DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi 10.1155/2019/8052497 8052497 Research Article Infinitely Many Positive Solutions for a Coupled Discrete Boundary Value Problem Li Liuming 1 http://orcid.org/0000-0001-5114-1418 Zhou Zhan 1 2 Anderson Douglas R. 1 School of Mathematics and Information Sciences Guangzhou University Guangzhou Guangdong 510006 China gzhu.edu.cn 2 Center for Applied Mathematics Guangzhou University Guangzhou Guangdong 510006 China gzhu.edu.cn 2019 1422019 2019 12 12 2018 27 01 2019 1422019 2019 Copyright © 2019 Liuming Li 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.

In this paper, we obtain some results for the existence of infinitely many positive solutions for a coupled discrete boundary value problem. The approach is based on variational methods.

National Natural Science Foundation of China 11571084 Program for Changjiang Scholars and Innovative Research Team in University IRT-16R16
1. Introduction

Let Z and R denote the sets of integers and real numbers, respectively. For a,bZ, define [a,b]={a,a+1,,b}, when ab.

In this paper, we consider the following coupled discrete boundary value problem:(1)-ΔpkΔuik-1+qkuik=λFuik,u1k,,umk,ui0=uiN+1=0,for all k[1,N], NZ+, where 1im, and λ is a positive parameter, Δui(k)=ui(k+1)-ui(k) is the forward difference operator, p(k)>0 for all k[1,N+1], and q(k)0 for all k[1,N], F(k,·,,·) is a C1-function in Rm satisfying F(k,0,,0)=0 for every k[1,N], and Fui denotes the partial derivative of F(k,u1,u2,,um) with respect to ui for i=1,,m.

As we know, results for existence of solutions for difference equations have been widely studied because of their applications to various fields of applied sciences, like mechanical engineering, control systems, computer science, economics, artificial or biological neural networks, and many others. Many scholars have studied such problems and main tools are fixed point methods, Brouwer degree theory, and upper and lower solution techniques; see  and references therein. In recent years, variational methods have been employed to study difference equation and various results have been obtained. See, for instance, .

More recently, especially, in , by starting from the seminal papers [32, 33], many results for the existence and multiplicity of solutions for discrete boundary value problems have been obtained also by adopting variational methods.

However, these papers only deal with a single equation. For instance, in , by studying the Dirichlet discrete boundary value problem(2)-Δ2uk-1=λfkuk,k1,N,u0=uN+1=0,the author obtained the existence of two positive solutions of the problem through appropriate variational methods. In this paper, we consider system (1) with m difference equations by using Ricceri’s variational principle proposed in . In Theorem 4 and Remark 5, we prove the existence of an interval (λ1,λ2) such that, for each λ(λ1,λ2), problem (1) admits a sequence of positive solutions which is unbounded in X. To the best of our knowledge, this is the first time to deal with coupled discrete boundary value problems. This method has already been used for the continuous counterparts . The monographs  are related books of the critical point theory and difference equations.

The rest of the paper is organized as follows: Section 1 consists of some definitions and mathematical symbols. In Section 2, we emphasize that a strong maximum principle (Lemma 1) is presented so that if Fui(k,u1,,ui-1,0,ui+1,,um)0, for all k[1,N] and i=1,,m, our results guarantee the existence of infinitely many positive solutions (Remark 5 in Section 3). Section 3 contains a more precise version (Lemma 3) of Ricceri’s variational principle, the statements and proofs of the main results (Theorem 4 and Remark 5), two corollaries (Corollaries 6 and 7), and an example (Example 10).

Throughout this paper, we let S={u:[0,N+1]R:u(0)=u(N+1)=0} and X be the Cartesian product of m Banach spaces S,,S; i.e., (3)XS××Smendowed with the norm(4)u=u1,,umi=1mui21/2,where (5)vk=1N+1pkΔvk-12+k=1Nqkvk21/2,forvS,which is a norm in S.

Put (6)Aliminfy+k=1Nmaxt1,,tmKyFk,t1,,tmy2,where (7)Kyt1,,tmRm:i=1mtiy,for every yR+, and (8)Blimsupi=1mti2+t1,,tmR+mk=1NFk,t1,,tmi=1mti2,where R+m={(t1,,tm)Rm:ti0 for all 1im}.

2. Preliminaries

First, we establish a strong maximum principle.

Lemma 1.

Fix uS such that either u(k)>0 or(9)ifuk0then-ΔpkΔuk-1+qkuk0.Then either u>0 in [1,N] or u0.

Proof.

Let j[1,N] such that(10)uj=minuk:k1,N.If u(j)>0, then it is clear that u>0 in [1,N].

If u(j)0, then by (9), we have(11)-ΔpjΔuj-1-Δqjuj0,that is (12)pj+1ΔujpjΔuj-1.On the other hand, by the definition of u(j), we see that(13)Δuj-10,Δuj0.Thus, by (12), we obtain that u(j+1)=u(j)=u(j-1).

By similar arguments applied to u(j-1) and u(j+1) and continuing in this way, we have u(k)=u(N+1)=u(0)=0,k[1,N].

Remark 2.

Let F:[1,N]×RmR be such that Fti(k,t1,,ti-1,0,ti+1,,tm)0 for all k[1,N] and i=1,,m.

Put(14)Ftik,t1,,tm=Ftik,t1,,tm,ifti>0,Ftik,t1,,ti-1,0,ti+1,,tm,ifti0. Clearly, Fti(k,·) is continuous in Rm for each k[1,N]. Owing to Lemma 1, all solutions of problem(15)-ΔpkΔuik-1+qkuik=λFuik,u1k,,umk,ui0=uiN+1=0,are either zero or positive and hence are also solutions for problem (1). Hence we emphasize that when (15) admits nontrivial solutions, then problem (1) admits positive solutions, independently of the sign of Fui.

3. Main results

Let X be a reflexive real Banach space and let Iλ:XR be a function satisfying the following structure hypothesis:

(Λ)Iλ(u)Φ(u)-λΨ(u) for all uX, where Φ,Ψ:XR are two functions of class C1 on X with Φ coercive; i.e., limu+Φ(u)=+, and λ is a real positive parameter.

Provided that infXΦ<r, put(16)φrinfuΦ-1-,rsupvΦ-1-,rΨv-Ψur-Φu, and(17)γliminfr+φr. Clearly, γ0. When γ=0, in the sequel, we agree to read 1/γ as +.

For the readers’ convenience, we recall a more precise version of Theorem 2.1 of  (see also Theorem 2.5 of ) which is the main tool used to investigate problem (1).

Lemma 3.

Assume that condition (Λ) holds. If γ<+, then, for each λ(0,1/γ), the following alternative holds: either

(a1)Iλ possesses a global minimum, or

(a2)there is a sequence {un} of critical points (local minima) of Iλ such that (18)limn+Φun=+

Put (19)Pp1+pN+1,Qk=1Nqk,pminpk:k1,N+1.Our main result is the following theorem.

Theorem 4.

Assume that

(i) F is nonnegative in [1,N]×R+m

(ii) A<p/m(P+Q)(N+1)B, where A and B are given by (6) and (8), respectively

Then, for each λ(P+Q/2B,p/2m(N+1)A), system (1) admits an unbounded sequence of solutions.

Proof.

For each u=(u1(k),,um(k))X, put (20)Φu12u2,Ψuk=1NFk,u1k,,umk, and (21)IλuΦu-λΨu.Standard arguments show that IλC1(X,R) and that critical points of Iλ are exactly the solutions of problem (1). In fact, Φ,ΨC1(X,R); that is, Φ and Ψ are continuously Fréchet differentiable in X. Using the summation by parts formula and the fact that ui(0)=ui(N+1)=0 for any uX, we get (22)Φuv=limt0Φu+tv-Φut=k=1N+1i=1mpkΔuik-1Δvik-1+k=1Ni=1mqkuikvik=i=1mpN+1uiNviN+i=1mpkΔuik-1vik-11N+1-k=1Ni=1mΔpkΔuik-1vik+k=1Ni=1mqkuikvik=-k=1Ni=1mΔpkΔuik-1vik-qkuikvikand (23)Ψuv=limt0Ψu+tv-Ψut=k=1Ni=1mFuik,u1k,,umkvik,for any u,vX.

Moreover, we obtain (24)Iλuv=k=1Ni=1m-ΔpkΔuik-1+qkuik-λFuik,u1k,,umkvik,for all u,vX.

Now we verify that γ<+. Let {bn} be a real sequence such that limn+bn=+, and(25)limn+k=1Nmaxt1,,tmKbnFk,t1,,tmbn2=A.It follows from  that(26)supk1,NuikN+11/22p1/2ui,for 1im. And put rn2pbn2/m(N+1) for all nZ+. Hence a computation ensures that i=1m|ui(k)|bn whenever uΦ-1(-,rn).

Taking into account the fact that k=1NF(k,0,,0)=0, one has (27)φrn=infuΦ-1-,rnsupvΦ-1-,rnΨv-Ψurn-Φu=infi=1mui2/2<rnsupi=1mvi2/2<rnk=1NFk,v1k,,vmk-k=1NFk,u1k,,umkrn-i=1mui2/2supi=1mvi2/2<rnk=1NFk,v1k,,vmkrnmN+12pk=1Nmaxt1,,tmKbnFk,t1,,tmbn2. Therefore, since from assumption (ii) one has A<+, we obtain(28)γliminfn+φrnmN+12pA<+.

Now fix λ(P+Q/2B,p/2m(N+1)A). We claim that Iλ is unbounded from below. Let {ξi,n} be m positive real sequences such that limn+(i=1mξi,n2)=+, and(29)limn+k=1NFk,ξ1,n,,ξm,ni=1mξi,n2=B,for all nZ+.

For each nZ+, let ωi,n(k)ξi,n for all k[1,N],ωi,n(0)=ωi,n(N+1)=0.

Clearly, ωn=(ω1,n,,ωm,n)X, and (30)ωi,n2=k=1N+1pkΔωi,nk-12+k=1Nqkωi,nk2=p1ωi,n1-ωi,n02++pN+1ωi,nN+1-ωi,nN2+k=1Nqkξi,n2=p1+pN+1+k=1Nqkξi,n2=P+Qξi,n2. Therefore, we have (31)Φωn-λΨωn=i=1mωi,n22-λk=1NFk,ξ1,n,,ξm,n=P+Q2i=1mξi,n2-λk=1NFk,ξ1,n,,ξm,n, for all nZ+.

If B<+, let εP+Q/2λB,1. By (29) there exists Nε such that(32)k=1NFk,ξ1,n,,ξm,n>εBi=1mξi,n2, for all n>Nε. Moreover, (33)Φωn-λΨωn<P+Q2i=1mξi,n2-λεBi=1mξi,n2=P+Q2-λεBi=1mξi,n2, for all n>Nε. Taking into account the choice of ε, we have(34)limn+Φωn-λΨωn=-. If B=+, let us consider M>P+Q/2λ. By (29) there exists NM such that(35)k=1NFk,ξ1,n,,ξm,n>Mi=1mξi,n2, for all n>NM. Moreover, (36)Φωn-λΨωn<P+Q2i=1mξi,n2-λMi=1mξi,n2=P+Q2-λMi=1mξi,n2, for all n>NM. Taking into account the choice of M, in this case we also have(37)limn+Φωn-λΨωn=-. Due to Lemma 3, for each λP+Q/2B,p/2m(N+1)A, the functional Iλ admits an unbounded sequence of critical points, and the conclusion is proven.

Remark 5.

When Fti(k,t1,,ti-1,0,ti+1,,tm)0 for all k[1,N] and i=1,,m, owing to Remark 2, the solutions in the conclusion of Theorem 4 are positive.

It is interesting to list some special cases of the above results.

Corollary 6.

Assume that

(i)F is nonnegative in [1,N]×R+m

(ii)A<B/m(2+Q)(N+1)

Then, for each λ(2+Q/2B,1/2m(N+1)A), the system(38)-Δ2uik-1+qkuik=λFuik,u1k,,umk,k1,N,ui0=uiN+1=0, for 1im, admits an unbounded sequence of solutions.

Corollary 7.

Let F:RmR be a C1-function and assume that

(i)F is nonnegative in R+m

(ii)C<p/m(P+Q)(N+1)D,

where (39)Climinfy+maxt1,,tmKyFt1,,tmy2and (40)Dlimsupi=1mti2+t1,,tmR+mFt1,,tmi=1mti2.Then, for each λ(P+Q/2D,p/2m(N+1)C), the system(41)-ΔpkΔuik-1+qkuik=λFuiu1k,,umk,k1,N,ui0=uiN+1=0, for 1im, admits an unbounded sequence of solutions.

Remark 8.

When Fti(k,t1,,ti-1,0,ti+1,,tm)0 for all k[1,N] and i=1,,m, owing to Remark 2, the solutions in the conclusion of Corollary 6 are positive.

Remark 9.

When Fti(t1,,ti-1,0,ti+1,,tm)0 for i=1,,m, owing to Remark 2, the solutions in the conclusion of Corollary 7 are positive.

Now we give an example to illustrate our results.

Example 10.

Let m=2,p(k)=q(k)=1 and consider the increasing sequence of positive real numbers given by (42)an2nn+1/2, for every nZ+.

Define the function F:R2R as follows: If (t1,t2)B((an,an),1) for some positive integer n, then(43)Ft1,t2=an21-t1-an2-t2-an22; otherwise, (44)Ft1,t2=0, where B((an,an),1) denotes the open unit ball of center (an,an).

By the definition of F, we see that it is nonnegative in R+2 and F(0,0)=0. Further it is a simple matter to verify that FC1(R2). We will denote by Ft1(t1,t2) and Ft2(t1,t2), respectively, the partial derivative of F(t1,t2) with respect to t1 and t2. Now, for every nZ+, the restriction F(t1,t2)|B((an,an),1) attains its maximum in (an,an) and one has F(an,an)=(an)2. Obviously,(45)limsupt12+t22+t1,t2R+2Ft1,t2t12+t22=12,owing to the fact that(46)limn+Fan,an2an2=12. On the other hand, by setting yn=an+1-1 for every nZ+, one has (47)maxt1,t2KynFt1,t2=an2,nZ+.Then(48)limn+maxt1,t2KynFt1,t2yn2=0,and hence(49)liminfy+maxt1,t2KyFt1,t2y2=0.Finally(50)0=C<12N+2N+1D=14N+2N+1.

The previous observations and computations ensure that all the hypotheses of Corollary 7 are satisfied. Then, for each λ(N+2,+), the problem(51)-Δ2u1k-1+u1k=λFu1u1k,u2k,k1,N,-Δ2u2k-1+u2k=λFu2u1k,u2k,k1,N,u10=u1N+1=u20=u2N+1=0,admits an unbounded sequence of solutions.

Taking partial derivative to F(t1,t2) with respect to t1 gives(52)Ft1t1,t2=-4t1-anan21-t1-an2-t2-an2, if (t1,t2)B((an,an),1) for some positive integer n; otherwise, Ft1(t1,t2)=0.

It is easy to see that(53)Ft10,t2=0.In a similar way, we obtain(54)Ft2t1,0=0.Consequently, according to Remark 9, problem (51) admits an unbounded sequence of positive solutions.

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

Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 11571084) and the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT-16R16).

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