MPEMathematical Problems in Engineering1563-51471024-123XHindawi10.1155/2020/83427358342735Research ArticleMultiple Solutions for Boundary Value Problems of p-Laplacian Difference Equations Containing Both Advance and RetardationWangZhenguo12https://orcid.org/0000-0001-5114-1418ZhouZhan1HuangChuangxia1School of Mathematics and Information ScienceGuangzhou UniversityGuangzhou 510006Chinagzhu.edu.cn2Department of MathematicsLüliang UniversityLüliang 033000Chinallhc.edu.cn202010820202020050520200106202010820202020Copyright © 2020 Zhenguo Wang and Zhan Zhou.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.

This paper concerns the existence of solutions for the Dirichlet boundary value problems of p-Laplacian difference equations containing both advance and retardation depending on a parameter λ. Under some suitable assumptions, infinitely many solutions are obtained when λ lies in a given open interval. The approach is based on the critical point theory.

National Natural Science Foundation of China11971126Program for Changjiang Scholars and Innovative ResearchIRT_16R16Scientific and Technologial Innovation Programs of Higher Education Institutions in Shanxi2019L0955
1. Introduction

Let and be the sets of integers and real numbers, respectively. For a,b, a,b denotes the discrete interval a,a+1,,b if ab.

Assume that N is a positive integer. In this paper, we consider the following boundary value problem of difference equation containing both advance and retardation:(1)ΔϕpΔuk1+2qkϕpuk=λfk,uk+1,uk,uk1,k1,N,u0=uN+1=0,where λ is a positive real parameter, the forward difference operator is defined as Δuk=uk+1uk. For every k1,N, qk0, fk,:3 is a continuous function, ϕps is the p-Laplacian operator, that is, ϕps=sp2s, and 1<p<+.

As it is to know, difference equations have been widely used in various research fields such as computer science, economics, network, and control system. Relevant examples and mathematical models can be found in . By means of critical point theory, many researchers devote themselves to the study of difference equations and achieve many excellent results, for example, the results on boundary value problems , periodic solutions , and homoclinic solutions  had been obtained.

Difference equations containing both advance and retardation have many applications in physical and biological phenomena . Yu et al.  obtained the existence of nontrivial homoclinic orbits for the following second-order difference equation containing both advance and retardation:(2)Lukωuk=fk,uk+T,uk,ukT,k.

The operator L is a second-order difference operator given by(3)Luk=akuk+1+ak1uk1+bkuk,where ak and bk are real valued for each k and ω.

Mei and Zhou  considered the existence of the periodic and subharmonic solutions of a 2nth order p-Laplacian difference equation containing both advances and retardations:(4)ΔnrknφpΔnukn=1nfk,uk+τ,,uk+1,uk,uk1,,ukτ,k.

There are more about the results of the difference equations containing both advance and retardation which can be seen in [13, 19, 22, 24].

Here, we are interested in using critical point theory to investigate infinitely many solutions for (1) containing both advance and retardation.

In fact, there are some papers which studied the existence of infinitely many solutions for the boundary value problems of difference equations. Bonanno and Candito  in 2009 proved the existence of infinitely many solutions of the following discrete boundary value problem:(5)ΔϕpΔuk1+qkϕpuk=λfk,uk,k1,N,u0=uN+1=0.

Recently, Zhou and Ling  studied the existence of infinitely many positive solutions for the following boundary value problem of the second order nonlinear difference equation with ϕc-Laplacian:(6)ΔϕcΔuk1=λfk,uk,k1,N,u0=uN+1=0.

However, to the best of our knowledge, no similar results are obtained for problem (1) containing both advance and retardation. The main difficulty is caused by the advance and retardation. In this paper, we obtain some sufficient conditions to guarantee the existence of infinitely many solutions of (1). In fact, under some assumptions, we prove the existence of infinitely many solutions of equation (1) for each λ2+2Q/pB,2/pN+1p1A in Theorem 1. Moreover, Theorem 2 guarantees the existence of infinity positive solutions for (1) by applying a strong maximum principle. Finally, we show that (1) possesses a sequence of distinct solutions which converges to zero for each λ2+2Q/pB,2p11+q¯/pN+1p1A in Theorem 3.

This paper is organized as follows. In Section 2, some definitions and preliminaries on difference equations are collected. In Section 3, our main results are established. Finally, two examples are given to illustrate our main results.

2. Preliminaries

Let E be a reflexive real Banach space and Iλ:E be a function satisfying the following structure hypothesis:H1 Assume that λ is a real positive parameter. Iλ:=ΦuλΨu, uE, where Φ,ΨC1E,, Φ is coercive, that is, limuΦu=+.

Provided that infEΦ<r, write(7)φr=infvΦ1,rsupvΦ1,rΨvΨurΦu,γ=liminfr+φr,δ=liminfrinfEΦ+φr.

Obviously, γ0 and δ0. When γ=0 or δ=0, we agree to read 1/γ or 1/δ as +.

Lemma 1 (see [<xref ref-type="bibr" rid="B34">34</xref>]).

Assume that condition H1 holds, and one has

For every r>infEΦ and every λ0,1/φr, the restriction of functional Iλ=ΦuλΨu to uΦ1,r admits a global minimum, which is a critical point (local minimum) of Iλ in E.

If γ<+, then for each λ0,1/γ, the following alternative holds:

b1Iλ possesses a global minimum

b2 There is a sequence un of critical points (local minimum) of Iλ such that limn+Φun=+.

If δ<+, then for each λ0,1/δ, the following alternative holds:

c1 There is a global minimum of Φ which is a local minimum of Iλ

c2 There is a sequence un of pairwise distinct critical points (local minimum) of Iλ, with limn+Φun=infEΦ, which weakly converges to a global minimum of Φ.

Consider the N-dimensional Banach space:(8)S=u:0,N+1 such that u0=uN+1=0,endowed with the norm(9)u=k=1N+1Δuk1p+2qkukp1/p.

We define another two norms on S as follows:(10)up=k=1N+1Δuk1p1/p,u=maxk1,Nuk.

According to Lemma 2.2 in , we have the following inequality:(11)u=maxk1,NukN+1p1/p2up,for every uS.

Lemma 2.

For every uS, one has(12)uN+1p1/p21+q¯1/pu,where q¯=minqk,k1,N.

Proof.

Making use of (11), we obtain(13)up+q¯up12N+12p1upp+k=1N+12qkukpN+1p12pup,then uN+1p1/p/21+q¯1/pu.

For every uS, put(14)Φu=upp,Ψu=k=1NFk,uk+1,uk,Iλu=ΦuλΨu,where Fk, satisfies the following conditions:

H2Fk,C12,, Fk,0,0=0, F0,.,.=0k1,N

H3Fk1,x2,x3/x2+Fk,x1,x2/x2=fk,x1,x2,x3, for k,x1,x2,x31,N×3

Direct computation ensures that Iλ is a functional of class C1 on S with(15)Iλuk=ΔϕpΔuk1+2qkϕpukλfk,uk+1,uk,uk1,k1,N.

Then, u is a critical point of Iλ on S if and only if(16)ΔϕpΔuk1+2qkϕpuk=λfk,uk+1,uk,uk1,k1,N.

That is, the function Iλ is just the variational framework of (1). For the reader’s convenience, we recall a consequence of strong maximum principle . The strong maximum principle is used to obtain positive solutions to (1), that is, uk>0 for every k1,N.

Lemma 3 (see [<xref ref-type="bibr" rid="B8">8</xref>]).

Fix uS such that(17)either uk>0 orΔϕpΔuk1+qkϕpuk0,k1,N.

Then, either u>0 or u0.

Remark 1.

Assume fk,x1,x2,x3:3 is continuous for each k1,N and fk,x1,0,x30 for all k,x1,x31,N×2. Put(18)fk,x1,x2,x3=fk,x1,0,x3,if x20,fk,x1,x2,x3,if x2>0,then fk,C3, for each k1,N.

Consider the following boundary value problem:(19)ΔϕpΔuk1+2qkϕpuk=λfk,uk+1,uk,uk1,k1,N,u0=uN+1=0.

From Lemma 3, all solutions of (19) are either zero or positive; hence, they are also solutions for (1). When (19) possesses nontrivial solutions, (1) possesses positive solutions, independently of the sign of f.

3. Main Results

Put(20)Q=k=1Nqk,κ=11+QN+1p1,B=limsupξ+k=1N1Fk,ξ,ξ+FN,0,ξξp.

Now, we consider the suitable oscillating behavior of Fk,x1,x2 when x1p+x2p goes to +. We have the following theorem.

Theorem 1.

Assume that H2 and H3 are satisfied and there exist two real sequences an and cn, with limn+cn=+, such that(21)anp<2p21+q¯cnp1+QN+1p1,for all n,(22)A=liminfn+k=1Nmaxx1p+x2pcnpFk,x1,x2k=1N1Fk,an,anFN,0,an2p21+q¯cnp1+QN+1p1anp<κB.

Then, problem (1) admits an unbounded sequence of solutions for each λ2+2Q/pB,2/pN+1p1A.

Proof.

Our aim is to apply Lemma 1(b) to prove our conclusion. Fix λ2+2Q/pB,2/pN+1p1A, which clearly H1 holds. Our conclusion needs to provide that γ<+. Put rn=2p11+q¯cnp/pN+1p1 for all nZ. Owing to (12), if uprn1/p, then uk+1p+ukpcnp for every k1,N, and we obtain(23)φrn=infuprn1/psupuprn1/pk=1NFk,uk+1,ukk=1NFk,uk+1,ukrnup/pinfuprn1/pk=1Nmaxuk+1p+ukpcnpFk,uk+1,ukk=1NFk,uk+1,uk2p11+q¯cnp/pN+1p1up/p.

Let gnk=an for every k1,N, and gn0=gnN+1=0. Clearly, gnS and gnp=21+Qanp. From (21), we obtain gnprn1/p, and one has(24)φrnpN+1p12x1p+x2pcnpNFk,x1,x2k=1N1Fk,an,anFN,0,an2p21+q¯cnp1+QN+1p1anp.

Hence, γliminfnφrnpN+1p1A/2<+.

Now, we verify that Iλ is unbounded from below. By (20), let dn be a positive real sequence with limndn=+ and(25)B=limn+k=1N1Fk,dn,dn+FN,0,dndnp.

First, we assume that B=+, and then we can fix M>2+2Q/pλ, and there exists νM such that(26)k=1N1Fk,dn,dn+FN,0,dnMdnp,n>νM.

We take a sequence sn in S such that snk=dn for every k1,N, sn0=snN+1=0, and one has(27)Iλsn=ΦsnλΨsn=snppλk=1N1Fk,dn,dn+FN,0,dn<2+2QpλMdnp,for all n>νM. That is, limn+Iλsn=.

Next, we assume that B<+. Since λ>2+2Q/pB, from (25), we can fix ε>0 such that ε<B2+2Q/pλ, arguing as before, there is a νε such that(28)k=1N1Fk,dn,dn+FN,0,dnBεdnp,n>νε.

By choosing sn in S as above, we obtain that(29)Iλsn<2+2QpλBεdnp,for all n>νε. Hence, one has limn+Iλsn=.

We have verified all assumptions of Lemma 1(b); then, there is a sequence un of critical points (local minima) of Iλ such that limn+Φun=+.

According to Theorem 1, it is easy to obtain the following corollary.

Corollary 1.

Let H2 and H3 be satisfied, and assume that(30)A=liminfξ+k=1Nmaxx1p+x2pξpFk,x1,x2ξp<2p21+q¯κB.

Then, problem (1) admits an unbounded sequence of solutions for each λ2+2Q/pB,2p11+q¯/pN+1p1A.

Proof.

Let cn be a real sequence with limncn=+ such that(31)A=limn+k=1Nmaxx1p+x2pcnpFk,x1,x2cnp.

We take an=0 for all n in (21); combining with H2, (30) and (31), we can apply Theorem 1 to reach the conclusion.

From the argument of Remark 1, we have the following theorem and corollary.

Theorem 2.

If fk,x1,0,x30 for every k,x1,x31,N×2 and the hypotheses of Theorem 1 hold. Then, problem (1) admits an unbounded sequence of positive solutions for each λ2+2Q/pB,2/pN+1p1A.

Corollary 2.

If fk,x1,0,x30 for every k,x1,x31,N×2 and the hypotheses of Corollary 1 are satisfied, then problem (1) admits an unbounded sequence of positive solutions for each λ2+2Q/pB,2p11+q¯/pN+1p1A.

Next, we consider the oscillating behavior of Fk,x1,x2 when x1p+x2p goes to 0. We obtain the following theorem.

Theorem 3.

Assume that H2 and H3 are satisfied and(32)A<2p21+q¯κB,where A=liminfξ0+k=1Nmaxx1p+x2pξpFk,x1,x2/ξp and B=limsupξ0+k=1N1Fk,ξ,ξ+FN,0,ξ/ξp. Then, problem (1) possesses a sequence of pairwise distinct solutions, which converges to zero for each λ2+2Q/pB,2p11+q¯/pN+1p1A.

Proof.

We will check the part c of Lemma 1. Fix λ2+2Q/pB,2p11+q¯/pN+1p1A, let cn be a positive real sequence such that limncn=0 and(33)A=limn+k=1Nmaxx1p+x2pcnpFk,x1,x2cnp.

As before, we let rn=2p11+q¯cnp/pN+1p1 for each nZ. In view of (12), note that uprn1/p implies uk+1p+ukpcnp for every k1,N, by the definition of φ, we have(34)φrn=infuprn1/psupuprn1/pk=1NFk,uk+1,ukk=1NFk,uk+1,ukrnup/ppN+1p12p11+q¯k=1Nmaxx1p+x2pcnpFk,x1,x2cnp,then δliminfnφrnpN+1p1A/2p11+q¯<+.

Clearly, 0 is a global minimum of Φ in S, and Iλ0=0 by H2.

Moreover, we can verify that 0 is not a local minimum of Iλ. Given a positive real sequence dn with limn+dn=0 such that(35)B=limn+k=1N1Fk,dn,dn+FN,0,dndnp,if B=+, fix M>2+2Q/pλ, and there exists νM such that(36)k=1N1Fk,dn,dn+FN,0,dn>Mdnp,n>νM.

Let vnS be a sequence satisfying vnk=dn for every k1,N, vn0=vnN+1=0. Since vn=2+2Q1/pdn0 as n, we have(37)Iλvn=ΦvnλΨvn=vnppλk=1N1Fk,dn,dn+FN,0,dn<2+2QpλMdnp<0,for all n>νM. Thus, 0 is not a local minimum of Iλ.

If B<+, since λ>2+2Q/pB, we can also find a positive real sequence dn with limn+dn=0 such that (35) holds. Fix 0<ε<B2+2Q/pλ, and there exists νε, such that(38)k=1N1Fk,dn,dn+FN,0,dn>Bεdnp,n>νε.

Arguing as before, there exist a sequence vnS, such that(39)Iλvn<2+2QpλBεdnp<0,for all n>νε. Obviously, 0 is not a local minimum of Iλ, and c2 of Lemma 1(c) is true.

A similar result to Theorem 2 is obtained as follows.

Theorem 4.

If fk,x1,0,x30 for every k,x1,x31,N×2 and the hypotheses of Theorem 3 hold, then problem (1) admits a sequence of pairwise distinct positive solutions, which converges to zero for each λ2+2Q/pB,2p11+q¯/pN+1p1A.

4. Examples

Finally, we give two examples to illustrate our results.

Example 1.

Consider the boundary value problem (1) with(40)fk,x1,x2,x3=1+ε+cosεln1+x2p+x3pεsinεln1+x2p+x3ppx2p2x2+1+ε+cosεln1+x1p+x2pεsinεln1+x1p+x2ppx2p2x2,for k1,N. Put(41)Fk,x1,x2=1+x1p+x2p1+ε+cosεln1+|x1|p+|x2|p2ε.

Obviously, Fk,x1,x2 satisfies conditions H2 and H3. It is easy to see that(42)maxx1p+x2pξpFk,x1,x2=1+ξp1+ε+cosεln1+ξp2ε.

Clearly,(43)A=liminfξ+k=1Nmaxx1p+x2pξpFk,x1,x2ξp=liminfξ+k=1N1+ξp1+ε+cosεln1+ξp2εξp=Nliminfξ+1+ξp1+ε+cosεln1+ξp2εξp=εN,and(44)B=limsupξ+k=1N1Fk,ξ,ξ+FN,0,ξξp=limsupξ+2N11+ε+2N1cosεln1+2ξp+cosεln1+ξp2N1ε+22+ε.

Let ε be sufficiently small, then 2+2Q/pB2+2Q/2+εp2p11+q¯/pN+1p1εN. Hence, for each λ2+2Q/pB,2p11+q¯/pN+1p1εN, (1) admits an unbounded sequence of solutions by Corollary 1. Moreover, we have fk,x1,0,x3=0 for all k,x1,x31,N×2; according to Corollary 2, (1) admits an unbounded sequence of positive solutions.

Example 2.

Consider the boundary value problem (1) with(45)fk,x1,x2,x3=sinεlnx2p+x3p+εcosεlnx2p+x3p+sinεlnx1p+x2p+εcosεlnx1p+x2p+2+2εpx2p2x2,if x12+x22+x320 and fk,0,0,0=0 for k1,N. Let(46)Fk,x1,x2=x1p+x2psinεlnx1p+x2p+1+ε,if x12+x220,0,if x12+x22=0,then H3 holds. Clearly, H2 holds and(47)maxx1p+x2pξpFk,x1,x2=ξpsinεlnξp+1+ε.

By computation, we obtain(48)A=liminfξ0+k=1Nmaxx1p+x2pξpFk,x1,x2ξp=liminfξ0+k=1Nξpsinεlnξp+1+εξp=Nliminfξ0+sinεlnξp+1+ε=εN,B=limsupξ0+k=1N1Fk,ξ,ξ+FN,0,ξξp=limsupξ0+2N11+ε+2N1sinεln2ξp+sinεlnξp2N1ε+22+ε.

Obviously, we have 2+2Q/pB2+2Q/2+εp2p11+q¯/pN+1p1εN, when ε be sufficiently small. On the contrary, fk,x1,0,x3=0 for all k,x1,x31,N×2.

The computations ensure that all the assumptions of Theorem 4 are satisfied. Then, for each λ2+2Q/pB,2p11+q¯/pN+1p1εN, (1) admits a sequence of pairwise distinct positive solutions, which converges to zero.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11971126), Program for Changjiang Scholars and Innovative Research Team in University (Grant no. IRT_16R16), and Scientific and Technologial Innovation Programs of Higher Education Institutions in Shanxi (Grant no. 2019L0955).