# Existence and Uniqueness of Positive Solitons for a Second-Order Difference Equation

Authors: Xinfu Li; Guang Zhang; Ying Wang
Journal: Discrete Dynamics in Nature and Society
 (2014)

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## Abstract

A discrete logistic steady-state equation with both positive and negative birth rate of population will be considered. By using sub- and upper-solution method, the existence of bounded positive solutions and the existence and uniqueness of positive solitons will be established. To this end, the Dirichlet eigenvalue problem with positive and negative coefficients is considered, and a general sub- and upper-solution theorem is also obtained.


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## Body

## 1. Introduction

In this paper, we consider the second-order difference equation(1)-Δ2xi-1=λpixi-xi1+α,i∈Z,
where Z is a set of all integers, α>0 and λ>0 are constants, Δxi=xi+1-xi, Δ2xi-1=ΔΔxi-1, and pii∈Z will be specified later.This equation is associated with the famous discrete logistic equation with diffusion:(2)xit+1-xit=dΔ2xi-1t+pixit-xit1+α
for i∈Z and t∈Z+={0,1,2,…}, where the parameter d>0 corresponds to the rate at which the population diffuses, and the unknown function x corresponds to the density of a population. The term -x1+α in the equation corresponds to the fact that the population is self-limiting and the p corresponds to the birth rate of population if the self-limitation is ignored. At points where pi>0 (<0) the population, ignoring self-limitation, has positive (negative) birth rate. Hence, we assume throughout this paper that p takes on both positive and negative values on Z; we further assume that the sequence pii∈Z is bounded and there exists an integer i0∈Z such that pi0>0.If we writeλ=1/d, we see that the steady-state solutions of (2) must satisfy (1). Since the sequence x represents a population density, it is nonnegative. Such solutions correspond to possible steady-state distributions of population. So our purpose in this paper is to establish the existence of positive solutions or the existence and uniqueness of positive solitons for (1). A discrete soliton is a spatially localized solution of (1) and decays to 0 at infinity; that is,
(3)lim⁡|i|→∞xi=0.Recently, the existence of nontrivial discrete solitons for the general equation(4)-Δ2xi-1=fi,xi,i∈Z,
has been extensively established by a number of authors [1–12]. Among the methods used are the principle of anticontinuity [1–3], variational methods [4–10], center manifold reduction [11], Nehari manifold approach [12], and so forth. On the other hand, a soliton of (4) is also its homoclinic orbit or homoclinic solution. By using the variational methods, the existence of homoclinic orbits or homoclinic solutions has also been extensively discussed by a number of authors; see [4–10, 13–16]. However, as far as we know, there are few papers concerned with the existence of positive solitons.Equation (2) is a discrete analogue of the well-known logistic equation of the form
(5)ut=Duxx+pxu-u1+α,x∈R,t∈R+.
Indeed, by means of standard finite difference methods, we set up a grid in the x, t plane with grid spacings Δx and Δt and then replace the second derivative uxx with a central difference and ut with a forward difference. By writing xi=iΔx, tn=nΔt, pi=pxi, and uin≈uxi,tn, a finite difference scheme for (5) is obtained:
(6)uin+1-uinΔtW=DΔx2ui+1n-2uin+ui-1n+piuin-uin1+α
or
(7)uin+1-uinW=DΔtΔx2ui+1n-2uin+ui-1n+Δtpiuin-Δtuin1+α,
which has the steady-state equation:
(8)-Δ2xi-1=Δx2Dpixi-xi1+α,i∈Z.
We note that the steady-state equation of (5) is
(9)Du′′+pxu-u1+α=0
or
(10)-u′′=μpxu-u1+α,
where μ=1/D. When α=1, (10) is reduced to
(11)-u′′=μpxu-u.
In [17], the authors considered the existence and uniqueness of positive solitons of (11) by using sub- and upper-solution method. The present work is motivated by [17].To obtain a positive subsolution of (1), we need a positive eigenvalue and its corresponding positive eigenfunction of the eigenvalue problem
(12)-Δ2xi-1=λpixi,i∈a,a+1,…,b,xa-1=0=xb+1.
Note that p takes on both positive and negative values; thus, it is indefinite in a very strong sense. The corresponding problem for ordinary differential equation had been considered in [18] by using variational method. However, such problem is new for the above discrete problem; see [19–21]. In Section 2, we will consider the above discrete eigenvalue problem by using the matrix and vector method.As far as we know, for the discrete problem (1) there is no general sub- and upper-solution theorem on the set Z. Thus, a general sub- and upper-solution theorem will be firstly obtained in Section 3 and such theorem will be used for (1). On the other hand, our solutions are classical; however, [17] cannot insure this fact because some points of their subsolution are not derivative.Our results will give some theory groundwork for the numerical calculation of (11). Note that λ=(Δx)2/D≠μ when (Δx)2≠1. This will lead to different results between (1) and (11).


## 2. Preliminaries

For anya,b∈Z with a<b, we consider the eigenvalue problem of the form
(13)-Δ2xi-1=λpixi,i∈a,b,xa-1=0=xb+1,
where a,b=a,a+1,…,b, pii=ab is a real sequence, and there exists i0∈a,b such that pi0>0.Denote(14)x=xaxa+1⋮xb,A=2-10⋯0-12-1⋯0⋯⋯⋯0⋯2-10⋯0-12b-a+1×b-a+1,(15)P=pa0⋯00pa+1⋯0⋯⋯0⋯0pbb-a+1×b-a+1.
Then problem (13) can be rewritten by matrix and vector in the form
(16)Ax=λPx.LetH be a set of all real sequences xii=a-1b+1 with xa-1=0=xb+1. For any u,v∈H, the inner product is defined as u,v=∑i=ab‍uivi and the norm · is defined as u=∑i=ab‍ui2.For anyv∈H, we define
(17)Qλv=vTAv-λ∑i=ab‍pivi2.
Consider the Rayleigh quotient
(18)Kv=vTAv∑i=ab‍pivi2
and let
(19)λ1=inf⁡Kv:v∈H,∑i=ab‍pivi2>0.For such definedλ1, we have the following important result.Lemma 1.
λ
1 is a positive eigenvalue of the problem (13) or (16). Moreover λ1 is simple and the corresponding eigenfunction φ can be chosen such that φi>0 for all i∈a,b.Proof.
First, we showλ1>0. Clearly
(20)Qλ1v=vTAv-λ1∑i=ab‍pivi2≥0
for all v∈H. Moreover, by the spectral theorem, (Av,v)≥γ1(v,v), where
(21)γ1=4sin2π2b-a+2
is the first eigenvalue of the eigenvalue problem
(22)-Δ2xi-1=γxi,i∈a,b,xa-1=0=xb+1.
Hence, if v∈H and ∑i=ab‍pivi2>0, then
(23)Kv=vTAv∑i=ab‍pivi2≥γ1v,v∑i=ab‍pivi2≥γ1max⁡pi,i=a,a+1,…,b>0.
Hence λ1>0.
Second, consider the linear eigenvalue problem(24)-Δ2ui-1-λ1piui=μui,i∈a,b,ua-1=0=ub+1,
and define Sui=-Δ2ui-1-λ1piui. It is easy to see that λ1 is an eigenvalue for (13) with corresponding eigenfunction φ if and only if 0 is an eigenvalue of S and so of (24) with corresponding eigenfunction φ. The minimal eigenvalue α1 of S is given by
(25)α1=inf⁡vTAv-λ1∑i=ab‍pivi2:v∈H,v=1=inf⁡Qλ1v:v∈H,v=1.
Note that Qλ1v≥0 for all v∈H; thus, we have α1≥0. Because of how we defined λ1, there exists a sequence vn∈H and ∑‍pi(vi(n))2>0 such that
(26)lim⁡n→∞vnTAvn∑i=ab‍pivin2=λ1
which implies that
(27)lim⁡n→∞Qλ1vn=0.
Thus, we have α1≤0. In this case, we know that α1=0 is the minimal eigenvalue of (24). By Lemma A.1 in the Appendix, α1 is simple and the corresponding eigenfunction can be chosen to be positive on H. Thus, the statement in this lemma is right.Letλ>λ1; consider the eigenvalue problem of the form
(28)-Δ2ui-1-λpiui=μui,i∈a,b,ua-1=0=ub+1.
We have the following result.Lemma 2.
Ifλ>λ1, then the minimal eigenvalue μ1 of (28) is negative and the corresponding eigenfunction φ(1) can be chosen so that φi(1)>0 for all i∈a,b.Proof.
Letφ be the eigenvector obtained in Lemma 1 and ∥φ∥=1; then we have
(29)φTAφ-λ∑i=ab‍piφi2=λλ1φT(A-λ1P)φ-λ-λ1λ1φTAφ=-λ-λ1λ1φTAφ<0.
Consequently,
(30)μ1=min⁡ψTAψ-λ∑i=ab‍piψi2∣ψ=1<0.
In view of Lemma A.1 in the Appendix, the eigenvector corresponding to μ1 can be chosen to be positive. This completes the proof.


## 3. Main Results

First of all, we introduce the definitions of the subsolution and upper-solution and give a general sub- and upper-solution theorem.For anya,b∈Z with a<b, consider the Dirichlet boundary value problem of the form
(31)-Δ2xi-1=fi,xifori∈a,b,xa-1=0=xb+1.
We will denote by (31)n the problem (31) when a=-n and b=n.Definition 3.
A sequencex¯=x¯ii=a-1b+1 is said to be an upper-solution of (31) if
(32)-Δ2x¯i-1≥fi,x¯ifori∈a,b,x¯i≥0fori=a-1,b+1.
A sequence x_=x_ii=-∞+ is said to be a subsolution of (31) if
(33)-Δ2x_i-1≤fi,x_ifori∈a,b,x_i≤0fori=a-1,b+1.Theorem 4.
Suppose thatx_,x¯:Z→R are functions with x_≤x¯ (where x_≤x¯ is defined by x_i≤x¯i for i∈Z) and x_|n, x¯|n are, respectively, a subsolution and an upper-solution of (31)n for all large n. Assume that f(i,z) is continuous with z∈[α,β] and there exists a constant k>0 such that
(34)fi,s2-fi,s1≥-ks2-s1
for all i∈Z and s2>s1 with s1,s2∈α,β (where α=inf⁡i∈Zx_i and β=sup⁡i∈Zx¯i). Then problem (4) has a solution x such that x_≤x≤x¯.Proof.
Sincex_ and x¯ provide sub- and upper-solutions for (31)n, there exists a solution φ of (31)n such that x_i≤φi≤x¯i for all |i|≤n.
In fact, for any sequenceu=uii=-nn, clearly, the problem
(35)-Δ2wi-1+kwi=fi,ui+kuifori∈[-n,n],w-n-1=0=wn+1,
has a unique solution w. This defines a mapping T:u→w. We claim that T is increasing on [x_,x¯]. In fact, for any sequences {u~i}i=-nn, {v~i}i=-nn∈[x_,x¯] with v~≤u~, we have
(36)ΓTu~-Tv~=f(i,u~i)+ku~i-f(i,v~i)-kv~i≥-k(u~i-v~i)+k(u~i-v~i)≥0,
where Γ is an operator defined as (Γu)i=-Δ2ui-1+kui for i∈[-n,n]. Since (Tu~)-n-1-(Tv~)-n-1≥0, (Tu~)n+1-(Tv~)n+1≥0, by the strong monotonicity of Γ, we obtain Tu~≥Tv~; see [21].
Letu(m)=Tu(m-1), u(0)=x_; v(m)=Tv(m-1), v(0)=x¯. In the following, we claim that
(37)x_=u0≤u1≤u2≤···≤v2≤v1≤v0=x¯.
First of all, by using (35) and the definition of subsolution, we see
(38)-Δ2ui-10-ui-11+kui0-ui1WW=-Δ2ui-10+kui0-fi,ui0+kui0WW=-Δ2x_i-1-fi,x_i≤0fori∈[-n,n],
and u-n-1(0)-u-n-11≤0, un+10-un+11≤0. This implies that u0-u1≤0 by the strong monotonicity of Γ. A similar argument gives v1≤v0. The monotonicity of Γ and T gives the rest. So there exist u and v such that
(39)lim⁡m→∞um=u,lim⁡m→∞vm=v.
By the definition of u and v, we see that u and v satisfy (31)n and x_≤u≤v≤x¯. Let x(n) denote such solution corresponding to (31)n.
Then standard a priori estimates and a diagonalization argument show that there exists a subsequence of{x(n)} which converges to solution x of (4) on every bounded subset of Z. Moreover, since x_≤x(n)≤x¯ for all n, it follows that x_≤x≤x¯ on Z. The proof is complete.In the following, we use the sub- and upper-solution theorem to establish the existence of positive solutions or the existence and uniqueness of positive solitons for (1).For problem (1), we have assumed that there exists i0∈Z such that pi0>0. For any n>|i0|, consider the eigenvalue problem (13) when a=-n and b=n and denote the corresponding positive eigenvalue λ1 defined in (19) by λ1n. Then
(40)λ1∞∶=lim⁡n→∞λ1n≥0.Theorem 5.
For anyλ>λ1∞, (1) has a positive solution, where λ1∞ is defined in (40).Proof.
For anyλ>λ1∞, clearly, there exists a positive integer n1≥|i0| such that i0∈-n1,n1 and λ1∞<λ1n1<λ. In view of Lemma 2, the eigenvalue problem
(41)-Δ2xi-1-λpixi=μxi,i∈-n1,n1,x-n1-1=0=xn1+1,
has a negative eigenvalue μ1n1 and its corresponding eigenfunction φn1 with ∥φn1∥=1 can be chosen so that φin1>0 for all i∈-n1,n1. For any ɛ>0, we define
(42)x_i=ɛφin1,i∈-n1,n1,0,i∈Z∖-n1,n1.
At this time, we have
(43)-Δ2ɛφi-1n1-λpiɛφin1+λɛφin11+αWW=μ1n1ɛφin1+λɛφin11+α≤0
for i∈-n1,n1 and sufficiently small ϵ. When i=n1+1,
(44)-Δ2x_i-1=-x_n1+2+2x_n1+1-x_n1=-ɛφn1n1<0
and -Δ2x_i-1≡0 for i>n1+1. Similarly, we can also prove that -Δ2x_i-1≤0 for i≤-n1-1. Thus, the sequence x_ defined in (42) is a subsolution of (1).
Note thatα>0 and the sequence pi is bounded. Thus, the sufficiently large positive constant M≥x_ is a supersolution of (1).
Forfi,xi=λ(pixi-xi1+α), we have
(45)fi,s2-fi,s1WW=λpis2-s21+α-λpis1-s11+αWW=λpi-1+αξαs2-s1,
where ξ∈s1,s2. Note that the sequences pi, x_i, and M are bounded; thus, (1) satisfies condition (34). By Theorem 4, problem (1) has a solution u such that 0≤x_≤u≤M. We claim that u>0. Suppose there exists i1∈Z such that ui1=0; then we obtain ui1-1+ui1+1=0 since -Δ2ui1-1=λ(pi1ui1-ui11+α). Thus we obtain ui1-1=ui1+1=ui1=0. So u≡0, which contradicts u≢0. The proof is complete.Theorem 6.
Assume thatλ>λ1∞ and there exists a constant C>6/λ and n0>i0 such that
(46)pi≤Cii+1α,α∈(0,1],-Ci(i+1),α>1
for |i|>n0. Then (1) has a positive soliton and there exists a constant M>0 and n2≥n0 such that
(47)xi≤Mii+1fori≥n2.Proof.
Letx_ be the subsolution of (1) obtained in Theorem 5 and
(48)ϕi=Mii+1,i≠0,±1,±2,
where M>0 is a constant determined later. By simple calculation, we have
(49)Δϕi-1=Mii+1-Mii-1=-2Mii2-1,-Δ2ϕi-1=2Mii+1i+2-2Mii2-1=-2Mii+1·3i-1i+2,-Δ2ϕi-1-λpiϕi+λϕi1+αW=λMi(i+1)-6λ(i-1)(i+2)+Mi(i+1)α-pi.
By the assumption on p, there exist n2≥n0 and M1>0 such that
(50)-6λ(i-1)(i+2)+Mi(i+1)α-pi≥0
for M≥M1 and |i|≥n2.
For anyM≥M1, define
(51)ψi=Mi(i+1),i≥n2,M,i<n2.
We will show that ψ is upper-solution of (1) by appropriate choice of M. In fact,(a)
for|i|>n2, we have -Δ2ψi-1-λpiψi+λψi1+α≥0 by the choice of M;(b)
for|i|<n2-1, we have
(52)-Δ2ψi-1-λpiψi+λψi1+α=-λpiM+λM1+α≥0by choosingM>M1 large enough since α>0 and p is bounded;(c)
similarly, fori=n2,n2-1,-n2,-n2+1, one can show that -Δ2ψi-1-λpiψi+λψi1+α≥0 for M>M1 large enough.
In view of (a)–(c), we construct upper-solutionψ of (1) with ψ≥x_. Using the sub- and upper-solution Theorem 4, we complete the proof.Theorem 7.
Assume that there exists a positive integerm>i0 such that pi≤0 for i≥m. Let x be a bounded positive solution of (1); then
(53)lim⁡|i|→∞xi=0.Proof.
At this time, we have(54)-Δ2xi-1=λpixi-xi1+α≤0fori≥m.
Hence Δxi-1 is an increasing sequence for i≥m and so there exists a positive integer n1 such that Δxi is one sign for i>n1. Thus x is eventually a monotone sequence for |i|>n1. Note that x is bounded. Thus, lim⁡n→+∞xn and lim⁡n→-∞xn exist.
Iflim⁡n→+∞xn≠0, we have
(55)Δ2xi-1≥λxi1+αfori≥m,Δxn-Δxn0-1≥λ∑i=n0n‍xi1+α,
which implies that
(56)lim⁡n→+∞xn=+∞.
This is impossible since x is bounded and so we must have lim⁡n→+∞xn=0. Similarly, we also have lim⁡n→-∞xn=0. The proof is complete.Theorem 8.
For anyλ≠0, there exists at most one positive solution of (1) such that lim⁡i→∞xi=0.Proof.
Suppose thatu and v are such two distinct solutions. As before we can construct an arbitrarily small subsolution and so there must exist a solution w of (1) such that w≤u and w≤v. Multiplying the u-equation by w and the w-equation by u, then we have
(57)∑i=-nn‍uiΔ2wi-1-wiΔ2ui-1=λ∑i=-nn‍uiwiwiα-uiα
or
(58)unΔwn-u-nΔw-n-1-wnΔun+w-nΔu-n-1WW=λ∑i=-nn‍uiwiwiα-uiα.
Let n→+∞; we have
(59)∑i=-∞∞‍uiwiwiα-uiα=0
which implies that u=w. Similarly, we can also prove that v=w. The proof is complete.Corollary 9.
Assume that all conditions of Theorem8 hold; then (1) has a unique positive soliton.


## 4. Conclusion

This paper studied a discrete logistic steady-state equation with both positive and negative birth rate of population. By using sub- and upper-solution method (Theorem4), the existence of positive solution and positive soliton is obtained (Theorems 5 and 6). Uniqueness of homoclinic type solution is also obtained (Theorem 8).



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