# Homoclinic Solutions for a Higher Orderϕc-Laplacian Difference Equation Containing Both Advance and Retardation

Authors: Peng Mei; Zhan Zhou
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

We consider a 2mth-order nonlinear difference equation containing both advance and retardation with ϕc-Laplacian. Using the critical point theory, some new and concrete criteria for the existence of homoclinic solutions with mixed nonlinearities are obtained.


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## Body

## 1. Introduction

Consider the following 2mth-order nonlinear difference equation containing both advance and retardation with ϕc-Laplacian:(1)−1mΔmϕcΔmun−m+ωnun=fn,un+1,un,un−1,n∈ℤ,where Δ is the forward difference operator defined by Δuk=uk+1−uk and Δjuk=ΔΔj−1uk for j≥2, ωn>0 is real valued for each n∈ℤ, f∈Cℤ×ℝ3,ℝ, ωn, and fn are T-periodic in n for a given positive integer T, and ϕcs is a special ϕ-Laplacian operator [1] defined by ϕcs=s/1+s2,s∈ℝ.It is well known thatϕcs is the mean curvature operator given by the corresponding potential Φcs=1+s2−1. And it is such that(2)ϕc−1v=v1−v2=∇ΦRv,v∈−1,1,where ΦRv=1−1−v2 may be seen as the acceleration in classical relativity. Actually, it is important to determine the existence and regularity properties of maximal and constant mean curvature hypersurfaces, as they provide Riemannian submanifolds that reflect spatiotemporal properties. There have been extensive study and application of nonlinear difference equations with ϕ-Laplacian. For example, scalar equations were considered in [2, 3] using topological methods, and T-periodic solutions have been studied in [1] by variational methods.Bonheure et al. [4] in 2007 studied classical and nonclassical solutions of a prescribed curvature equation:(3)−u′1+u′2′=λft,u,t∈ℝ,withu0=u1=0.This problem depends on the behavior at the origin and at infinity of the potential∫0uft,sds. Their method is essentially variational and is based on a regularization of the action functional associated with the curvature problem.Due to their wide range of applications, there has been tremendous interest in studying some properties of the nonlinear discreteϕc-Laplacian equations, including periodic solutions [1], boundary value problems [5, 6], and homoclinic orbits [7, 8]. For instance, using the mountain pass lemma, Zhou and Su [5] obtained some sufficient conditions for the first time on the existence of solutions of the 2nth-order ϕc-Laplacian difference equation with the boundary value conditions, and by the critical point theory, Zhou and Ling [6] obtained some sufficient conditions on the existence of infinitely many positive solutions of the boundary value problems for a second-order ϕc-Laplacian difference equation. The importance of nonlinear difference equations is described in [9, 10], with applications involving statistics, computing, electrical circuit analysis, dynamical systems, and economics. And they appear naturally as discrete analogue of differential equations which model various diverse phenomena.We assume thatfn,0,0,0=0 for each n∈ℤ, then un=0 is a solution of (1), which is called the trivial solution. As usual, we say that a solution u=un of (1) is homoclinic (to 0) if limn⟶∞un=0. In addition, if un≠0, then u is called a nontrivial homoclinic solution.In recent years, the existence and multiplicity of homoclinic solutions for various discrete systems have been investigated by many researchers. For some recent work, the reader can refer to [11–22] and the reference therein. For example, by using the symmetric mountain pass theorem, Chen and Tang [12] studied the existence of infinitely many homoclinic orbits for the fourth-order difference systems containing both advance and retardation:(4)Δ4un−2+qnun=fn,un+1,un,un−1,n∈ℤ.Kong [20] applied the critical point theory to study the existence of at least three homoclinic solutions for a p-Laplacian difference equation of higher order with both advance and retardation:(5)−1nΔnak−nϕpΔnuk−n+bkϕpuk=λfk,uk+1,uk,uk−1,k∈ℤ.However, to the best of our knowledge, there is no work on the existence of homoclinic solutions forϕc-Laplacian difference equations containing both advance and retardation which have important background and significance in the field of cybernetics and biological mathematics [23, 24]. Motivated by the reasons aforementioned, in this paper, we will establish some new sufficient conditions for the existence of homoclinic solutions to (1) in a very general case of the nonlinear functional.On the other hand, we notice that the nonlinear termf is only either superlinear or asymptotically linear at ∞ which plays an important role in the existence of homoclinic solutions when the similar arguments were considered in many references [25–28]. But in this paper, the nonlinearities can be mixed superlinear with asymptotically linear at ∞, see Remark 1 for details. In fact, our conditions on the potential are rather relaxed, and some existing results in the literature are improved (see Remark 2). Moreover, we give an example to illustrate the main result.Now, we establish the main result.Theorem 1.
Assume that the following conditions hold:T1 there exists a functional Fn,v1,v2∈C1ℤ×ℝ2,ℝ with Fn+T,v1,v2=Fn,v1,v2 and it satisfies(6)∂Fn−1,v2,v3∂v2+∂Fn,v1,v2∂v2=fn,v1,v2,v3.T2(7)limsupv12+v22⟶0Fn,v1,v2v12+v22=0,limsupv12+v22+v32⟶0fn,v1,v2,v3v12+v22+v32=0.T3 there exists a real sequence an such that(8)liminfv12+v22⟶∞Fn,v1,v2v12+v22=an≤∞.T4 ∂Fn,v1,v2/∂v1v1+∂Fn,v1,v2/∂v2v2−2Fn,v1,v2>0 for n∈ℤ,v1,v2∈ℝ0.T5 ∂Fn,v1,v2/∂v1v1+∂Fn,v1,v2/∂v2v2−2Fn,v1,v2⟶+∞ as v12+v22⟶+∞.
If2an>ωn for each n∈ℤ, then (1) has at least a nontrivial solution u in l2.Remark 1.
The conditionT3 shows that the nonlinear terms mix superlinear nonlinearities with asymptotically linear ones at ∞.Remark 2.
Ifm=1 and fn,un+1,un,un−1=gn,un, Theorem 1 can reduce to Theorem 2 when ϕ-Laplacian is ϕc-Laplacian in [8]. And our sufficient conditions are based on the limit superior and limit inferior, which are more applicable.
This paper is organized as follows: In Section2, we establish the variational framework associated with (1) and recall some related fundamental results for convenience. In Section 3, some lemmas are proved, and then we complete the proof of our results by the mountain pass lemma. Finally, we illustrate our main result with an example in Section 4.


## 2. Preliminaries

This section is to establish the corresponding variational framework for (1) and cite some basic conclusions for the coming discussion.Let(9)lp=u=un:n∈ℤ,un∈ℝ,ulp=∑n∈ℤunp1/p<∞,then, the corresponding inner product denoted by ⋅,⋅ in l2 and the corresponding norm in l2 is denoted by ⋅=⋅l2.LetS be the set of all two-sided sequences, that is,(10)S=u=un∣un∈ℝ,n∈ℤ.Then,S is a vector space with au+bv=aun+bvn for u,v∈Sanda,b∈ℝ. For any fixed positive integer k, we define the subspace Ek of S as(11)Ek=u=un∈S∣un+2kT=un,n∈ℤ.Obviously,Ek is isomorphic to ℝ2kT, and hence Ek can be equipped with the inner product ⋅,⋅k and norm ⋅k as(12)u,vk=∑n=−kTkT−1unvn,u,v∈Ek,uk=∑n=−kTkT−1un21/2,u∈Ek,respectively. We also define a norm ⋅k∞ in Ek by(13)uk∞=maxun:−kT≤n≤kT−1,u∈Ek.Consider the functionalJk in Ek defined by(14)Jku=∑n=−kTkT−1ΦcΔmun+12ωnun2−Fn,un+1,un,and the Fréchet derivative is given by(15)Jk′u,v=∑n=−kTkT−1ϕcΔmunΔmvn+ωnunvn−fn,un+1,un,un−1vn,u,v∈Ek.Equation (15) implies that (1) is the corresponding Euler–Lagrange equation for Jk. Therefore, we have reduced the problem of finding a nontrivial solution of (1) to that of seeking a nonzero critical point of the functional Jk.It is easy to see that the critical points ofJk in Ek are exactly 2kT-periodic solutions of system (1).LetH be a Hilbert space, and C1H,ℝ denote the set of functionals that are Fréchet differentiable and their Fréchet derivatives are continuous on H. We conclude this section with some known results.Definition 1.
LetJ∈C1H,ℝ. A sequence xj⊂H is called a Cerami sequence (C sequence for short) for J, if Jxj⟶c for some c∈ℝ and 1+xjJ′xj⟶0 as j⟶∞. We say, J satisfies the Cerami condition (C condition for short) if any C sequence for J possesses a convergent subsequence.
LetBr be the open ball in H with radius r and center 0, and let ∂Br denote its boundary.Lemma 1.
(mountain pass lemma [29]). If J∈C1H,ℝ and satisfies the following conditions: there exist e∈H\0 and r∈0,e such that maxJ0,Je<infu=rJu. Then, there exists a C sequence un for the mountain pass level c which is defined by(16)c=infh∈Γmaxs∈0,1Jhs,where(17)Γ=h∈C0,1,H:h0=0,h1=e.


## 3. Proof of Main Result

In order to prove Theorem 1, we need several lemmas.Let(18)ω∗=minn∈ℤωn.Lemma 2.
Under the assumptions of Theorem 1, the functionalJk satisfies the C condition.Proof 1.
Letuj⊂Ek be a C sequence for Jk. We need to show that uj has a convergent subsequence. Since Ek is finite dimensional, it suffices to show that ujk is bounded. Since Jkuj⟶cand1+ujkJk′uj⟶0, then there exists M>2c+1 such that Jkuj≤M−1/2 and 1+ujkJk′uj<1 for j∈ℕ. So, we have ujkJk′uj≤1+ujkJk′uj<1 for j∈ℕ. Then, by (14), (15), and(19)2Φcu=21+u2−1≥u1+u2u=ϕcuu≥0,foru∈ℝ,we have(20)∑n=−kTkT−1un+1j+∂Fn,un+1j,unj∂un+1jun+1j−2Fn,un+1j,unj≤2Jkuj−Jk′uj,uj≤2Jkuj+ujkJk′uj≤M.
FromT4 and T5, there exists δ>0 such that(21)∂Fn,u1,u2∂u1u1+∂Fn,u1,u2∂u2u2−2Fn,u1,u2>M,forn∈ℤandu12+u22>δ2,then, (20) implies that unj≤δ for n∈ℤ, that is,(22)ujk∞≤δ.
SinceEk is finite dimensional, ⋅k and ⋅k∞ are equivalent. Then, (22) implies that ujk is bounded. The proof is completed.Lemma 3.
Under the assumptions of Theorem 1, there existsn0∈ℕ such that Jk has at least a nonzero critical point uk in Ek for each k≥n0.Proof 2.
We first show thatJk satisfies the conditions of Lemma 1. From T2, there exists r>0 such that(23)Fn,u1,u2≤18ω∗u12+u22,foru12+u22≤r2.
Then, foru∈Ek with uk≤r,(24)Jku≥12∑n=−kTkT−1ωnun2−∑n=−kTkT−1Fn,un+1,un≥12∑n=−kTkT−1ωnun2−∑n=−kTkT−118ω∗un+12+un2≥14ω∗uk2.
Takinga=1/4ω∗r2, then Jk∂Br≥a>0.
Since2an>ωn, there exists d>0 such that(25)an−ωn2>d,forn∈ℤ.
Letε∈0,1 satisfy(26)2ε−d<0.
Since−Δ2 is a bounded semipositive self-adjoint linear operator in l2 and 0∈σ−Δ2, there exists e=en∈l2 with ∑n=−∞+∞en2=1 such that ∑n=−∞+∞Δmen2<ε. Let n0 be large enough such that(27)∑n=−n0Tn0T−1Δmen2<ε,12≤∑n=−n0Tn0T−1en2≤1.
Fork≥n0, define ek∈Ek by(28)enk=en,−n0T≤n≤n0T−1,0,−kT≤n≤−n0T−1orn0T≤n≤kT−1.
By (T3), there exists μ0>maxr,1, such that(29)Fn,μen+1,μen≥an−εμ2en+12+en2,for−n0T≤n≤n0T−1,μ≥μ0.
Then, forμ≥μ0,(30)Jkμek=∑n=−kTkT−1ΦcμΔmenk+ωn2μenk2−Fn,μen+1k,μenk≤∑n=−kTkT−1μΔmenk+ωn2μenk2+ε−anμ2en+1k2+enk2≤εμ+ωn2μ2+ε−anμ2≤2ε−dμ2.
Thus,(31)Jkμ0ek≤2ε−dμ02<0.
It can easily be seen thatJk0=0 for each k∈ℤ. Then, we have r∈0,μ0ekk and(32)maxJk0,Jkμ0ek=0<a≤infuk=rJku.
Now that we have verified all assumptions of Lemma 1, and then we knowJk possesses a C sequence ujk for the mountain pass level αk≥a with(33)αk=infh∈Γkmaxs∈0,1Jkhs,where(34)Γk=h∈C0,1,Ek:h0=0,h1=μ0ek.
According to Lemma 2,ujk has a convergent subsequence ujmk such that ujmk⟶uk as jm⟶+∞ for some uk∈Ek. Since Jk∈C1Ek,ℝ, we have(35)Jkujmk⟶Jkuk,1+ujmkkJ′ujmk⟶1+ukkJ′uk,as jm⟶+∞. By the uniqueness of the limit, we obtain that uk is a critical point of Jk corresponding to αk. Moreover, uk is nonzero as αk≥a>0.Lemma 4.
There existp,q>0 such that(36)p≤ukk∞≤q,holds for every critical point uk obtained by Lemma 3, of Jk in Ek with k≥n0, where n0 is defined in Lemma 3.Proof 3.
Fork≥n0 and s∈0,1, we define hk∈Γk as hks=sμ0ek; similarly to the derivation of [5], we can find(37)Jkuk≤maxs∈0,1Jksμ0ek≤maxs∈0,1∑n=−n0Tn0T−1Δmsμ0en+ωn2sμ0en2−Fn,sμ0en+1,sμ0en≤maxs∈0,12m2n0T∑n=−n0Tn0T−1sμ0en21/2+∑n=−n0Tn0T−1ωn2sμ0en2−Fn,sμ0en+1,sμ0en.
Let(38)M0=maxs∈0,12m2n0T∑n=−n0Tn0T−1sμ0en21/2+∑n=−n0Tn0T−1ωn2sμ0en2−Fn,sμ0en+1,sμ0en,then,(39)Jkuk≤M0.
Obviously,M0>0 is independent of k.
Sinceuk is a critical point of Jk, by (14), (15), and (39), we have(40)∑n=−kTkT−1∂Fn,un+1k,unk∂un+1kun+1k+∂Fn,un+1k,unk∂unkunk−2Fn,un+1k,unk≤2M0.
FromT5, there exists q>0 such that(41)∂Fn,u1,u2∂u1u1+∂Fn,u1,u2∂u2u2−2Fn,u1,u2>2M0,forn∈ℤ,u12+u22>q2.
By combining (40), it implies that unk≤q for each n∈ℤ, that is,(42)ukk∞≤q.
From (15), we have(43)∑n=−kTkT−1ωnunk2≤∑n=−kTkT−1fn,un+1k,unk,un−1kunk.
ByT2, there exists p>0 such that(44)fn,v1,v2,v3≤36ω∗v12+v22+v32,forv12+v22+v32<3p2,which combining with (43) gives(45)∑n=−kTkT−1ωnunk2≤∑n=−kTkT−136ω∗un+1k2+unk2+un−1k2unk≤36ω∗∑n=−kTkT−1un+1k2+unk2+un−1k21/2ukk=12ω∗ukk2.
Arguing by a contradictionω∗=minn∈ℤωn, we have(46)ukk∞≥p.
The proof is complete.
With the help of the preceding lemmas we can prove Theorem 1.Proof of Theorem 1.
For everyk>n0, Jk has a critical point uk=unk∈Ek, which is obtained by Lemma 3. Moreover, there exists nk∈ℤ such that(47)p≤unkk≤q.
Note that(48)−1mΔmϕcΔmun−mk+ωnunk=fn,un+1k,unk,un−1k,n∈ℤ.
By the periodicity ofωn and fn,un+1,un,un−1, we see that un+Tk is also a solution of (48). Without the loss of generality, we may assume that 0≤nk≤T−1 in (47). Moreover, passing to a subsequence of uk, if necessary, we can also assume that nk=n∗ for k≥n0 and some integer n∗ such that 0≤n∗≤T−1. It follows from (47) that we can choose a subsequence, still denoted by uk, such that(49)unk⟶unask⟶∞,n∈ℤ.
Then,u=un is a nonzero sequence as (47) which implies un∗≥p. It remains to show that u=un∈l2, and it is a solution of (1).
Let(50)Ak=n∈ℤ:un+1k<p,unk<p,un−1k<p,−kT≤n≤kT−1,Bk=n∈ℤ:un+1k≥porunk≥porun−1k≥p,−kT≤n≤kT−1.
SinceFn,u1,u2 is continuously differentiable in the second and third variables and T-periodic in n, let(51)c1=max∂Fn−1,u2,u3∂u2u2+∂Fn,u1,u2∂u2u2:p2≤u12+u22+u32≤3q2,n∈ℤ,c2=min12∂Fn−1,u2,u3∂u2u2+∂Fn,u1,u2∂u2u2−Fn,u1,u2:p2≤u12+u22+u32≤3q2,n∈ℤ,it implies c1,c2>0. Thus, for n∈Bk,(52)∂Fn−1,unk,un−1k∂unkunk+∂Fn,un+1k,unk∂unkunk≤c1c212∂Fn−1,unk,un−1k∂unkunk+∂Fn,un+1k,unk∂unkunk−Fn,un+1k,unk.
Then, combining (40), (43), and (45) gives us(53)∑n=−kTkT−1ωnunk2≤∑n∈Akfn,un+1k,unk,un−1kunk+∑n∈Bkfn,un+1k,unk,un−1kunk≤12ω∗ukk2+∑n∈Bk∂Fn−1,unk,un−1k∂unkunk+∂Fn,un+1k,unk∂unkunk≤12ω∗ukk2+c1c2∑n∈Bk12∂Fn−1,unk,un−1k∂unkunk+∂Fn,un+1k,unk∂unkunk−Fn,un+1k,unk≤12ω∗ukk2+c1M0c2.
Thus, we have(54)ukk2≤2c1M0c2ω∗.
For eachϱ∈ℕ, let k>maxϱ,n0, and it follows from (54) that(55)∑n=−ϱϱunk2≤ukk2≤2c1M0c2ω∗.
It is clear that∑n=−ϱϱun2≤2c1M0/c2ω∗ as k⟶∞ and u=un∈l2 by the arbitrariness of ϱ.
Indeed, for eachn∈ℤ, letting k⟶∞ in (48) gives us(56)−1mΔmϕcΔmun−m+ωnun=fn,un+1,un,un−1,n∈ℤ,that is, u=un satisfies (1).
Consequently, we infer thatu=un is a nontrivial solution of (1). We have thus proved Theorem 1.


## 4. An Example

As an application of Theorem 1, we give an example to illustrate our main result.Example 1.
Consider the difference equation (1), with(57)fn,v1,v2,v3=θv22+cosnπTv12+v22θ/2−1+2+cosn−1πTv22+v32θ/2−1,where θ>2 and T is a given positive integer. Then,(58)Fn,v1,v2=2+cosnπTv12+v22θ/2,∂Fn−1,v2,v3∂v2+∂Fn,v1,v2∂v2=θv22+cosnπTv12+v22θ/2−1+2+cosn−1πTv22+v32θ/2−1.
We can see that(59)Fn,v1,v2v12+v22=2+cosnπTv12+v22θ/2−1,due to θ/2−1>0, it follows that(60)limv12+v22⟶0Fn,v1,v2v12+v22=0,limv12+v22⟶∞Fn,v1,v2v12+v22=∞.
And we know(61)fn,v1,v2,v3v12+v22+v32−1/2≤θv2v12+v22+v32−1/22+cosnπTv12+v22θ/2−1+2+cosn−1πTv22+v32θ/2−1≤6θv12+v22+v32θ/2−1.
Then, it implies that(62)limv12+v22+v32⟶0fn,v1,v2,v3v12+v22+v32=0.
It is easy to verify that all 1.1 are satisfied. Consequently, (1) has at least a nontrivial solution u in l2.


## 5. Conclusion

In this work, we have considered a higher order nonlinear difference equation containing both advance and retardation withϕc-Laplacian. To the best of our knowledge, this is the first time to discuss the existence of homoclinic solutions for such equations. Applying the variational method and mountain pass theorem with Cerami’s condition, we have derived some new sufficient conditions for the existence of homoclinic solutions to (1) in a very general case of the nonlinear functional. By setting up the variational framework with (1), we have reduced the problem of finding a nontrivial solution of (1) to that of seeking a nonzero critical point of the functional Jk. It is worth mentioning that the nonlinear term f can be mixed superlinear with asymptotically linear at ∞. In fact, our conditions on the potential are rather relaxed, and some existing results in the literature are improved. Finally, we have given an example to illustrate the main result.



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*Source: 3624842-2019-10-17.xml*