# Uniqueness and Multiplicity of Solutions for  a Second-Order

Authors: Xi-Lan Liu; Jian-Hua Wu
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

This paper is concerned with the existence of unique and multiple solutions to the boundary value problem of a second-order difference equation with a parameter, which is a complement of the work by J. S. Yu and Z. M. Guo in 2006.


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## Body

## 1. Introduction and Preliminaries

In this paper, we consider the
existence, uniqueness, and multiplicity of solutions for a second-order
discrete boundary value problem(1.1)p(n+1)u(n+1)+c(n)u(n)+p(n)u(n−1)=λf(n,u(n)),n∈Z(1,k),u(0)+αu(1)=A,u(k+1)+βu(k)=B,  where λ∈ℝ is a parameter.
Our technique is based on critical point theory, which is successfully used to
deal with the existence of solutions for discrete problems (see
[1–9]), especially in
[7, 9]. Similarly to [7], we denote by ℕ,ℤ,  and ℝ the sets of all
natural numbers, integers, and real numbers, respectively. For a,b∈ℤ, Z(a)={a,a+1,…}, Z(a,b)={a,a+1,…,b} when a≤b. We assume that p(n) is nonzero and
real-valued for each n∈Z(1,k), c(n) is real-valued
for each n∈Z(1,k), and f(n,u) is real-valued
for each (n,u)∈Z(1,k)×ℝ and continuous
in u. Let ℝk be the real
Euclidean space with dimension k. For any u,v∈ℝk, ∥u∥ and (u,v), denote the usual norm and inner product in ℝk, respectively.Consider the functional defined onℝk,(1.2)J(u)=12(Mu,u)+(η,u)−λF(u),u=(u(1),u(2),…,u(k))T∈ℝk,where (⋅)T is the
transpose of a vector in ℝk,(1.3)M=(c(1)−αp(1)p(2)0⋯00p(2)c(2)p(3)⋯000p(3)c(3)⋯00⋯⋯⋯000⋯c(k−1)p(k)000⋯p(k)c(k)−βp(k+1)),η=(p(1)A0⋮0p(k+1)B),F(u)=∑j=1k∫0u(j)f(j,s)ds.It is easy to see thatJ(u) is Fréchet
differentiable with Fréchet derivative(1.4)J'(u)=Mu+η−λf(u),where f(u)=(f(1,u(1)),f(2,u(2)),…,f(k,u(k)))T, and there is a one-to-one correspondence between the
critical point of functional J and the
solution of BVP (1.1). Furthermore, u=(u(1),u(2),…,u(k))T is a critical
point of J if and only if {u(t)}t=0k+1=(u(0),u(1),…,u(k+1))T is a solution
of (1.1), where u(0)=A−αu(1),u(k+1)=B−βu(k) [7].Recently, Yu and Guo [7] studied the
BVP,(1.5)p(n+1)u(n+1)+c(n)u(n)+p(n)u(n−1)=f(n,u(n)),u(0)+αu(1)=A,u(k+1)+βu(k)=B.They obtained some existence
results for (1.5). One of the main results is as follows.Theorem (A0).
Suppose  thatf(n,z) satisfies the
following assumption:
(f3) there exist constantsa1>0,  a2>0,  R>0,  and β>2 such
that(1.6)∑j=1k∫0u(j)f(j,s)ds≥a1∥u∥β−a2foru=(u(1),u(2),…,u(k))T∈ℝk,∥u∥≥R.Then BVP (1.5) has at least one
solution.Equation (1.6) shows that F(u)=∑j=1k∫0u(j)f(j,s)ds>0 for ∥u∥ large enough.
Since
F(u) may be
negative, then the conclusion of Theorem 1.1
cannot be
drawn, which motivates us to consider (1.1). Note that if we take λ=−1 in (1.1), then F(u)<0. Under the similar condition to (1.6) when λ<0, we not only obtain the existence of solutions for
(1.1), but also the multiplicity.LetE be a Banach
space with a direct sum decomposition E=X⊕Y. The functional I∈C1(E,ℝ) has a local
linking at 0 if for some ρ>0,I(u)≤0,u∈X, ∥u∥≤ρ, and I(u)≥0,u∈Y, ∥u∥≤ρ. The functional I∈C1(E,ℝ) is said to
satisfy the (PS) condition if any sequence xn⊂X for which I(xn) is bounded, and I'(xn)→0(n→∞)   possesses a
convergent subsequence in E.Theorem 1 A (see[10]).
LetE be a Banach
space. Suppose that I∈C1(E,ℝ) satisfies the
(PS) condition and has a local linking at 0. Assume that I is bounded
below and infEI<0. Then I has at least
two nontrivial critical points.Theorem 1 B (see[6, 11]).
LetX be a real
Banach space, I∈C1(X,ℝ) with I even, bounded
from below, and satisfying (PS) condition. Suppose I(0)=0. There is a set K⊂X such that K is homeomorphic
to a unit sphere Sn−1 in ℝn (n∈ℕ) by an odd map, and supKI<0. Then I possesses at
least n distinct pairs
of critical points.


## 2. Main Results

Following conditions will be useful to prove our main
results.(H1)
There exist
numbersα1>2 and a1>0 such
that(2.1)F(u)≥a1∥u∥α1foru=(u(1),u(2),…,u(k))T∈ℝk.(H2)
limu→0(F(u)/∥u∥2)=0foru=(u(1),u(2),…,u(k))T∈ℝk.Theorem 2.1.
Suppose thatM is positive
definite, f(j,0)=0, f(j,u)u<0  for  j∈Z(1,k) and u≠0, and that p(1)A=p(k+1)B=0. Then (1.1) has only trivial solution for λ>0.Proof.
Note
that(2.2)(J'(u),u)=(Mu,u)+(η,u)−λ(f(u),u)≥λ1∥u∥2−λ∑j=1kf(j,u(j))u(j)≥λ1∥u∥2>0for u≠0, where λ1 is the least
eigenvalue of M, which means that the Nahari manifold is empty. Thus
(1.1) has only trivial solution.Theorem 2.2.
Suppose that (H1) and (H2) hold,p(1)A=p(k+1)B=0 and M is neither
positive definite nor negative definite. Then (1.1) has at least two nontrivial
solutions for λ<0.Proof.
We
will prove that the functionalJ(u) satisfies all
conditions of Theorem A by two steps.
Step 1.
J is bounded from
below and satisfies (PS) condition. Let λ−l,λ−l+1,…,λ−1,λ1,λ2,…,λm denote all the
eigenvalues of M, where λ−l≤λ−l+1≤⋯≤λ−1<0<λ1≤λ2≤⋯≤λm and l+m=k. For any j∈Z(−l,−1)⋃Z(1,m), set ξj to  be an
eigenvector of M corresponding
to the eigenvalue λj, j=−l,−l+1,…,−1,1,2,…,m,  such
that(2.3)(ξi,ξj)={0fori≠j,1fori=j.
LetX and Y be subspaces of ℝk defined
by(2.4)X={x∈ℝk:x=∑j=1mxjξj,xj∈ℝ,j∈Z(1,m)},Y={y∈ℝk:y=∑j=−l−1yjξj,yj∈ℝ,j∈Z(−l,−1)},respectively. Then ℝk has the direct
sum decomposition ℝk=X⊕Y. In view of (H1), we have(2.5)J(u)≥12λ−l∥u∥2−λa1∥u∥α1≥−12|λ−l|(1−2/α1)(|λ−l|/(−α1λa1))2/(α1−2).The second inequality follows
from the elementary inequality −ax2+bxq≥−a(1−2/q)((2a)/(qb))2/(q−2), where a>0,b>0,x>0, and q>2, which can be
easily obtained by the fact that the function h(x)=−ax2+bxq(a,b>0,x≥0)     attains its
minimum at ((2a)/(qb))1/(q−2). Thus J(u) is bounded from
below.
Equation (2.5) shows that J(u) is coercive, so
we can obtain that any (PS) sequence must be bounded in ℝk and, by a standard
argument, has a convergent subsequence.
Step 2.
J has a local
linking at 0. Indeed, by (H2) for given λ<0 and sufficiently
small ε>0 such that −λε+(1/2)λ−1<0, there exists r>0 small enough
such that for ∥u∥<r,(2.6)F(u)<ε∥u∥2holds. Then for y∈Y such that 0<∥y∥<r, we have(2.7)J(y)≤12λ−1∥y∥2−λε∥y∥2=(12λ−1−λε)∥y∥2<0.
On the other hand, forx∈X with ∥x∥<r, we have(2.8)J(x)≥12λ1∥x∥2−λa1∥x∥α1≥0.The application of
Theorem A
finishes our proof.Remark 2.3.
By the above proof, we see that replacing (H1) with (f3), and
by adding the condition thatF(u)≥0 for u∈ℝk, Theorem 2.2 still holds, where (f3) is the same as
in Theorem 1.1. Indeed, we have(2.9)J(u)≥{12λ−lR2for∥u∥≤R,−12|λ−l|(1−2/β)(|λ−l|/(−βλa1))2/(β−2)+λa2for∥u∥>R,which means that J(u) is bounded from
below. The fact that J(u) has local
linking at 0 may be verified
similarly.If we  further impose some condition onf(u) and matrix M, then the following result can be derived.Theorem 2.4.
Suppose (H1) and (H2) hold,p(1)A=p(k+1)B=0,f(u) is odd in u, that is, f(j,−u)=−f(j,u) for (j,u)∈Z(1,k)×ℝ, and that M is neither
positive definite nor negative definite and has l distinct
negative eigenvalues. Then (1.1) has at least l distinct pairs
of solutions for λ<0.Proof.
By
the proof of Theorem2.2, J(u) is bounded from
below and satisfies (PS) condition. In addition, J(0)=0, J(u) is even.
Consider the subset K of ℝk:(2.10)K={y∈Y:y=∑j=−l−1yjξj,∑j=−l−1yj2=ρ2,yj∈ℝ,j∈Z(−l,−1)},where ρ is a positive
number small enough to be determined later, Y is defined by
Theorem 2.2 similarly. Define the mapping T:K→Sl−1 by(2.11)T(y)=T(y=∑j=−l−1yjξj)=(−y−lρ,−y−l+1ρ,…,−y−1ρ),where Sl−1 is
a unit sphere in ℝl. Then T is a
homeomorphism between K and Sl−1, and K is a subset of
the finite dimensional space Y equipped with
the Euclidian norm. We can choose ρ>0 and ε>0 small enough
such that (1/2)λ−1−λε<0 for y∈K, and then we have(2.12)J(y)≤12λ−1∥y∥2−λF(u)≤12λ−1ρ2−λερ2≤(12λ−1−λε)ρ2.For above ρ>0, we have(2.13)supKJ<0,which together with Theorem B
concludes the proof.Remark 2.5.
The condition thatJ(u) is bounded from
below is crucial to prove both Theorems 2.2 and 2.4.
As in Remark 2.3,
if we replace (H1) by (f3) and the condition that F(u)≥0 for u∈ℝk, then Theorem 2.4 is also true.



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