On Nonresonance Problems of Second-Order Difference Systems

Let T be an integer with T ≥ 3, and let T : {1, . . . , T}. We study the existence and uniqueness of solutions for the following two-point boundary value problems of second-order difference systems: Δu t − 1 f t, u t e t , t ∈ T, u 0 u T 1 0, where e : T → R and f : T × R → R is a potential function satisfying f t, · ∈ C1 R and some nonresonance conditions. The proof of the main result is based upon a mini-max theorem.


Introduction
The existence and uniqueness of solutions of nonresonance problems of differential equations have been studied extensively see 1-5 , and the references therein .However, very few results have been established for nonresonance problems of differential equations.Although we have seen some results of the existence of solutions of discrete equations subjected to diverse boundary conditions, such as in 6-13 , none of them addresses the nonresonance problems.
In this paper, we consider nonlinear boundary value problems of second-order difference systems of the form where e : T → R n and f : T × R n → R n is a potential vector-valued function for t ∈ T.
Why do we pay attention to the discrete problem 1.1 ?Note that the continuous eigenvalue problem Advances in Difference Equations y t ηu t 0, t ∈ 0, 1 , u 0 u 1 0 1.2 has a sequence of eigenvalues while the discrete eigenvalue problem Δ 2 y t − 1 μy t 0, t ∈ T, y 0 y T 1 0 1.4 has exactly T real eigenvalues where T is an integer with T ≥ 3, T : {1, . . ., T}, and T : {0, 1, . . ., T 1}.Thus, the study of nonresonance problems near the largest eigenvalue μ T is new and interesting.Furthermore, the eigenspace corresponding to any eigenvalue in 1.5 is one-dimensional; see 14 for more extensive discussion of these topics.For every e 1 : T → R 1 , the corresponding nonhomogeneous problem has a unique solution if μ / ∈ {μ 1 , . . ., μ T }.The purpose of this paper is to provide some nonresonance conditions which guarantee the existence and uniqueness of solutions of 1.1 .Especially, we allow that the nonlinearity may be superlinear at ∞.The main tool in this paper is a mini-max theorem due to Lazer 1 .

Statement of the main result
In this section, we state our main result.First, we need to introduce some notations and preliminary results.
Let •, • n be the usual scalar product in R n .Let LS R n be the set of all symmetric n × n real matrices.For A, B ∈ LS R n , we say that A B if n be the eigenvalues of A, and let w k be the eigenvector corresponding to λ A k .Then, there exists γ k such that Bw k γ k w k , k 1, . . ., n.

2.2
Proof.Since A and B are commutative, we have that Then, there exists γ k such that Bw k γ k w k .
Remark 2.2.It is worth remarking that the conditions of Lemma 2.1 cannot guarantee that Therefore, 4, Assumption H2.2 is not suitable.
and γ 1 , γ 2 , . . ., γ n be the eigenvalues of A and B, respectively.Let w k be the eigenvector corresponding to both λ A k and γ k .Then, Proof.From the fact that we have This implies that γ k ≥ λ A k for k 1, . . ., n.

Theorem 2.5. Let (H1) and (H2) hold. Assume that
H3 there exist two diagonal matrices A and B: 11 n such that one of the following conditions holds: Then, the boundary value problem 1.1 has exact one solution for every e : T → R n .Remark 2.6.In a in H3 , we use the revised interval λ A k , γ k to replace the interval λ 1 k , λ 2 k which was used in 4, Assumption H2.2 .
Remark 2.7.c in H3 allows that the nonlinearity f may be superlinear at ∞ and −∞.

The main tools
Lemma 3.1 see 1 .Let X and Y be two closed subspaces of a real Hilbert space H such that X is finite-dimensional and H X ⊕ Y .Let f : H → R be a functional and let ∇f and D 2 f denote the gradient and Hessian of f, respectively.Suppose that there exist two positive constants m 1 and m 2 such that Then, f has a unique critical point.Moreover, this critical point of f is characterized by the equality In order to introduce the other tool, we give firstly some notations.Let E denote real Banach space with norm • E .If E * is the topological dual of E, then the symbol •, • will denote the duality pair between E and E * .
Let {u n } be a sequence in E. We say that u n converges weakly to u, written as Let g : E → R be a functional.We say that g is weakly continuous if for every {u n } ⊂ E with u n u, we have that g u lim n→∞ g u n .

3.3
We say that g is weakly lower semicontinuous if {u n } ⊂ E and u n u imply that We Then, there exists x 0 ∈ E such that

Preliminary lemmas
In this section, we give and prove some preliminary lemmas which are necessary for the proof of the main result, Theorem 2.5.Let Proof.Using "summation by parts" see 14, Theorem 2.8 , we have

4.8
Define a functional J : From now on we assume that the eigenfunction ϕ k corresponding to the eigenvalue μ k satisfies The following result is a special case of 14, Theorem 7.2 .
Lemma 4.4.T t 1 ϕ k t ϕ j t 0 for k, j ∈ T with k / j.

Proof of the main result
Now, we give the proof of Theorem 2.5.We divide the proof into three cases.

5.1
For u ∈ H with we define the orthogonal projectors P : By a in H3 , Let us consider the functional J : H → R which is defined in 4.9 : u t , e t n .

5.6
It is easy to check that for h, k ∈ H,

5.7
Now, from a in H3 and Lemma 4.4, for u ∈ H and x ∈ X with 5.9 Thus,  Then, it is easy to show that the condition a in H3 holds.According to Theorem 2.5, the boundary value problem 6.1 has a unique solution for every e : {1, 2, 3} → R 2 .
Remark 6.2.Note that γ 1 > γ 2 in B; this case cannot be handled in 4 .
Let (H1) and (H2) hold.Then, J : H → R n is weakly semicontinuous and J ∈ C 2 .Proof.The proof is standard; so we omit it.