AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation35156210.1155/2011/351562351562Research ArticleExistence of Homoclinic Orbits for a Class of Asymptotically p-Linear Difference Systems with p-LaplacianZhangQiongfenTangX. H.ZhouYongSchool of Mathematical Sciences and Computing TechnologyCentral South UniversityChangshaHunan 410083Chinacsu.edu.cn20111082011201130032011020620112011Copyright © 2011 Qiongfen Zhang and X. H. Tang.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.

By applying a variant version of Mountain Pass Theorem in critical point theory, we prove the existence of homoclinic solutions for the following asymptotically p-linear difference system with p-Laplacian Δ(|Δu(n-1)|p-2Δu(n-1))+[-K(n,u(n))+W(n,u(n))]=0, where p(1,+), n, uN, K,W:×N are not periodic in n, and W is asymptotically p-linear at infinity.

1. Introduction

Consider the following p-Laplacian difference system:Δ(|Δu(n-1)|p-2Δu(n-1))+[-K(n,u(n))+W(n,u(n))]=0,nZ, where Δ is the forward difference operator defined by Δu(n)=u(n+1)-u(n), Δ2u(n)=Δ(Δu(n)), p(1,+), n, uN, K, W: ×N are not periodic in n, W is asymptotically p-linear at infinity, and K and W are continuously differentiable in x. As usual, we say that a solution u(n) of (1.1) is homoclinic (to 0) if u(n)0 as n±. In addition, if u(n)0, then u(n) is called a nontrivial homoclinic solution.

When p=2, (1.1) can be regarded as a discrete analogue of the following second-order Hamiltonian system:ü(t)+[-K(t,u(t))+W(t,u(t))]=0,tR.

The existence of homoclinic orbits for Hamiltonian systems is a classical problem and its importance in the study of the behavior of dynamical systems has been recognized by Poincaré . If a system has the transversely intersected homoclinic orbits, then it must be chaotic. If it has the smoothly connected homoclinic orbits, then it cannot stand the perturbation and its perturbed system probably produces chaotic phenomenon. For the existence of homoclinic solutions of problem (1.2), one can refer to the papers .

Difference equations usually describe evolution of certain phenomena over the course of time. For example, if a certain population has discrete generations, the size of the (n+1)th generation x(n+1) is a function of the nth generation x(n). In fact, difference equations provide a natural description of many discrete models in real world. Since discrete models exist in various fields of science and technology such as statistics, computer science, electrical circuit analysis, biology, neural network, and optimal control, it is of practical importance to investigate the solutions of difference equations. For more details about difference equations, we refer the readers to the books .

In some recent papers , the authors studied the existence of periodic solutions and subharmonic solutions of difference equations by applying critical point theory. These papers show that the critical point theory is an effective method to the study of periodic solutions for difference equations. Along this direction, several authors  used critical point theory to study the existence of homoclinic orbits for difference equations. Motivated by the above papers, we consider the existence of homoclinic orbits for problem (1.1) by using the variant version of Mountain Pass Theorem. Our result is new, which seems not to have been considered in the literature. Here is our main result.

Theorem 1.1.

Suppose that K and W satisfy the following conditions.

There are two positive constants b1 and b2 such that b1|x|pK(n,x)b2|x|p,(n,x)Z×RN.

There is a positive constant b3 such that b3|x|p(K(n,x),x)|K(n,x)||x|pK(n,x),  (n,x)Z×RN.

W(n,0)=0, W(n,x)=o(|x|p-1) as |x|0 uniformly for n.

There exists a constant R>0 such that |W(n,x)||x|p-1R,        nZ,  xRN.

There exists a function Vl(,+) such that lim|x||W(n,x)-V(n)|x|p-2x||x|p-1=0        uniformly for  nZ,infZV(n)>max{1,pb2}.

W̃(n,x)=(W(n,x),x)-pW(n,x), lim|x|W̃(n,x)=+uniformly  for  nZ, and for any fixed 0<c1<c2<+, infnZ,  c1|x|c2W̃(n,x)|x|p>0.

Then problem (1.1) has at least one nontrivial homoclinic solution.

Remark 1.2.

The function W(n,x) in this paper is asymptotically p-linear at infinity. The behavior of the gradient of W(n,x) at infinity is like that of a function V(n)|x|p-2x, where V(n) is a real function but not a matrix function. To the best of our knowledge, similar results of this kind of p-Laplacian difference systems with asymptotically p-linear W(n,x) at infinity cannot be found in the literature. From this point, our result is new.

2. Preliminaries

Let S={{u(n)}nZ:u(n)RN,nZ},E={uS:nZ[|Δu(n-1)|p+|u(n)|p]<+}, and for uE, let u={nZ[|Δu(n-1)|p+|u(n)|p]<+}1/p. Then E is a uniform convex Banach space with this norm. As usual, for 1p<+, let lp(Z,RN)={uS:nZ|u(n)|p<+},l(Z,RN)={uS:supnZ|u(n)|<+}, and their norms are given by ulp=(nZ|u(n)|p)1/p,ulp(Z,RN),u=sup{|u(n)|:nZ},ul(Z,RN), respectively.

For any uE, letφ(u)=1pnZ|Δu(n-1)|p-nZ[-K(n,u(n))+W(n,u(n))].

To prove our results, we need the following generalization of Lebesgue's dominated convergence theorem.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B21">29</xref>]).

Let {fk(t)} and {gk(t)} be two sequences of measurable functions on a measurable set A, and let |fk(t)|gk(t),a.e.  tA. If limkfk(t)=f(t),limkgk(t)=g(t),a.e.  tA,limkAgk(t)dt=Ag(t)dt<+, then limkAfk(t)dt=Af(t)dt.

Lemma 2.2.

For uE, uulp2u.

Proof.

Since uE, it follows that lim|n||u(n)|=0. Hence, there exists n* such that u=|u(n*)|=maxnZ|u(n)|. Hence, we have u(nZ|u(n)|p)1/p=ulp=(nZ|u(n)-u(n-1)+u(n-1)|p)1/p(nZ(|u(n)-u(n-1)|+|u(n-1)|)p)1/p(2pnZ(|u(n)-u(n-1)|p+|u(n-1)|p))1/p=2(nZ(|Δu(n-1)|p+|u(n-1)|p))1/p=2(nZ(|Δu(n-1)|p+|u(n)|p))1/p=2u.

Lemma 2.3.

Suppose that (K1), (K2), and (W2) hold. If uku in E, then K(n,uk)K(n,u) and W(n,uk)W(n,u) in lp(,N), where p>1 satisfies 1/p+1/p=1.

Proof.

From (K1) and (K2), we have |K(n,x)|pb2|x|p-1,(n,x)Z×RN. Hence, from (2.12), we have |K(n,uk(n))-K(n,u(n))|p[pb2(|uk(n)|p-1+|u(n)|p-1)]p[pb22p-1|uk(n)-u(n)|p-1+pb2(1+2p-1)|u(n)|p-1]p2pp(pb2)p|uk(n)-u(n)|p+2p(pb2)p(1+2p-1)p|u(n)|p:=gk(n). Moreover, since uku in lp(,N) and uk(n)u(n) for almost every n, hence, limkgk(n)=2p(pb2)p(1+2p-1)p|u(n)|p:=g(n),a.e.nZ,limknZgk(n)=limknZ[2pp(pb2)p|uk(n)-u(n)|p+2p(pb2)p(1+2p-1)p|u(n)|p]=2pp(pb2)plimknZ|uk(n)-u(n)|p+2p(pb2)p(1+2p-1)pnZ|u(n)|p=2p(pb2)p(1+2p-1)pnZ|u(n)|p=nZg(n)<+. It follows from Lemma  2.1, (2.13), and the previous equations that limknZ|K(n,uk(n))-K(n,u(n))|p=0. This shows that K(n,uk)K(n,u) in lp(,N). By a similar proof, we can prove that W(n,uk)W(n,u) in lp(,N). The proof is complete.

Lemma 2.4.

Under the conditions of Theorem 1.1, one has φ(u),v=nZ[|Δu(n-1)|p-2(Δu(n-1),Δv(n-1))+(K(n,u(n))-W(n,u(n)),v(n))|Δu(n-1)|p-2] for u,vE, which yields that φ(u),u=nZ[|Δu(n-1)|p+(K(n,u(n)),u(n))-(W(n,u(n)),u(n))]. Moreover, φ is continuously Fréchet-differential defined on E; that is, φC1(E,) and any critical point u of φ on E is classical solution of (1.1) with u(±)=0.

Proof.

Firstly, we show that φ:E. Let uE, by (2.9) and (K1), we have nZK(n,u(n))nZb2|u(n)|pb22pup. By (W2), we get |W(n,x)|=|01(W(n,sx),x)ds|R|x|p,(n,x)Z×RN. Hence, from (2.9) and (2.19), we have |nZW(n,u(n))|nZ|W(n,u(n))|nZR|u(n)|pR2pup. It follows from (2.5), (2.18), and (2.20) that φ:E. Next we prove that φC1(E,). Rewrite φ as follows: φ(u)=φ1(u)+φ2(u)-φ3(u), where φ1(u):=1pnZ|Δu(n-1)|p,  φ2(u):=nZK(n,u(n)),  φ3(u):=nZW(n,u(n)). It is easy to check that φ1C1(E,) and φ1(u),v=nZ|Δu(n-1)|p-2(Δu(n-1),Δv(n-1)),u,vE. Next, we prove that φiC1(E,),  i=2,  3, and φ2(u),v=nZ(K(n,u(n)),v(n)),u,vE,φ3(u),v=nZ(W(n,u(n)),v(n)),u,vE. For any u,vE and for any function θ:(0,1), by (K2), we have nZmaxh[0,1]|(K(n,u(n)+θ(t)hv(n)),v(n))|pb2nZmaxh[0,1]|u(n)+θ(t)hv(n)|p-1|v(n)|2p-1pb2nZ(|u(n)|p-1+|v(n)|p-1)|v(n)|2p-1pb2[ulpp-1vlp+vlpp]<+. Then by the previous equations and Lebesgue's dominated convergence theorem, we have φ2(u),v=limh0+φ2(u+hv)-φ2(u)h=limh0+1hnZ[K(n,u(n)+hv(n))-K(n,u(n))]=limh0+nZ(K(t,u(n)+θ(t)hv(n)),v(n))=nZ(K(n,u(n)),v(n)),u,vE. Similarly, we can prove that (2.25) holds by using (W2) instead of (K2). Finally, we prove that φiC1(E,),  i=2,3. Let uku in E; then by Lemma 2.3, we have |φ2(uk)-φ2(u),v|=|nZ(K(n,uk(n))-K(n,u(n)),v(n))|nZ|K(n,uk(n))-K(n,u(n))v(n)|v[nZ|K(n,uk(n))-K(n,u(n))|p]1/p0,        k,  vE. This shows that φ2C1(E,). Similarly, we can prove that φ3C1(E,). Furthermore, by a standard argument, it is easy to show that the critical points of φ in E are classical solutions of (1.1) with u(±)=0. The proof is complete.

Lemma 2.5 (see [<xref ref-type="bibr" rid="B5">30</xref>]).

Let E be a real Banach space with its dual space E* and suppose that φC1(E,) satisfies max{φ(0),φ(e)}η0<ηinfu=ρφ(u), for some η0<η, ρ>0, and eE with e>ρ. Let cη be characterized by c=infΥΓmax0τ1φ(Υ(τ)), where Γ={ΥC([0,1],E):Υ(0)=0,Υ(1)=e} is the set of continuous paths joining 0 to e; then there exists {uk}kE such that φ(uk)c,(1+uk)φ(uk)E*0  as  k.

3. Proof of Theorem <xref ref-type="statement" rid="thm1.1">1.1</xref>Proof of Theorem <xref ref-type="statement" rid="thm1.1">1.1</xref>.

We divide the proof of Theorem 1.1 into three steps.

Step 1.

From (W1), there exists ρ0>0 such that W(n,x)C12p|x|p-1,nZ,  |x|ρ0, where C1=min{1/p,b1}. From (3.1), we have W(n,x)=01(W(n,sx),x)ds01C12p|x|psp-1ds=C1p2p|x|p,nZ,  |x|ρ0. Let ρ=ρ0/2 and S={uEu=ρ}; then from (2.9), we obtain uρ0,ulp2ρ,uS, which together with (2.9), (3.2), and (K1) implies that φ(u)=1pnZ|Δu(n-1)|p-nZ[-K(n,u(n))+W(n,u(n))]1pnZ|Δu(n-1)|p+b1nZ|u(n)|p-nZC1p2p|u(n)|pmin{1p,b1}up-C1p2pulppC1up-C1pup=(p-1)C1pup=α1>0,uS.

Step 2.

From (K1), we have φ(u)=1pnZ|Δu(n-1)|p-nZ[-K(n,u(n))+W(n,u(n))]1pnZ|Δu(n-1)|p+b2nZ|u(n)|p-nZW(n,u(n))max{1p,b2}up-nZW(n,u(n))C2up-nZW(n,u(n)). By (W2) and (W3), we get lim|x|pW(n,x)|x|p=V(n)      uniformly  for  nZ. Let W¯(n,x)=pW(n,x)-V(n)|x|p; it follows from (W2), (W3), (2.19), and (3.6) that W¯(n,x)(pR+supZV(n))|x|p,xRN,lim|x|W¯(n,x)|x|p=0. Define E1:={u(n)=xe-|n|:xN,  n}E with infZV(n)>max{1,pb2}(1+|1-e|n|-|n-1||p). By an easy calculation, we have up=(1+|1-e|n|-|n-1||p)ulpp. In what follows, we prove that for some uE1 with u=1, φ(su)- as s. Otherwise, there exist a sequence {sk} with sk as k and a positive constant C3 such that φ(sku)-C3 for all k. From (3.5), we obtain -C3skpφ(sku)skpC2-1pnZW¯(n,sku(n))skp-1pnZV(n)|u(n)|pC2-1pnZW¯(n,sku(n))skp-1pinfZV(n)ulpp. It follows from (3.7) that W¯(n,sku(n))skp(pR+supZV(n))|u(n)|p,W¯(n,sku(n))|sk|p0        as  k. Hence, from Lebesgue's dominated theorem and (3.11), we have nZW¯(n,snu(n))|sk|p0  as  k. It follows from (3.8), (3.9), (3.10), and (3.12) that 0-C3skpC2-1p(1+|1-e|n|-|n-1||p)infZV(n)<0  as  k, which is a contradiction. Hence, there exists eE with e>ρ such that φ(e)0.

Step 3.

From Step 1, Step 2, and Lemma 2.5, we know that there is a sequence {uk}kE such that φ(uk)c,(1+uk)φ(uk)E*0as  k, where E* is the dual space of E. In the following, we will prove that {uk}k is bounded in E. Otherwise, assume that uk as k. Let zk=uk/uk; we have zk=1. It follows from (2.5), (2.16), (3.14), and (K2) that C4pφ(uk)-φ(uk),uk=nZ[(W(n,uk(n)),uk(n))-pW(n,uk(n))]+nZ[pK(n,uk(n))-(K(n,uk(n)),uk(n))]nZ[(W(n,uk(n)),uk(n))-pW(t,uk(n))]:=nZW̃(n,uk(n)). Set Ωk(α,β)={n:α|uk(n)|β} for 0<α<β. Then from (3.15), we have C4nΩk(0,α)W̃(n,uk(n))+nΩk(α,β)W̃(n,uk(n))+nΩk(β,+)W̃(n,uk(n)). From (K1), (K2), and (3.14), we get o(1)=φ(uk),uk=nZ[|Δuk(n-1)|p+(K(n,uk(n))-W(n,uk(n)),uk(n))]nZ[|Δuk(n-1)|p+b3|uk(n)|p-(W(n,uk(n)),uk(n))]min{1,b3}ukp-nZ(W(n,uk(n)),uk(n)):=C5ukp-nZ(W(n,uk(n)),uk(n))=ukp(C5-nZ(W(n,uk(n)),uk(n))ukp), which implies that limsupknZ(W(n,uk(n)),uk(n))|uk(n)|p|zk(n)|p=limsupknZ(W(n,uk(n)),uk(n))ukpC5. Let 0<ɛ<C5/3. From (W1), there exists αɛ>0 such that |W(n,x)|ɛ2p|x|p-1for  |x|αɛ  uniformly  for  nZ. Since zk=1, it follows from (2.9) and (3.19) that nΩk(0,αɛ)|W(n,uk(n))||uk(n)|p-1|zk(n)|pnΩk(0,αɛ)ɛ2p|zk(n)|pɛ,  kN. For s>0, let h(s):=inf{W̃(n,x)nZ,    xRN  with  |x|s}. Thus, from (W4), we have h(s)+ as s+, which together with (3.16) implies that meas(Ωk(β,+))C6h(β)0,  as  β+. Hence, we can take βɛ sufficiently large such that nΩk(βɛ,+)|zk(n)|p<ɛR. The previous inequality and (W2) imply that nΩk(βɛ,+)|W(n,uk(n))||uk(n)|p-1|zk(n)|pRnΩk(βɛ,+)|zk(n)|p<ɛ,kN. Next, for the previous 0<αɛ<βɛ, let cɛ:=inf{W̃(n,x)|x|p:nZ,  xRN  with  αɛ|x|βɛ},dɛ:=max{|W(n,x)||x|p-1:nZ,  xRN  with  αɛ|x|βɛ}. From (W4), we have cɛ>0 and W̃(n,uk(n))cɛ|uk(n)|p,        nΩk(αɛ,βɛ). From (3.15) and (3.26), we get nΩk(αɛ,βɛ)|zk(n)|p=1ukpnΩk(αɛ,βɛ)|uk(n)|p1ukpnΩk(αɛ,βɛ)1cɛW̃(n,uk(n))C4cɛukp0        as  k, which implies that nΩk(αɛ,βɛ)|W(n,uk(n))||uk(n)|p-1|zk(n)|pdɛnΩk(αɛ,βɛ)|zk(n)|p0  as  k. Therefore, there exists k0>0 such that nΩk(αɛ,βɛ)|W(n,uk(n))||uk(n)|p-1|zk(n)|pɛ,kk0. It follows from (3.20), (3.24), and (3.29) that nZ(W(n,uk(n)),uk(n))|uk(n)|p|zk(n)|pnZ|W(t,uk(n))||uk(n)|p-1|zk(n)|p<3ɛ<C5,kk0, which implies that limsupnnZ(W(n,uk(n)),uk(n))|uk(n)|p|zk(n)|p<C5, but this contradicts to (3.18). Hence, uk is bounded in E.

Going to a subsequence if necessary, we may assume that there exists uE such that uku as k. In order to prove our theorem, it is sufficient to show that φ(u)=0. For any a with a>0, let χa(t)=1 for t[-a,a] and let χa(t)=0 for t(-,-a)(a,). Then from (2.16), we have φ(uk)-φ(u),χa(uk-u)=nZ[-a,a]|Δuk(n-1)|p-2(Δuk(n-1),Δuk(n-1)-Δu(n-1))-nZ[-a,a]|Δu(n-1)|p-2(Δu(n-1),Δuk(n-1)-Δu(n-1))+nZ[-a,a](K(n,uk(n))-K(n,u(n)),uk(n)-u(n))-nZ[-a,a](W(n,uk(n))-W(n,u(n)),uk(n)-u(n))ΔuklpZ[-a,a]p+ΔulpZ[-a,a]p-nZ[-a,a]|Δuk(n-1)|p-1|Δu(n-1)|-nZ[-a,a]|Δu(n-1)|p-1|Δuk(n-1)|+nZ[-a,a](K(n,uk(n))-K(n,u(n)),uk(n)-u(n))-nZ[-a,a](W(n,uk(n))-W(n,u(n)),uk(n)-u(n))ΔuklpZ[-a,a]p+ΔulpZ[-a,a]p-ΔulpZ[-a,a]ΔuklpZ[-a,a]p-1-ΔuklpZ[-a,a]ΔulpZ[-a,a]p-1+nZ[-a,a](K(n,uk(n))-K(n,u(n)),uk(n)-u(n))-nZ[-a,a](W(n,uk(n))-W(n,u(n)),uk(n)-u(n))=(ΔuklpZ[-a,a]p-1-ΔulpZ[-a,a]p-1)(ΔuklpZ[-a,a]-ΔulpZ[-a,a])+nZ[-a,a](K(n,uk(n))-K(n,u(n)),uk(n)-u(n))-nZ[-a,a](W(n,uk(n))-W(n,u(n)),uk(n)-u(n)). Since φ(uk)0 as k+ and uku in E, it follows from (3.14) that φ(uk)-φ(u),χa(uk-u)0  as  k,nZ[-a,a](K(n,uk(n))-K(n,u(n)),uk(n)-u(n))0  as  k,nZ[-a,a](W(n,uk(n))-W(n,u(n)),uk(n)-u(n))0  as  k. It follows from (3.32) and (3.33) that Δuklp[-a,a]Δulp[-a,a] as k+.

For any wC0(,N), and assume that for some A with A>0, supp(w)[-A,A]. Since limkΔuk(n-1)=Δu(n-1),a.e.    nZ,|(|Δuk(n-1)|p-2Δuk(n-1),Δw(n-1))|p-1p|Δuk(n-1)|p+1p|Δw(n-1)|p,nZ,k=1,2,,limknZ[-A,A][p-1p|Δuk(n-1)|p+1p|Δw(n-1)|p]=p-1plimkΔuklpZ[-A,A]p+1pΔwlpZ[-A,A]p=p-1pΔulpZ[-A,A]p+1pΔwlpZ[-A,A]p=nZ[-A,A][p-1p|Δu(n-1)|p+1p|Δw(n-1)|p]<+, then, we have nZ[-A,A](|Δuk(n-1)|p-2Δuk(n-1),Δw(n-1))nZ[-A,A](|Δu(n-1)|p-2Δu(n-1),Δw(n-1)) as k. Noting that nZ[-A,A](K(n,uk(n)),w(n))nZ[-A,A](K(n,u(n)),w(n))as  k,nZ[-A,A](W(n,uk(n)),w(n))nZ[-A,A](W(n,u(n)),w(n))as  k. Hence, we have φ(u),w=limkφ(uk),w=0, which implies that φ(u)=0; that is, u is a critical point of φ. From (K1) and (W1), we know that u0. In fact, if u=0, we have from (2.5), (K1), and (W1) that φ(u)=0. On the other hand, from Step 1, Step 2, and Lemma 2.5, we know that φ(u)=c>0. This is a contradiction. The proof of Theorem 1.1 is complete.

4. An ExampleExample 4.1.

In problem (1.1), let p=3/2, and K(n,x)=(1+1|x|3/2+1)|x|3/2,        W(n,x)=a(n)|x|3/2(1-1(ln(e+|x|))1/2), where al(,+) with infa(n)>3. One can easily check that K satisfies conditions (K1) and (K2) with b1=1, b2=2, and b3=3/2. An easy computation shows that W(n,x)=32a(n)|x|-1/2x(1-1(ln(e+|x|))1/2)+a(n)|x|1/2x2(e+|x|)(ln(e+|x|))3/2,(W(n,x),x)-32W(n,x)=a(n)|x|5/22(e+|x|)(ln(e+|x|))3/2. Then it is easy to check that W satisfies (W1)–(W4). Hence, K(n,x) and W(n,x) satisfy all the conditions of Theorem 1.1 and then problem (1.1) has at least one nontrivial homoclinic solution.

Acknowledgment

This work is partially supported by the NNSF (no. 10771215) of China.