+|∫r∞a(s)g(H∫s∞f(τ,c)Δτ)Δs|≤2∫t2∞a(s)g(H∫s∞f(τ,c)Δτ)Δs≤2ε.
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
(2.21)|(Γx)(t)-(Γx)(r)|=|∫t∞a(s)g(∫s∞f(τ,xσ(τ))Δτ)Δs-∫r∞a(s)g(∫s∞f(τ,xσ(τ))Δτ)Δs|≤|∫tra(s)g(H∫s∞f(τ,c)Δτ)Δs|
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
(2.22)x(t)=c-∫t∞a(s)g(∫s∞f(τ,xσ(τ))Δτ)Δs,t∈[t1,∞)T.
Set
(2.23)y(t)=∫t∞f(τ,xσ(τ))Δτ,t∈[t1,∞)T.
Then limt→∞y(t)=0 and yΔ(t)=-f(t,xσ(t)). On the other hand,
(2.24)x(t)=c-∫t∞a(s)g(y(s))Δs,
which implies that limt→∞x(t)=c and xΔ(t)=a(t)g(y(t)). The proof is complete.

Corollary 2.2.

Suppose that g is nondecreasing and that (H_{1}) and (H_{2}) hold. Then system (1.1) has a nonoscillatory solution (x(t),y(t)) such that limt→∞x(t)=α≠0 and limt→∞y(t)=0 if and only if for some c≠0(2.25)∫t0∞a(t)g(∫t∞|f(s,c)|Δs)Δt<∞.

Theorem 2.3.

Suppose that limt→∞A(t0,t)=∞ and g is nondecreasing. Suppose further that (H_{3}) holds. Then system (1.1) has a nonoscillatory solution (x(t),y(t)) such that limt→∞(x(t)/A(t0,t))=α≠0 and limt→∞y(t)=β≠0 if and only if for some c≠0(2.26)∫t0∞|f(t,cA(t0,σ(t)))|Δt<∞.

Proof.

Suppose that (x(t),y(t)) is a nonoscillatory solution of (1.1) such that limt→∞(x(t)/A(t0,t))=α≠0 and limt→∞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 yΔ(t)<0,y(t)>β for t∈[t1,∞)𝕋. By condition (H_{3}), there exists a constant h>0 such that f(t,xσ(t))≥hf(t,lA(t0,σ(t))) for t∈[t1,∞)𝕋. According to the second equation in (1.1), we have
(2.27)∞>y(t1)-β=∫t1∞f(t,xσ(t))Δt≥h∫t1∞f(t,lA(t0,σ(t)))Δt,
which implies that (2.26) holds with c=l.

Conversely, Let (2.26) holds for some c=2p, where p>0. By (H_{3}), 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
(2.28)H∫t1∞f(t,cA(t0,σ(t)))Δt≤d,
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∥=supt∈[t1,∞)𝕋(|x(t)|/A(t1,t)), and the usual pointwise ordering ≤.

Define
(2.29)Ω={x∈BC[t1,∞)T∶g(d)A(t1,t)≤x(t)≤g(2d)A(t1,t)}.
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:
(2.30)(Γx)(t)=∫t1ta(s)g(d+∫s∞f(τ,xσ(τ))Δτ)Δs,t∈[t1,∞)T.

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
(2.31)x(t)=∫t1ta(s)g(d+∫s∞f(τ,xσ(τ))Δτ)Δs,t∈[t1,∞)T.
Set
(2.32)y(t)=d+∫t∞f(τ,xσ(τ))Δτ,t∈[t1,∞)T.
Then limt→∞y(t)=d and yΔ(t)=-f(t,xσ(t)). On the other hand, by L'Hôpital's Rule (see [15, Lemma 2.11]), we have
(2.33)limt→∞x(t)A(t0,t)=limt→∞a(t)g(d+∫t∞f(τ,xσ(τ))Δτ)a(t)=limt→∞g(d+∫t∞f(τ,xσ(τ))Δτ)=g(d)≠0,
and xΔ(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.

There exists a continuous nondecreasing function φ:ℝ→ℝ such that
(3.1)sgnφ(u)=sgnu,∫±∞dug(φ(u))<∞,

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.

There exists a continuous nondecreasing function φ:[-M,M]→ℝ,M>0 being a constant, such that
(3.2)sgnφ(v)=sgnv,∫0±Mdvφ(g(v))<∞,

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 limt→∞A(t0,t)=∞ and g is nondecreasing. Suppose further that (H_{1}), (H_{2}) and (H_{4}) hold. Then system (1.1) is oscillatory if and only if for all c≠0(3.3)∫t0∞a(t)g(∫t∞|f(s,c)|Δs)Δt=∞.

Proof.

If (3.3) does not hold, by Theorem 2.1, system (1.1) has a nonoscillatory solution (x(t),y(t)) such that limt→∞x(t)=α≠0 and limt→∞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 limt→∞A(t0,t)=∞, it is easy to show that y(t)>0,t∈[t1,∞)𝕋. From the second equation in (1.1), we have yΔ(t)<0,t∈[t1,∞)𝕋. Hence, limt→∞y(t)≥0. It follows from the first equation in (1.1) that xΔ(t)>0,t∈[t1,∞)𝕋, and limt→∞x(t)=∞ by Theorem 2.1. Integrating the second equation in (1.1) from t to ∞ yields that
(3.4)y(t)≥∫t∞f(s,xσ(s))Δs,t∈[t1,∞)T.
By (3.4), (H_{2}) and in view of nondecreasing φ, it follows that
(3.5)xΔ(t)g(φ(xσ(t)))=a(t)g(y(t))g(φ(xσ(t)))≥a(t)g(∫t∞f(s,xσ(s))Δs)g(φ(xσ(t)))≥ka(t)g(∫t∞f(s,xσ(s))φ(xσ(s))Δs)
for t∈[t1,∞)𝕋.

Since (H_{4}) holds and limt→∞x(t)=∞, there is t2∈[t1,∞)𝕋 and l>0 such that
(3.6)f(t,xσ(t))φ(xσ(t))≥f(t,l)
for t∈[t2,∞)𝕋. From (3.5) and (3.6), we get
(3.7)xΔ(t)g(φ(xσ(t)))≥ka(t)g(∫t∞f(s,l)Δs).
Integrating (3.7) from t2 to t, we have
(3.8)∫t2txΔ(s)g(φ(xσ(s)))Δs≥k∫t2ta(s)g(∫s∞f(τ,l)Δτ)Δs.
Since g,φ, and x are nondecreasing, we obtain
(3.9)∫t2txΔ(s)g(φ(xσ(s)))Δs≤∫t2txΔ(s)g(φ(x(s)))Δs.
By (H_{4}), (3.8) and (3.9), we get
(3.10)k∫t2ta(s)g(∫s∞f(τ,l)Δτ)Δs≤∫x(t2)x(t)dug(φ(u))<∞,
which contradicts (3.3) when t→∞. The proof is complete.

Theorem 3.2.

Suppose that limt→∞A(t0,t)=∞ and (H_{5}) 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(3.11)∫t0∞|f(t,cA(t0,σ(t)))|Δt=∞.

Proof.

If (3.11) does not hold, by Theorem 2.3, system (1.1) has a nonoscillatory solution (x(t),y(t)) such that limt→∞(x(t)/A(t0,t))=α≠0 and limt→∞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 xΔ(t)>0,yΔ(t)<0,y(t)>0 eventually. We claim that (3.11) implies limt→∞y(t)=0. In fact, if limt→∞y(t)=β>0, then y(t)≥β for t∈[t1,∞)𝕋. According to the first equation in (1.1), we get
(3.12)xσ(t)=x(t1)+∫t1σ(t)a(s)g(y(s))Δs≥∫t1ta(s)g(y(s))Δs+∫tσ(t)a(s)g(y(s))Δs≥g(y(t))[A(t1,t)+μ(t)a(t)]=g(y(t))A(t1,σ(t))≥g(β)A(t1,σ(t)).
Integrating the second equation in (1.1) from t1 to ∞, we have
(3.13)β-y(t1)=-∫t1∞f(s,xσ(s))Δs≤-∫t1∞f(s,g(β)A(t1,σ(s)))Δs=-∞,
which is a contradiction. Hence, limt→∞y(t)=0.

By (H_{5}), we have
(3.14)kf(t,A(t1,σ(t)))≤kf(t,xσ(t)g(y(t)))≤f(t,xσ(t))φ(g(y(t)))=-yΔ(t)φ(g(y(t))).
From (3.14), it follows
(3.15)∫t1tkf(s,A(t1,σ(s)))Δs≤-∫t1tyΔ(s)φ(g(y(s)))Δs.
Hence,
(3.16)k∫t1tf(s,A(t1,σ(s)))Δs≤-∫y(t1)y(t)duφ(g(u)).
In view of (H_{5}) 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:
(3.17)xΔ(t)=|y(t)|1/α-1y(t),yΔ(t)=-tv|xσ(t)|γ-1xσ(t)1+tu|xσ(t)|m,
where 𝕋=aℕ={an∣n∈ℕ},a,m,γ,u,α>0 and v are constants as well as γ>m.

Let
(3.18)a(t)=1,g(y)=|y|(1/α)-1y,f(t,x)=tv|x|γ-1x1+tu|x|m.
It is easy to see that g(y) is increasing and for 0<l≤x≤L,γ≥m,
(3.19)f(t,l)≤f(t,x)≤f(t,L),f(t,lt)≤f(t,xt)≤f(t,Lt).
For u>v+2α, we have
(3.20)∫t0∞a(t)g(∫t∞|f(s,c)|Δs)Δt=∫t0∞(∫t∞sv|c|γ1+su|c|mΔs)1/αΔt≤|c|(γ-m)/α∫t0∞(∫t∞sv-uΔs)1/αΔt≤|c|(γ-m)/αa(v-u+α+1)/α∑n=n0∞∑r=n∞r(v-u)/α=|c|(γ-m)/αa(v-u+α+1)/α∑n=n0∞n(v-u+α)/α<∞,
that is, (2.25) holds. By Corollary 2.2, system (3.17) has a nonoscillatory solution (x(t),y(t)) such that limt→∞x(t)≠0 and limt→∞y(t)=0.

On the other hand, For u+m>v+γ+1, we obtain
(3.21)∫a∞|f(t,cA(a,σ(t)))|Δt=∫a∞tv+γ|c|γ1+tu+m|c|mΔt≤|c|γ-m∫a∞tv+γ-u-mΔt=|c|γ-mav+γ-u-m+1∑n=1∞nv+γ-u-m<∞.
Hence, (2.26) holds. By Theorem 2.3, system (3.17) has a nonoscillatory solution (x(t),y(t)) such that limt→∞(x(t)/t)≠0 and limt→∞y(t)≠0.

Example 3.5.

Consider the system:
(3.22)xΔ(t)=1ty5(t),yΔ(t)=-t3|xσ(t)|4/3xσ(t)1+t3(xσ(t))2,
where 𝕋=aℕ0,ℕ0={0}∪ℕ and a>1.

Let
(3.23)a(t)=1t,g(y)=y5,f(t,x)=t3|x|4/3x1+t3x2.
Obviously, f(t,x) is increasing in x for fixed t, and taking φ(u)=|u|-2/3u, we have
(3.24)|f(t,x)|≥|f(t,sgnx)||x|1/3,|x|≥1,∫±∞dug(φ(u))=∫±∞duu5/3<∞.
On the other hand, we obtain
(3.25)∫1∞a(t)g(∫t∞|f(s,c)|Δs)Δt=∫1∞1t(∫t∞s3|c|7/31+s3|c|2Δs)5Δt≤(a-1)6|c|35/3∑n=0∞(∑r=n∞a4r1+a3r|c|2)5=∞,
that is, (3.3) holds. Hence, system (3.22) is oscillatory by Theorem 3.1.

Example 3.6.

Consider the system:
(3.26)xΔ(t)=y(t),yΔ(t)=-b(t)|xσ(t)|λsgnxσ(t)
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
(3.27)∫t0∞σλ(t)b(t)Δt=∞.
On the other hand, system (3.26) can be written in the Emden-Fowler equation:
(3.28)xΔΔ(t)+b(t)|xσ(t)|λsgnxσ(t)=0.
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.

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