DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 137471 10.1155/2012/137471 137471 Research Article Oscillation and Nonoscillation Criteria for Nonlinear Dynamic Systems on Time Scales Zhu Shanliang Sheng Chunyun Cao Jinde College of Mathematics and Physics Qingdao University of Science and Technology Qingdao 266061 China qust.edu.cn 2012 19 7 2012 2012 10 04 2012 23 06 2012 2012 Copyright © 2012 Shanliang Zhu and Chunyun Sheng. 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.

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 is a continuous function with ug(u)>0 for u0, and f(t,u)[t0,)𝕋× is continuous as a function of u with sign property uf(t,u)>0 for u0 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,yCrd1([t0,)𝕋,) 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)|ttx}>0 for any txt0. 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 ). 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.  and references cited therein. A book on the subject of time scales (see ) summarizes and organizes much of time scale calculus. The reader is referred to Chapter 1 in  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  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  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 [48, 1418] and the references therein. Erbe and Mert [14, 17] obtained some oscillation results for the system (1.1). Fu and Lin  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 ), 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 [79, 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 (H1) holds. Then system (1.1) has a nonoscillatory solution (x(t),y(t)) such that limtx(t)=α0 and limty(t)=0 if and only if for any positive constant d(2.4)t0a(t)g(dt|f(s,c)|Δs)Δt<forsome  c0.

Proof.

Suppose that (x(t),y(t)) is a nonoscillatory solution of (1.1) such that limtx(t)=α0 and limty(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 lx(t)L for t[t1,)𝕋. Condition (H1) 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 limsy(s)=0, we have (2.7)y(t)=tf(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)>limtx(t)-x(t1)=t1a(s)g(y(s))Δs=t1a(s)g(sf(τ,xσ(τ))Δτ)Δst1a(s)g(hsf(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 (H1), there is a constant H>0 such that c/2x(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)t1a(s)g(Hsf(τ,c)Δτ)Δsc2.         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={xBC[t0,)T:c2x(t)c}. Define an operator Γ:ΩBC[t0,)𝕋 as follows: (2.12)(Γx)(t)={c-ta(s)g(sf(τ,xσ(τ))Δτ)Δs,t[t1,)T,c-t1a(s)g(sf(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 ΓxΩ for any xΩ. In fact, for any xΩ and t[t1,)𝕋, in view of (2.10), we get (2.13)c(Γx)(t)=c-ta(s)g(sf(τ,xσ(τ))Δτ)Δsc-t1a(s)g(Hsf(τ,c)Δτ)Δsc-c2=c2.

Similarly, we can prove that c/2(Γx)(t)c for any xΩ and t[t0,t1)𝕋. Hence, ΓxΩ for any xΩ.

We prove that Γ is a completely continuous mapping. First, we consider the continuity of Γ. Let xnΩ and xn-x0 as n, then xΩ 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(tf(τ,xnσ(τ))Δτ)-g(tf(τ,xσ(τ))Δτ)]|=0.

From (2.9), we obtain that (2.15)a(t)|g(tf(τ,xnσ(τ))Δτ)-g(tf(τ,xσ(τ))Δτ)|2a(t)g(Htf(τ,c)Δτ).         On the other hand, from (2.12) we have (2.16)|(Γxn)(t)-(Γx)(t)|t1a(s)|g(sf(τ,xnσ(τ))Δτ)-g(sf(τ,xσ(τ))Δτ)|Δs for t[t0,t1]𝕋 and (2.17)|(Γxn)(t)-(Γx)(t)|ta(s)|g(sf(τ,xnσ(τ))Δτ)-g(sf(τ,xσ(τ))Δτ)|Δs for t[t1,]𝕋. Therefore, from (2.16) and (2.17), we have (2.18)Γxn-Γxt1a(s)|g(sf(τ,xnσ(τ))Δτ)-g(sf(τ,xσ(τ))Δτ)|Δs. Referring to Chapter 5 in , 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 limnΓxn-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)t2a(s)g(Hsf(τ,c)Δτ)Δsε. Then for any xΩ and t,r[t2,)𝕋, we have (2.20)|(Γx)(t)-(Γx)(r)||ta(s)g(Hsf(τ,c)Δτ)Δs|+|ra(s)g(Hsf(τ,c)Δτ)Δs|2t2a(s)g(Hsf(τ,c)Δτ)Δs2ε. 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 xΩ, we have |(Γx)(t)-(Γx)(r)|0 for t,r[t0,t1]𝕋 and (2.21)|(Γx)(t)-(Γx)(r)|=|ta(s)g(sf(τ,xσ(τ))Δτ)Δs-ra(s)g(sf(τ,xσ(τ))Δτ)Δs||tra(s)g(Hsf(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 xΩ. 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 xΩ such that Γx=x. Therefore, we have (2.22)x(t)=c-ta(s)g(sf(τ,xσ(τ))Δτ)Δs,t[t1,)T. Set (2.23)y(t)=tf(τ,xσ(τ))Δτ,t[t1,)T. Then limty(t)=0 and yΔ(t)=-f(t,xσ(t)). On the other hand, (2.24)x(t)=c-ta(s)g(y(s))Δs, which implies that limtx(t)=c and xΔ(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 limtx(t)=α0 and limty(t)=0 if and only if for some c0(2.25)t0a(t)g(t|f(s,c)|Δs)Δt<.

Theorem 2.3.

Suppose that limtA(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 limt(x(t)/A(t0,t))=α0 and limty(t)=β0 if and only if for some c0(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))=α0 and limty(t)=β0. 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 (H3), 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=t1f(t,xσ(t))Δtht1f(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 (H3), there exists a constant H>0 such that pu2p 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)Ht1f(t,cA(t0,σ(t)))Δtd, 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={xBC[t1,)Tg(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:ΩBC[t1,)𝕋 as follows: (2.30)(Γx)(t)=t1ta(s)g(d+sf(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 xΩ such that x=Γx, this is (2.31)x(t)=t1ta(s)g(d+sf(τ,xσ(τ))Δτ)Δs,t[t1,)T. Set (2.32)y(t)=d+tf(τ,xσ(τ))Δτ,t[t1,)T. Then limty(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)limtx(t)A(t0,t)=limta(t)g(d+tf(τ,xσ(τ))Δτ)a(t)=limtg(d+tf(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 [79, 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: 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 l0 with sgnl=sgnu.

There exists a continuous nondecreasing function φ:[-M,M],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,u0,0<|v|<v0 for some positive constant k>0 and v0>0.

Theorem 3.1.

Suppose that limtA(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 c0(3.3)t0a(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 limtx(t)=α0 and limty(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 limtA(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, limty(t)0. It follows from the first equation in (1.1) that xΔ(t)>0,t[t1,)𝕋, and limtx(t)= by Theorem 2.1. Integrating the second equation in (1.1) from t to yields that (3.4)y(t)tf(s,xσ(s))Δs,t[t1,)T. By (3.4), (H2) 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(tf(s,xσ(s))Δs)g(φ(xσ(t)))ka(t)g(tf(s,xσ(s))φ(xσ(s))Δs) for t[t1,)𝕋.

Since (H4) holds and limtx(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(tf(s,l)Δs). Integrating (3.7) from t2 to t, we have (3.8)t2txΔ(s)g(φ(xσ(s)))Δskt2ta(s)g(sf(τ,l)Δτ)Δs. Since g,φ, and x are nondecreasing, we obtain (3.9)t2txΔ(s)g(φ(xσ(s)))Δst2txΔ(s)g(φ(x(s)))Δs. By (H4), (3.8) and (3.9), we get (3.10)kt2ta(s)g(sf(τ,l)Δτ)Δsx(t2)x(t)dug(φ(u))<, which contradicts (3.3) when t. The proof is complete.

Theorem 3.2.

Suppose that limtA(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 c0(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))=α0 and limty(t)=β0.

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 limty(t)=0. In fact, if limty(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))Δst1ta(s)g(y(s))Δs+tσ(t)a(s)g(y(s))Δsg(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)=-t1f(s,xσ(s))Δs-t1f(s,g(β)A(t1,σ(s)))Δs=-, which is a contradiction. Hence, limty(t)=0.

By (H5), 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)kt1tf(s,A(t1,σ(s)))Δs-y(t1)y(t)duφ(g(u)). 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: (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=a={ann},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<lxL,γm, (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)t0a(t)g(t|f(s,c)|Δs)Δt=t0(tsv|c|γ1+su|c|mΔs)1/αΔt|c|(γ-m)/αt0(tsv-uΔs)1/αΔt|c|(γ-m)/αa(v-u+α+1)/αn=n0r=nr(v-u)/α=|c|(γ-m)/αa(v-u+α+1)/αn=n0n(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 limtx(t)0 and limty(t)=0.

On the other hand, For u+m>v+γ+1, we obtain (3.21)a|f(t,cA(a,σ(t)))|Δt=atv+γ|c|γ1+tu+m|c|mΔt|c|γ-matv+γ-u-mΔt=|c|γ-mav+γ-u-m+1n=1nv+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 limty(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=a0,0={0} 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)1a(t)g(t|f(s,c)|Δs)Δt=11t(ts3|c|7/31+s3|c|2Δs)5Δt(a-1)6|c|35/3n=0(r=na4r1+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)t0σλ(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 . Theorem 3.2 generalizes and improves Theorem 7 of .

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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