# Existence of Positive Solutions for Second-Order Neumann Difference System

Authors: Yanqiong Lu; Ruyun Ma
Journal: Journal of Function Spaces
 (2014)

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## Abstract

This paper studies a class of Neumann difference system which is not cooperative but its linear part is and this makes it possible to establish existence and nonexistence results for nonnegative
solutions of the system in terms of the principal eigenvalue of the corresponding linearized system.


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## Body

## 1. Introduction

There has been a long history of interest in difference equations and difference systems; for example, see [1–8] and the references therein. In particular, a wide variety of nonlinear difference systems have been studied because they model numerous real-life problems in biology, physics, population dynamics, economics, and so on.One of the difference equations that has attracted some attention is(1)u(k+2)=f(k,u(k),u(k+1)),k∈{0,1,…,N-1},Δu(0)=A,Δu(N)=B,
where Δu(k)=u(k+1)-u(k) for all k∈ℤ, A,B∈ℝ. In 2003, Cabada and Otero-Espinar [4] studied the existence of solution of the problem (1) and obtained optimal existence results by lower and upper solutions methods.Besides, we note that some systems of discrete boundary value problems are investigated by several authors in recent years; see [5, 7, 8] and the references therein. For example, Sun and Li [5] studied the following boundary value problem of discrete system:
(2)Δ2u(k)+f(k,u(k),v(k))=0,k∈{0,1,2,…,N},N>3,Δ2v(k)+g(k,u(k),v(k))=0,k∈{0,1,2,…,N},u(0)=u(N+2)=0,v(0)=v(N+2)=0.
Under some assumptions on f,g, they obtained some sufficient conditions for the existence of one or two positive solutions to the system by using nonlinear alternative of Leray-Schauder and the fixed point theorem in cones.Henderson et al. [7] considered the following system of three-point discrete boundary value problem:
(3)Δ2u(k-1)+λa(k)f(u(k),v(k))=0,k∈{1,2,…,N-1},N>4,Δ2v(k-1)+μb(k)g(u(k),v(k))=0,k∈{1,2,…,N-1},u(0)=βu(η),u(N)=αu(η),v(0)=βv(η),v(N)=αv(η),
where η∈{1,2,…,N-1}, α, β>0, f, g, a, b are nonnegative functions and λ, μ>0. They deduced the existence of the eigenvalues λ and μ yielding at least one positive solution to the system (3) under some assumptions on f, g, a, and b with weakly coupling behaviors. Their main tool is the fixed point theorem in cones.However, very little work has been done for the existence of positive solutions of second-order Neumann difference systems. Inspired by the above works, we study the existence of positive solutions of the following second order Neumann difference system:(4)-Δ2u(k-1)=a(k)v(k)-c(k)u(k)-eu(k)[u(k)+v(k)],k∈𝕀,-Δ2v(k-1)=b(k)u(k)-d(k)v(k)-fv(k)[u(k)+v(k)],k∈𝕀,Δu(0)=Δu(n)=0,Δv(0)=Δv(n)=0,
where 𝕀={1,2,…,n}, the coefficients a, b, c, and d are positive functions on 𝕀, and e, f are positive constants. Through careful analysis, we have found that (4) is not cooperative but its linear part is and this makes it possible to establish existence and nonexistence results for nonnegative solutions of (4) in terms of the principal eigenvalue of the corresponding linearized system. These conditions are different from those given in [5, 7].Although system (4) is very simple, it contains an interesting mathematical feature. It is well known that cooperative systems can be analysed by using lower and upper solutions method and, in general, possesses many of the properties of scalar equations. Although system (4) is not cooperative, its linear part
(5)-Δ2u(k-1)=a(k)v-c(k)u,k∈𝕀,-Δ2v(k-1)=b(k)u-d(k)v,k∈𝕀,Δu(0)=Δu(n)=0,Δv(0)=Δv(n)=0
is and the cooperative nature of the linear system plays a key role in formulating and helping to prove our existence results for (4). For recent works in the literature on continuous Neumann boundary value problem as well as system, we refer to [9–15] and the references therein.Sufficiently motivated, the paper is organized as follows: in Section2, we discuss the properties of cooperative system which we will require. In Section 3, we establish necessary and sufficient conditions for the existence of a positive solution of (4) in terms of the principal eigenvalue of the associated linear system. Finally, an example is presented to illustrate the main result.


## 2. Cooperative Systems

The difference system(6)-Δ2u(k-1)=f(u(k),v(k)),k∈𝕀,-Δ2v(k-1)=g(u(k),v(k)),k∈𝕀,Δu(0)=Δu(n)=0,Δv(0)=Δv(n)=0
is said to be cooperative if v→f(u,v) is a nondecreasing function for any fixed u and u→g(u,v) is a nondecreasing function for any fixed v. Because of maximum principle arguments cooperative systems possesses many of the properties of single difference equation, in this section, we will discuss the results that we require for linear cooperative systems. We are primarily interested in the existence of principal eigenvalues for such systems and on the monotone behavior of such eigenvalues with respect to coefficients of the system. Our proofs depend on the connection between the existence of positive upper solutions for the system and the maximum principle. The continuous case has been obtained by Sweers [13], López-Gómez and Molina-Meyer [14], and Brown and Zhang [15], but as far as we known, there is no result for the discrete case.Throughout the rest of this section, we will consider the cooperative system(7)-Δ2u(k-1)=A(k)u(k)+B(k)v(k),k∈𝕀,-Δ2v(k-1)=C(k)u(k)+D(k)v(k),k∈𝕀,
where A, B, C, and D are functions on 𝕀 with A(k), D(k)<0 and B(k), C(k)>0 for k∈𝕀.Let𝕀^={0,1,…,n,n+1} and X={u∣u:𝕀^→ℝ} be a Banach space with the norm ∥u∥=maxt∈𝕀^|u(t)|. In addition, we introduce the notation u≫0 on 𝕀^ that means that u(t)>0 for t∈𝕀^.In the following, we give some important Lemmas to show the main result.Lemma 1.
Assume thatq:𝕀→(0,∞), h:𝕀→[0,∞) and R1≤0,R2≥0 are constants. Then the boundary value problem
(8)-Δ2u(k-1)+q(k)u(k)=h(k),k∈𝕀,Δu(0)=R1,Δu(n)=R2
has the solution u that satisfies either u≡0 or u≫0 on 𝕀^.Proof.
Letφ(k),ψ(k) be the solution of initial value problems
(9)-Δ2φ(k-1)+q(k)φ(k)=0,k∈𝕀,φ(0)=1,Δφ(0)=0,-Δ2ψ(k-1)+q(k)ψ(k)=0,k∈𝕀,ψ(n)=1,Δψ(n)=0,
respectively. It is easy to compute and show that(i)
φ(k)=1+∑j=1k-1(k-j)q(j)φ(j)>0 and φ is increasing on 𝕀^;(ii)
ψ(k)=1+∑j=k+1n(j-k)q(j)ψ(j)>0 and ψ is decreasing on 𝕀^.
It is not difficult to verify that the problem (8) has the solution
(10)u(k)=∑j=1n‍G(k,j)h(j)+R1Δψ(0)ψ(k)+R2Δφ(n)φ(k),k∈𝕀^,
where
(11)G(k,j)={φ(k)ψ(j),k<j,φ(j)ψ(k),k≥j.
Clearly, it follows from the properties of φ, ψ together with the above that either u≡0 or u≫0 on 𝕀^.Theorem 2.
Suppose that there exist functionsu0, v0∈X such that u0≫0, v0≫0 and
(12)-Δ2u0(k-1)≥A(k)u0(k)+B(k)v0(k)fork∈𝕀,-Δ2v0(k-1)≥C(k)u0(k)+D(k)v0(k)fork∈𝕀,Δu0(0)≤0,Δu0(n)≥0,Δv0(0)≤0,Δv0(n)≥0,
where equality does not hold in all of the equations in (12). Then (7) satisfies the maximum principle; that is, if u,v∈X such that
(13)-Δ2u(k-1)≥A(k)u(k)+B(k)v(k)fork∈𝕀,-Δ2v(k-1)≥C(k)u(k)+D(k)v(k)fork∈𝕀,Δu(0)≤0,Δu(n)≥0,Δv(0)≤0,Δv(n)≥0,
then either (i) u≡v≡0 on 𝕀 or (ii) u≫0 and v≫0 on 𝕀.Proof.
Suppose on the contrary that there existu1, v1 not both identically zero satisfying inequalities (13) but not satisfying (ii) in the conclusion of the theorem. For 0≤τ≤1, define uτ=(1-τ)u0+τu1 and vτ=(1-τ)v0+τv1. Then there exists τ0, 0<τ0≤1 such that uτ, vτ≫0 for 0≤τ<τ0 and either uτ0 or vτ0 has a zero in 𝕀^. Without loss of generality, we may assume that there exists k1∈𝕀^ such that uτ0(k1)=0; then,
(14)-Δ2uτ0(k-1)≥A(k)uτ0(k)+B(k)vτ0(k)fork∈𝕀,-Δ2uτ0(k-1)-A(k)uτ0(k)≥0,k∈𝕀.
Moreover, Δuτ0(0)≤0,Δuτ0(n)≥0 and so it follows from Lemma 1 that uτ0≡0 or uτ0≫0 on 𝕀^. Obviously, uτ0(k1)=0, so uτ0≡0. Hence, by (14), it also follows that vτ0≡0. From u1,v1 are not both identically zero for τ0<1, this goether with uτ0=(1-τ0)u0+τ0u1≡0 and vτ0=(1-τ0)u0+τ0v1≡0, implies that u1 and v1 equal the same negative multiples of u0 and v0, respectively. This is impossible as u0, v0 satisfy (12) and u1,v1 satisfy (13) and so the proof is complete.Corollary 3.
Suppose thatA(k)+B(k)<0 and C(k)+D(k)<0 for all k∈𝕀. Then the cooperative system (7) satisfies the maximum principle.Proof.
The result follows from Theorem2 by choosing u0≡v0≡K where K is any positive number.Theorem 4.
The cooperative system (7) has a principal eigenvalue; that is, there exists Λ∈ℝ and u,v∈X such that u,v≫0 and
(15)-Δ2u(k-1)-A(k)u(k)-B(k)v(k)=Λu(k),k∈𝕀,-Δ2v(k-1)-C(k)u(k)-D(k)v(k)=Λv(k),k∈𝕀,Δu(0)=Δu(n)=0,Δv(0)=Δv(n)=0.Proof.
Let(16)E={(uv)∈X×X∣Δu(0)=Δu(n)=0,Δv(0)=Δv(n)=0}
and Y=X×X. Define L:E→Y by
(17)L(uv)=(-Δ2u(k-1)-Δ2v(k-1))
and define the matrix M(k) by
(18)M(k)=(A(k)B(k)C(k)D(k)).
It is well know that if K>0 is sufficiently large, then L-M+K is an invertible operator such that (L-M+K)-1 is compact. If, moreover, K is chosen sufficiently large to ensure that A(k)+B(k)-K<0 and C(k)+D(k)-K<0 for all k∈𝕀, it follows from Corollary 3 that (L-M+K)-1 is strongly positive.
Since(L-M+K)-1 is compact and strongly positive, (L-M+K)-1 has a positive principal eigenvalue μ. Thus there exists ϕ=(u0v0)∈Y with u0,v0≫0 such that (L-M+K)-1ϕ=μϕ. Hence (L-M)ϕ=(1/μ-K)ϕ and L-M has a principal eigenvalue.For convenience, we will denote the principal eigenvalue ofL-M by λ1(M).Corollary 5.
Suppose thatM1(k) and M2(k) are cooperative matrices (i.e., matrices with positive entries in the off-diagonal elements) such that M1(k)≥M2(k) (i.e., the (i,j)th element of M1≥ the (i,j)th element of M2) for all k∈𝕀 but M1(k)≢M2(k)). Then λ1(M1)<λ1(M2).Proof.
There existsϕ0=(u0v0)∈E such that u0,v0≫0 and [L-M1-λ1(M1)]ϕ0=0. Then
(19)[L-M2-λ1(M1)]ϕ0=[L-M1-λ1(M1)]ϕ0+(M1-M2)ϕ0,k∈𝕀,=(M1-M2)ϕ0≥0,
but (M1-M2)ϕ0≢0 and so by Theorem 2, the cooperative system [L-M2-λ1(M1)]ϕ≥0 with ϕ=(uv)∈E satisfies the maximum principle. Hence, if μ denotes the principal eigenvalue for the system L-M2-λ1(M1)I, it follows that μ>0. Clearly, L-M2 has principal eigenvalue λ1(M2)=λ1(M1)+μ>λ(M1) and so λ1(M2)>λ1(M1).


## 3. Existence and Nonexistence of Positive Solutions

System (4) can be rewritten as
(20)L(uv)=M(k)(u(k)v(k))-N(u(k)v(k)),
where M(k)=(-c(k)a(k)b(k)-d(k)) and N:Y→Y such that
(21)N(u(k)v(k))=(eu[u+v]fv[u+v]).
Although M(k) is a cooperative matrix, (20) is not a cooperative system. The following general theorem describes how the method of lower and upper solutions must be modified to deal with general, possibly non-cooperative system such as (6). The proofs of the method of lower and upper solutions are based upon the connectivity properties of the solution sets of parameterized families of compact vector fields; see [16, Theorem 1.0].Lemma 6.
Letq:𝕀→(0,∞), and suppose that there exist functions u¯,u_∈X such that u_≤u¯ and, for all k∈𝕀,
(22)-Δ2u_(k-1)+q(k)u_(k)≤f(k,u_(k)),Δu_(0)≥0,Δu_(n)≤0,-Δ2u¯(k-1)+q(k)u¯(k)≥f(k,u¯(k)),Δu¯(0)≤0,Δu¯(n)≥0.
Then the problem
(23)-Δ2u(k-1)+q(k)u(k)=f(k,u(k)),Δu(0)=0,Δu(n)=0
has a solution u(k) that satisfies
(24)u_(k)≤u(k)≤u¯(k),∀k∈𝕀^.Proof.
Let us consider the auxiliary problem(25)-Δ2u(k-1)+q(k)u(k)=f*(k,u(k)),Δu(0)=0,Δu(n)=0,
where f*(k,u)=f(k,p(k,u)) and p(k,u) is defined by
(26)p(k,u)={u¯(k),u(k)>u¯(k),u(k),u_(k)≤u(k)≤u¯(k),u_(k),u(k)≤u_(k).
Clearly, to show that (23) has a solution u(k) satisfies u_(k)≤u(k)≤u¯(k), it is enough to prove that (25) has a solution that u*(k) satisfies u_(k)≤u*(k)≤u¯(k).
Define the operator(27)Tu(k)=∑j=1n‍G(k,j)f*(j,p(j,u)),k∈𝕀.
By the standard of compact operator argument, it is easy to show that T is a completely continuous operator. Obviously, f* is a bounded function; this combined with Lemma 1 and Brouwer fixed point theorem and we can get that T has a fixed point u*; that is, u* is a solution of the problem (25).
Next, we will show thatu_(k)≤u*(k)≤u¯(k).
We only have to deal withu*(k)≤u¯(k) and the case u_(k)≤u*(k) can be proved by a similar argument. Suppose on the contrary that there exists a point k0∈𝕀 such that u*(k0)>u¯(k0). Let w(k)=u¯(k)-u*(k); then, there exists k1∈𝕀 such that w(k1)=mink∈𝕀w(k)<0 and Δw(k1)≥0, Δw(k1-1)≤0. Subsequently,
(28)-Δ2w(k1-1)+q(k1)w(k1)<0.
On the other hand, we have that
(29)-Δ2w(k1-1)+q(k1)w(k1)≥f(k1,u¯(k1))-f*(k1,p(k1,u*(k1)))=0,
which is a contradiction. So u*(k)≤u¯(k). The proof is complete.Theorem 7.
Suppose that there exist functionsu¯, v¯, u_, v_∈X such that u_≤u¯ and v_≤v¯ and, for all k∈𝕀,
(30)-Δ2u_(k-1)+q(k)u_(k)≤f(u_(k),v),wheneverv_(k)≤v≤v¯(k),-Δ2v_(k-1)+q(k)v_(k)≤g(u,v_(k)),wheneveru_(k)≤u≤u¯(k),Δu_(0)≥0,Δu_(n)≤0,Δv_(0)≥0,Δv_(n)≤0,-Δ2u¯(k-1)+q(k)u¯(k)≥f(u¯(k),v),wheneverv_(k)≤v≤v¯(k),-Δ2v¯(k-1)+q(k)v¯(k)≥g(u,v¯(k)),wheneveru_(k)≤u≤u¯(k),Δu¯(0)≤0,Δu¯(n)≥0,Δv¯(0)≤0,Δv¯(n)≥0.
Then there exists a solution (uv) of the system
(31)-Δ2u(k-1)+q(k)u(k)=f(u(k),v(k)),k∈𝕀,-Δ2v(k-1)+q(k)v(k)=g(u(k),v(k)),k∈𝕀,Δu(0)=Δu(n)=0,Δv(0)=Δv(n)=0
such that u_(k)≤u(k)≤u¯(k) and v_(k)≤v(k)≤v¯(k) for all k∈𝕀.Proof.
From Lemma6, it is easy to show that for any fixed v_≤v≤v¯, there exists the solution set
(32)(v)≔{u∈X∣(u,v)satisfythefirstequationin(31)}≠∅.
Moreover, by the connectivity properties of the solution sets of parameterized families of compact vector fields [16, Theorem 1.0], the set
(33)𝒮≔{(u,v)T∈E∣(u,v)satisfythefirstequationin(31)}
contains a connected subset Σ which joins {v_}×W(v_) and {v¯}×W(v¯) with v_≤v¯.
For everyu∈W(v_), we have that
(34)-Δ2v_(k-1)+q(k)v_(k)≤g(u,v_(k)),
and for any u∈W(v¯), we get that
(35)-Δ2v¯(k-1)+q(k)v¯(k)≥g(u,v¯(k)).
Therefore, by the connectivity of Σ, there exist v0∈[v_,v¯] and w(v0)∈W(v0) such that (w(v0),v0)∈Σ and satisfies
(36)-Δ2v0(k-1)+q(k)v0(k)=g(w(v0)(k),v0(k)),k∈𝕀.
Thus (w(v0),v0) is solution of the system (31).
We can proceed in the same way, proving that there existsu0∈[u¯,u_], p(u0)∈P(u) such that (u0,p(u0)) is a solution of the system (31), where
(37)P(u)≔{v∈X∣(u,v)satisfythesecondequationin(31)}≠∅.
The proof is complete.We can give necessary and sufficient conditions for the existence of a positive solution of the system (4).Theorem 8.
There exists a positive solution of system (4) if and only if λ1(M)<0.Proof.
Supposeλ1(M)<0. Then there exist ϕ1,ϕ2≫0 such that (L-M)(ϕ1ϕ2)=λ1(M)(ϕ1ϕ2); that is,
(38)-Δ2ϕ1(k-1)=a(k)ϕ2(k)-c(k)ϕ1(k)+λ1(M)ϕ1(k),-Δ2ϕ2(k-1)=b(k)ϕ1(k)-d(k)ϕ2(k)+λ1(M)ϕ2(k),Δϕ1(0)=Δϕ1(n)=0,Δϕ2(0)=Δϕ2(n)=0.
Let (u_v_)=ϵ(ϕ1ϕ2) and (u¯v¯)=(KK). We will show that (u_v_) and (u¯v¯) satisfy the hypotheses of Theorem 7 provided that ɛ>0 is chosen sufficiently small and K is chosen sufficiently large. Let
(39)K=max{maxk∈𝕀a(k)e,maxk∈𝕀b(k)f}.
Then for all k∈𝕀, a(k)-eK≤0 and so
(40)a(k)v-c(k)u¯(k)-eu¯(k)[u¯(k)+v]=[a(k)-eK]v-c(k)K-eK2≤0=-Δ2u¯(k-1),
whenever v≥0. Similarly,
(41)-Δ2v¯(k-1)≥bu-dv¯(k)-fv¯(k)[u+v¯(k)]
whenever u≥0.
Let(42)ε0=min{mina(k)emaxϕ1(k),minb(k)fmaxϕ2(k)}.
Then if ɛ<ε0, we get that a(k)-ɛeϕ1(k)≥0 and b(k)-ɛfϕ2(k)≥0 for all k∈𝕀. Hence, when ɛ<ε0,u_(k)=ɛϕ1(k) and v(k)≥ɛϕ2(k), we have
(43)-Δ2u_(k-1)-a(k)v+c(k)u_(k)+eu_(k)[u_(k)+v]=ɛ[-Δ2ϕ1(k-1)+c(k)ϕ1-a(k)ϕ2]+a(k)[ɛϕ2(k)-v]+eɛϕ1(k)[ɛϕ1(k)+v]=ɛλ1(M)ϕ1(k)+ɛa(k)ϕ2(k)-[a(k)-ɛeϕ1(k)]v+ɛ2e[ϕ1(k)]2≤ɛλ1(M)ϕ1(k)+ɛa(k)ϕ2(k)-[a(k)-ɛeϕ1(k)]ɛϕ2(k)+ɛ2e[ϕ1(k)]2=ɛλ1(M)ϕ1(k)+ɛ2eϕ1(k)[ϕ1(k)+ϕ2(k)]<0,
when ɛ is sufficiently small.
Similarly, ifv_(k)=ɛϕ2(k),u≥ɛϕ1(k), and ɛ is sufficiently small, we have
(44)-Δ2v_(k-1)-b(k)u+d(k)v_(k)+fv_(k)[u+v_(k)]<0∀k∈𝕀.
Hence, by Theorem 7, there exists a positive solution of system (4).
Suppose now that system (4) has a solution (u0v0) with u0,v0≫0. Then (u0v0) is a solution of the system
(45)-Δ2u(k-1)+[c(k)+eq(k)]u(k)-a(k)v(k)=0,k∈𝕀,-Δ2v(k-1)-b(k)u(k)+[d(k)+fq(k)]v(k)=0,k∈𝕀,Δu(0)=Δu(n)=0,Δv(0)=Δv(n)=0,
where q(k)=u0(k)+v0(k). Hence (u0v0) may be regarded as the principal eigenfunction corresponding to the principal eigenvalue λ=0 of the system (L-Mq(k))u_=0, where Mq(k)=(-c(k)-eq(k)a(k)b(k)-d(k)-fq(k)). Hence λ1(Mq)=0. As Mq≤M but Mq≢M, it follows from Corollary 5 that λ1(M)<λ1(Mq)=0 and the proof is complete.Remark 9.
From Theorem8, it is easy to see that the system (4) has no positive solution if λ1(M)≥0.Example 10.
Let us consider the Neumann difference system(46)-Δ2u(k-1)=3v(k)-4u(k)+u(k)(u(k)+v(k)),k∈{1,2,3,4,5},-Δ2v(k-1)=2u(k)-3v(k)+2v(k)(u(k)+v(k)),k∈{1,2,3,4,5},Δu(0)=Δu(5)=0,Δv(0)=Δv(5)=0.
Obviously, M=(-4I53I52I5-3I5); here I5 denote the unit matrix of order 5. By simple computing, λ1(M)=-1<0. From Theorem 8, the problem (46) has at least one positive solution u on {0,1,…,6}.



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