# Multiple Positive Solutions of Second-Order Nonlinear Difference Systems with Repulsive Singularities

Authors: Shengjun Li; Fang Zhang
Journal: Journal of Function Spaces
 (2021)

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## Abstract

We study the existence of positive solutions for second-order nonlinear repulsive singular difference systems with periodic boundary conditions. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on a fixed point theorem in cones and a nonlinear alternative principle of Leray-Schauder; the result is applicable to the case of a weak singularity as well as the case of a strong singularity. An example is given; some recent results in the literature are improved and generalized.


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## Body

## 1. Introduction

Difference systems are widely used in modeling real-life phenomena [1] and references therein. In this paper, we establish the existence positive solutions for the following nonlinear difference systems:(1)−∆pn−1∆xn−1+qnxn=fn,xn+en,with the boundary conditions:(2)x0=xT,p0∆x0=pT∆xT,where qn=diagq1n,q2n,⋯,qNn, pn=diagp1n,p2n,⋯,pNn, e=e1,e2,⋯,eNT, and f=f1,f2,⋯,fNT, N≥1. By a periodic solution, we mean a function x=x1,x2,⋯,xNT, solving (1) and (2) and such that xn≠0 for all n. We call boundary condition (2) the periodic boundary conditions which are important representatives of nonseparated boundary conditions. For convenience, we denote by ℤ, ℕ, and ℝ the sets of all integer numbers, natural numbers, and real numbers, respectively. For a,b∈ℤ, let ℤa=a,a+1,⋯,ℤa,b=a,a+1,⋯,b when a≤b. As usual, ∆ denotes the forward difference operator defined by(3)∆xn=xn+1−xn.In particular, the nonlinearityfx,xn:ℕ×ℝN\0⟶ℝN may have a repulsive singularity at x=0, from the physical explanation, which means that limx⟶0fin,x=+∞, uniformly in n∈ℤ1,T,i=1,2,⋯,N.Such repulsive singularity appears in many problems of applications such as the Brillouin focusing systems and nonlinear elasticity [2].System (1) can be viewed as a discretization of the following more general class of the Sturm singular second-order differential system:(4)−pty00+qty=ft,y+et.Such systems, even in casep≡1, where they are referred to as being of Klein-Gordon or Schrödinger type, appear in many scientific areas including fluid mechanics, gas dynamics, and quantum field theory. During the last few decades, the study of the existence of periodic solutions for singular differential equations has deserved the attention of many researchers [3–11]. Tracing back to 1987, Lazer and Solimini [5] investigated the singular model:(5)x′′+htxλ=gt,where λ>0,h,g are T-periodic functions and the mean value of g is negative, g¯<0. One of the common conditions to guarantee the existence of positive periodic solution is a so-called strong force condition (corresponds to the case λ≥1 in (5)) [11, 12]. For example, if we consider the system:(6)x¨+∇Vxt,x=ft,with Vt,x=1/xa; the strong force condition holds for α≥2. On the other hand, the existence of positive periodic solutions of the singular differential equations has been established with a weak force condition (corresponds to the case 0<λ<1 in (5)) [13–15].From then on, some classical tools have been used to study singular differential equations in the literature, including the degree theory [6, 11, 16], the method of the upper and lower solutions [8, 17], Schauder’s fixed point theorem [14], some fixed point theorems in cones for completely continuous operators [13, 18], and a nonlinear Leray-Schauder alternative principle [19].For the existence of periodic solutions of difference equations, some results have been obtained using the variational methods or the topological methods [1, 20–25]. For example, by minimax principle, Guo and Yu [23] discussed the existence of periodic solutions for difference equation:(7)−∆2xn−1+fn,xn=0,where the nonlinearity f is of superlinear or sublinear growth at infinity. Based on the method of the upper and lower solutions, Atici and Cabada [21] studied the existence of periodic solutions for difference equation:(8)−∆2xn−1+qnxn=fn,xn.In [26], Zhou and Liu investigated the following autonomous difference equations:(9)∆2xn−1+fxn=0.By Conley index theory, the author showed that the suitable assumptions of asymptotically linear nonlinear are enough to guarantee the existence of periodic solutions.In this paper, we establish two different existence results of positive periodic solutions for (1) and (2) and proof of the existence of positive solutions; the first one is based on an application of a nonlinear alternative of Leray-Schauder, which has been used by many authors [19, 27, 28] and references therein; the second one is based on a fixed point theorem in cones. Our main motivation is to obtain new existence results for positive periodic solutions of the system:(10)−∆p1n−1∆xn−1+q1nxn=x2+y2−a/2+μx2+y2β/2+e1n,−∆p2n−1∆yn−1+q2nyn=x2+y2−a/2+μx2+y2β/2+e2n.Here, we emphasize that the new results are applicable to the case of a strong singularity as well as the case of a weak singularity and thate does not need to be positive.The rest of this paper is organized as follows. In Section2, some preliminary results will be given. In Section 3, we will state and prove the main results. We will use the notation ℝ+N=x∈ℝN:xi≥0 for each i=1,2,⋯,N, for x=x1,⋯,xN,y=y1,⋯,yN, we write x≥y, if x−y=x1−y1,⋯,xN−YN∈ℝ+N. We say that a function φ:ℝN⟶ℝ is nondecreasing if φx≥φy for x,y∈ℝN with x≥y. For a given function p defined on ℤ0,T, we denote its maximum and minimum by p∗ and p∗, respectively.


## 2. Preliminaries

Fori=1,2,⋯,N, let us denote by φin and ψin the solutions of the corresponding homogeneous equations:(11)−∆pin−1∆xn−1+qinxn=0,n∈ℤ1,T,satisfying the initial conditions:(12)φi0=φi1=0;ψi0=0,pi0ψi1=1.Let(13)Di=φiT+piT∆ψiT−2.Throughout this paper, we always assume that(H) For eachi=1,2,⋯,N,pin>0,qin≥0,qi·6≢0,n∈Z1,TLemma 1 (see [29]).
If (H) holds, thenDi>0.Lemma 2 (see [29]).
Assume (H) holds. For the solution of the problem:(14)−∆pin−1∆xn−1+qinxn=ein,n∈ℤ1,T,x0=xT,pi0∆x0=piT∆xT,the formula(15)xn=∑s=1TGin,seis,holds, where(16)Gin,s=ψiTDiφinφis−piT∆φiTDiψinψis+piT∆ψiT−1Diφinψis−φiT−1Diφisψin,0≤s≤n≤T+1,piT∆ψiT−1Diφisψin−φiT−1Diφinψis,0≤n≤s≤T+1,is the Green’s function; the number Di is defined by (13).Lemma 3 (see [29]).
Under condition (H), the Green’s functionGin,s of the boundary value problem (14) is positive, i.e., Gin,s>0 for n,s∈Z0,T.
We denote(17)Ai=minn,s∈Z0,TGin,s,Bi=maxn,s∈Z0,TGin,s,σ=Ai/Bi.
Obviously,Bi>Ai>0 and 0<σi<1.Remark 4.
Ifpit=1,qit=α>0, then Green’s function Gin,s of the boundary value problem (14) has the form:(18)Gin,s=βn−s+βs−n+Nβ−β−1βn−1,0≤s≤n≤T+1,βs−n+βn−s+Nβ−β−1βn−1,0≤n≤s≤T+1,where β=α+2+αα+2/2. If n is even, a direct calculation shows that(19)Ai=2βT/2β−β−1βT−1,Bi=1+βTβ−β−1βT−1,σi=2βT/21+βT<1.


## 3. Main Results

In this section, we state and prove the new existence results for (1). In order to prove our main results, the following nonlinear alternative of Leray-Schauder is needed, which can be found in [30].Lemma 5.
AssumeΩ is a relatively compact subset of a convex set E in a normed space X. Let A:Ω¯⟶E be a compact map with 0∈Ω. Then, one of the following two conclusions holds:(i)
T has at least one fixed point in Ω(ii)
There existu∈∂Ω and 0<λ<1 such that u=λAu
Let(20)X1=x:ℤ0,T+1⟶ℝx0=xT,p0∆x0=pT∆xT.
Then,X1 is a Banach space with the norm(21)x=maxn∈Z1,Txn.
We take(22)X=X1×X1×⋯×X1Ncopies,with the norm(23)x=maxx1,x2,⋯,xN.
Define(24)γin=∑s=1TGin,seis,which corresponds to the unique solution of (14), and the operator A:X⟶X by Ax=A1x,A2x,⋯,ANxT, where(25)Aixn=∑s=1TGin,sfis,xs+γs,i=1,2,⋯,N.Now, we present the first existence result of the positive solution to problem (1).Theorem 6.
Suppose that condition (H) holds andγ∗≥0. Furthermore, we assume that
(H1) For each constant L>0, there exists a function φLn>0 for all n∈ℤ1,T such that each component fi of f satisfies fin,x≥∅Ln for all n,x∈ℤ1,T×0,L
(H2) For each component fi of f, there exist nonnegative functions gix, hix, and kin such that(26)0≤fin,x≤gix+hixkinforalln,x∈ℤ1,T×ℝ+N0,and gix> 0 is nonincreasing and hix/gix is nondecreasing in x
(H3) There exists a positive number r such that σr+γ∗>0 and(27)rgiγ∗,⋯,γ∗,σir+γ∗,γ∗,⋯,γ∗1+hir+γ∗,⋯,r+γ∗/gir+γ∗,⋯,r+γ∗>Ki∗,for all i=1,2,⋯,N. Here,(28)Kin=∑s=1TGin,skis,σ=mini=1,2,⋯,Nσi,γ∗=mini,nγn,γ∗=maxi,nγn.Then, (1) and (2) has at least one positive periodic solution x with xn>γn for all n∈Z0,T and 0<x−γ<r.Proof.
We first show that(29)−∆pn−1∆xn−1+qnxn=fn,xn+γn,together with (2) has a positive solution x satisfying xn+γn for n∈Z0,T and 0<x<r. If this is true, it is easy to see that un=xn+γn will be a positive solution of (1) and (2) with 0<u−γ<r since(30)−∆pn−1∆un−1+qnun=−∆pn−1∆xn−1+γn−1+qnxn+γn=fn,xn+γn+en=fn,un+en.
Since (H3) holds, let J0=j0,j0+1,⋯, we can choose j0∈1,2,⋯ such that 1/J0≤σr+γ∗ and(31)giγ∗,⋯,γ∗,σir+γ∗,γ∗,⋯,γ∗1+hir+γ∗,⋯,r+γ∗gir+γ∗,⋯,r+γ∗Ki∗+1j0<r,for all i=1,2,⋯,N.
Fixj∈J0. Consider the family of systems(32)−∆pn−1∆xn−1+qnxn=λfjn,xn+γn+qnj,n∈ℤ1,N,where λ∈0,1 and for each i=1,2,⋯,N,(33)fijn,x=fin,x,ifx≥1j,fin,x1,⋯,xi−1,1j,xi+1,⋯,xN,ifx≤1j.
Problem (29) and (2) are equivalent to the following fixed point problem:(34)xin=λ∑s=1TGin,sfijs,xs+γs+1j=λAijxn+1j,for each i=1,2,⋯,N, here, we used the fact(35)∑s=1TGin,sqis≡1,i=1,2,⋯,N.
We claim that any fixed pointx of (34) for any λ∈0,1 must satisfy x≠r. Otherwise, assume that x is a fixed point of (34) for some λ∈0,1 such that x=r. Without loss of generality, we assume that xl=r for some l=1,2,⋯,N.
Thus, we have(36)xln−1j=λ∑s=1NGln,sfljn,xs+γsds≥λAl∑s=1Tfljn,xs+γsds=σlBlλ∑s=1Tfljn,xs+γsds≥σlmaxnλ∑s=1TGln,sfljn,xs+γsds=σlxl−1j.
Hence, for alln∈Z1,T, we have(37)xln≥σlxl−1j+1j≥σlxl−1j+1j≥σlr.
Therefore,(38)xln+γln≥σlr+γ∗>1j.
Using (34), we have from condition (H2), for all n∈Z1,T,(39)xln=λ∑s=1TGln,sfljs,xs+γs+1j=λ∑s=1TGln,sfls,xs+γs+1j≤∑s=1TGln,sfls,xs+γs+1j≤∑s=1TGln,sklsglxs+γs1+h1xs+γsg1xs+γs≤glγ∗,⋯,γ∗,σlr+γ∗,γ∗,⋯,γ∗1+hlr+γ∗,⋯,r+γ∗glr+γ∗,⋯,r+γ∗Kl∗+1j0.
Therefore,(40)r=xl≤glγ∗,⋯,γ∗,σlr+γ∗,γ∗,⋯,γ∗1+hlr+γ∗,⋯,r+γ∗glr+γ∗,⋯,r+γ∗Kl∗+1j0.
This is a contradiction to the choice ofj0, and the claim is proved.
From this claim, the nonlinear alternative of Leray-Schauder guarantees that(41)xn=Ajxn+1j,has a fixed point, denoted by xjn, in Br=x∈X:x<r, i.e.,(42)−∆pn−1∆xn−1+qnxn=fjn,xn+γn+qnj,has a periodic solution xj with xj<r.
Next, we claim that these solutionsxjn+γn have a uniform positive lower bound, that is, there exists a constant δ>0, independent of j∈J0, such that(43)mini,nxijn+γin≥δ,for all j∈J0. To see this, we know from (H1) that there exists a continuous function ϕr+γ∗n such that each component fi of f satisfies fin,x≥ϕr+γ∗n for all n,x∈ℤ1,T×0,r+γ∗. Now, let xr+γ∗n be the unique solution to(44)−∆pn−1∆xn−1+qnxn=Φn,with (2), here Φn=ϕr+γ∗n,⋯,ϕr+γ∗nT. Then, we have(45)xir+γ∗n+γin=∑s=1TGin,sϕr+γ∗s+γin≥Φ∗+γ∗>0for each i=1,⋯,N, here(46)Φ∗=infnΦin,Φin=∑s=1TGin,sϕr+γ∗s.
Next, we show that (43) holds for δ=Φ∗+γ∗>0. To see this, for each i=1,⋯,N, since xijn+γin≤r+γ∗ and xijn+γ∗≥1/j, we have(47)xijn+γin=∑s=1TGin,sfijs,xjs+γs+γin+1j≥∑s=1TGin,s∅r+γ∗+γin≥Φ∗+γ∗≔δ.
The factxn<r and (43) show that for each i=1,2,⋯,N, xijj∈J0 is a bounded family on ℤ1,T. Moreover, we have(48)xij0=xijT,pi0∆xij0=piT∆xijT,which implies that(49)xijT+1=pi0piT∆xij0+xijT,j∈J0.
Thus, the Arzela–Ascoli theorem guarantees thatxijj∈J0 has a subsequence, xijkjk∈J0,k∈N converging uniformly on ℤ0,T+1 to a function xi. Let x=x1,⋯,xN, xn satisfies δ≤xin+γin<r+γ∗ for all n∈ℤ1,T and i=1,⋯,N. Moreover, xijk satisfies the integral equation:(50)xijkn=∑s=1TGin,sfis,xjks+γs+1jk,i=1,⋯,N.
Lettingk⟶∞, we arrive at(51)xin=∑s=1TGin,sfis,xs+γs,i=1,2,⋯,N,here, we have used the fact that fn,x is with respect to n,x with n∈ℤ1,T and x>0 satisfying δ≤x≤r+γ∗. Therefore, x is a positive periodic solution of (1) and satisfies 0<∣x∣≤rCorollary 7.
Assume that (H) holds,α>0, β≥0. Then, for each e1,e2 with γ∗≥0, we have(i)
ifβ<1, then (10) has at least one positive periodic solution for each μ>0(ii)
ifβ≥1, then (10) has at least one positive periodic solution for each 0<μ<μ1, where μ1 is some positive constantProof.
We will apply Theorem6. To this end, assumption (H1) is fulfilled by ∅L=2L−α. If we take(52)g1x,y=g2x,y=x2+y2−α/2,h1x,y=h2x,y=μx2+y2β/2,and k1n=k2n=1, then (H2) is satisfied.
Let(53)ω1n=∑s=1TG1n,s,ω2n=∑s=1TG2n,s.
Then, the existence condition (H3) becomes(54)μ<rσir+γ∗2+γ∗22/α−ωi∗2α+β/2r+γ∗α+β,i=1,2,for some r>0. So, (10) has at least one positive periodic solution for(55)0<μ<μ1≔mini=1,2supr>0rσir+γ∗2+γ∗22/α−ωi∗2α+β/2r+γ∗α+β,i=1,2.
Note thatμ1=∞ if β<1 and μ1<∞ if β≥1. We have (i) and (ii).In more general, we can obtain the following result.Corollary 8.
Assume that (H) holds and there exist functionsa,a^,b,b^ and α,β>0 such that, for i=1,2,⋯,N,(56)αnxα+bnxβ≤fin,x≤α^nxα+μb^nxβ.
Then, for eache with γ∗≥0, we have(i)
ifβ<1, then (10) has at least one positive periodic solution for each μ>0(ii)
ifβ≥1, then (10) has at least one positive periodic solution for each 0<μ<μ2, where μ2 is some positive constantBy using a fixed point theorem for compact maps on conical shells [31], we established the second positive periodic solution for (1). Recall that a compact operator means an operator which transforms every bounded set into a relatively compact set and introducing the definition of a cone.Definition 9.
LetX be a Banach space and let K be a closed, nonempty subset of X. K is a cone if(i)
αu+βv∈K for all u,v∈K and all α,β>0(ii)
u,−u∈K implies u=0Lemma 10 (see [31]).
LetX be a Banach space and K a cone in X. Assume Ω1,Ω2 are open subsets of X with 0∈Ω1,Ω¯1∈Ω2. Let(57)Φ:K∩Ω¯2\Ω1⟶Kbe a continuous and completely continuous operator such that(i)
(58)Φx≤xforx∈K∩∂Ω1(ii)
There existψ∈K\0 such that x≠Φx+λψ for x∈K∩∂Ω2 and λ>0
Then,F has a fixed point in K∩Ω¯2\Ω1. The same conclusion remains valid if (i) holds on K∩∂Ω2, and (ii) holds on K∩∂Ω1.Define(59)K=x=x1,⋯,xN∈X:min0≤n≤Txin≥σixiforalln∈Z0,T,i=1,⋯,N.Then, one can readily verify thatK is a cone in X.Theorem 11.
Suppose conditions (H), (H1)–(H3) hold. Furthermore, assume that the following two conditions are satisfied:
(H4) There exist continuous, nonnegative functions g1x, h1x and k1n such that(60)fin,x≥gi1x+hi1xki1nforalln,x∈0,T×ℝ+n0,where gi1x>0 is nonincreasing and hi1x/gi1x is nondecreasing in x
(H5) There exists R>r such that(61)σRgi1R+γ∗,⋯,R+γ∗1+hi1σ1R+γ∗,⋯,σnR+γ∗/gi1σ1R+γ∗,⋯,σnR+γ∗≤Ki∗.
Then, problems (1) and (2) have another one positive periodic solution x~ with r<x~−γ≤R.Proof.
LetAx=A1x,⋯,ANxT, Aix is given by (25), then, it is easy to verify that A is well defined and maps X into K. Moreover, A is continuous and completely continuous, and let K be a cone in X defined by (59). Define the(62)Ω1=x∈X:∣x∣<r,Ω2=x∈X:x<R.
As in the proof of Theorem 3.1, we only need to show that (29) has a positive periodic solution u∈X with un+γn>0 and r<u≤R. We claim that(i)
(63)Ax≤xforx∈K∩∂Ω1(ii)
There existψ∈K\0 such that x≠Ax+λψ for x∈K∩∂Ω2 and λ>0
We start with (i). In fact, ifx∈K∩∂Ω1, then x=r and σir+γ∗≤xin+γn≤r+γ∗ for all t∈0,T. Fix i∈1,2,⋯,N, thus, we have(64)Aixt=∑s=1TGin,sfis,xs+γs≤∑s=1TGin,skisgixs+γs1+hixs+γsgixs+γs≤gixs+γs1+hixs+γsgixs+γs∑s=1TGin,skis≤giγ∗,⋯,γ∗,σir+γ∗,γ∗,⋯,γ∗1+hir+γ∗,⋯,r+γ∗gir+γ∗,⋯,r+γ∗Ki∗<r=x.
Therefore,Aix≤x for each i=1,2,...,N. This implies that (i) holds.
Next, we show that (ii) holds. Letψ≡1,⋯,1, then ψ∈K\0. Suppose that there exists x∈K∩∂Ω2 and λ>0 such that x=Ax+λψ. Since x∈K∩∂Ω2, then σiR+γ∗≤xin+γn≤R+γ∗ for all n∈Z0,T. As a result, it follows from (H4) and (H5) that, for all n∈Z0,T,(65)xin=Aixn+λ=∑s=1TGin,sfis,xs+γsds+λ≥∑s=1TGin,skilsg1lxs+γs1+hilxs+γsgilxs+γs+λ≥gilxs+γs1+hilxs+γsgilxs+γs∑s=1TGin,skils+λ≥gilR+γ∗,⋯,R+γ∗1+hilσ1R+γ∗,⋯,σNR+γ∗gilσ1R+γ∗,⋯,σNR+γ∗Ki∗+λ≥σR+λ.
Hence,min0≤n≤Txin>σR; this is a contradiction and we prove the claim.
Now, Lemma 3.7 guarantees thatA has at least one fixed point x∈K∩Ω¯2\Ω1 with r≤x≤R.Let us consider again the example (10) in Corollary 7 for the superlinear case.Corollary 12.
Assume in (10) that pi,qii=1,2 satisfy (H), for each e1,e2 with γ∗≥0,β>1. Then, for each μ with 0<μ<μ1, where μ1 is given as in Corollary 7, problem (10) has at least two different positive solutions. To verify (H4), one may take(66)g11x,y=g21x,y=12x2+y2−α/2,h21x,y=12x2+y2β/2,and k1n1=k21n=1. If β>1, then the existence condition (H5) becomes(67)μ≥2α+2/2R+γ∗ασR−2ωi∗σ1R+γ∗2+σ2R+γ∗2α+β/2ωi∗,i=1,2.Sinceβ > 1, the right-hand side goes to 0 as R⟶+∞. Thus, for any given 0<μ<μ1, it is always possible to find such R≫r that (67) is satisfied. Thus, (10) has an additional positive periodic solution x~.Remark 13.
We emphasize that our results are applicable to the case of a strong singularity as well as the case of a weak singularity since we only needα>0. Moreover, e does not need to be positive. In fact, using the assumption that the Green function is positive, one may readily verify that γ∗≥0 is equivalent to the ∑i=1Nein≥0,i=1,2,⋯,N.
Let us consider the 2-dimensional system(68)−Δ2xn−1+xn=αnxα+μbnxβ+e1n,−Δ2yn−1+yn=αnxα+μbnxβ+e2n,with(69)ein=ndi−n,di∈ℝ,i=1,2.Example 1.
Assume thatα>0, β>1, an, and bn are positive functions, ein,i=1,2 are given by (69) with(70)d1+2d2≥5.
Then, the results in Corollary12 hold.Proof.
We only need showγ∗≥0, which is equivalent to(71)∑i=12ein≥0,
Sinced1+2d2≥5, a direct computation show that(72)∑i=12ein≥0∑i=12=ndi−n≥0.


## 4. Conclusions

In this paper, we study the periodic problem for nonlinear difference systems with a singularity of repulsive type in the case ofγ∗≥0. The proofs of main results are based on a nonlinear alternative principle of Leray-Schauder and a fixed point theorem in cones. It is interesting that the singularity f is applicable to the case of a weak singularity as well as the case of a strong singularity. In the next research, we will continue to study the periodic problem to the difference systems like (10) where f may have attractive singularity at x=0, and whether the condition γ∗≥0 can be removed.



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