Boundary Value Problem for a Second-Order Difference Equation with Resonance

where the forward difference operator is defined by Δu(k) � u(k + 1) − u(k); p(k) and q(k) are real valued on Z, and p(k) is nonzero; and f(k, ·) ∈ C1(R,R) satisfies f(k, 0) � 0 for each k ∈ Z(1, T). Clearly, problem (1) has the trivial solution u � 0. In the recent years, the existence of solutions for nonlinear difference equations has been widely studied by many authors. We note that these results were usually obtained by means of critical point theory, for example, the existence of ground state solutions [1], homoclinic orbits [2–5], the boundary value problem, and periodic solutions [6–8]. (e boundary value problem (1) may be regarded as a discrete analogue of the following boundary value problem: p(t)u′(t) ( 􏼁′ + q(t)u(t) + f(t, u(t)) � 0, 0< t< 1,


Introduction
Let Z and R be the sets of integers and real numbers, respectively. For a, b ∈ Z, Z(a, b) denotes the discrete interval a, a + 1, . . . , b { } if a ≤ b. In this paper, we consider the existence and multiplicity of nontrivial solutions for the following discrete Dirichlet boundary value problem: where the forward difference operator is defined by Δu(k) � u(k + 1) − u(k); p(k) and q(k) are real valued on Z, and p(k) is nonzero; and f(k, ·) ∈ C 1 (R, R) satisfies f(k, 0) � 0 for each k ∈ Z(1, T). Clearly, problem (1) has the trivial solution u � 0.
In the recent years, the existence of solutions for nonlinear difference equations has been widely studied by many authors. We note that these results were usually obtained by means of critical point theory, for example, the existence of ground state solutions [1], homoclinic orbits [2][3][4][5], the boundary value problem, and periodic solutions [6][7][8].
e boundary value problem (1) may be regarded as a discrete analogue of the following boundary value problem: p(t)u ′ (t) ′ + q(t)u(t) + f(t, u(t)) � 0, 0 < t < 1, which has been successfully applied to the modeling of astrophysics, gas dynamics, and chemically reacting system, see [9][10][11]. By various methods and techniques, many authors studied the similar second-order difference equation under various boundary value conditions. For example, Agarwal in [10] considered the existence of solutions for the boundary value problem (1) by the contraction mapping principle and Brouwer fixed point theorems in Euclidean space. Yu and Guo in [12] first used the critical point theory to study the following discrete boundary value problem: where A and B are the constants, and they proved the existence of solutions of problem (3) when the nonlinear term f is sublinear or superlinear. We refer the readers to [13][14][15][16] and the reference therein for more information.
It is well known that the resonance exists in many realworld applications, and the equations with resonance have been extensively studied in various fields [17][18][19][20][21]. In fact, it is more difficult to study the boundary value problem under the case of resonance because the resonance case can change the local geometric properties of the critical points [17]. In this paper, we study the existence and multiplicity of nontrivial solutions for problem (1) at resonance by means of the interaction between the nonlinearity and the spectrum of the symmetric matrix P + Q, where P + Q denotes a matrix whose elements are given by p(k) and q(k) for k ∈ Z(1, T).
Note that in [12], the authors obtained the existence of one solution for (3) via the variational methods or the saddle point theorem. In the case where A � B, the solution could be trivial, which was not considered in [12]. In this paper, we will show that the boundary value problem (1) has at least one nontrivial solution. In fact, our eorem 1 and Corollary 1 extend and complement the existence results given in [12] and establish the existence of multiple solutions.
Our results are obtained by combining the Morse theory [22][23][24], critical group computation, and the minimax methods including the local linking [17,25]. We prove the existence of nontrivial solutions for problem (1) at resonance by the relationship between these nonlinear techniques and methods. In order to apply these ideas to identify the unknown critical point, we compute the corresponding critical groups of functional J. Obviously, when we consider the existence of nontrivial solutions of problem (1) at resonance, we need to overcome much more difficulties than those in the literature [12].
We consider the T-dimensional Banach space: e space S is a Hilbert space with the inner product endowed with the norm Now, we define the C 1 -functional J on S as follows: We can compute the Frećhet derivative as for all u, v ∈ S. It is clear that the critical points of J are the solutions of problem (1).
For convenience, we identify u ∈ S with u � (u(1), u(2), . . . , u(T)) ∈ R T . us, we rewrite J(u) and 〈J ′ (u), v〉 as further, where u τ denotes the transpose of u and P and Q are the T × T symmetric matrices given by

Complexity
Let E be a real Banach space and In order to obtain the critical points of the functional J on E, we recall some basic concepts and results of the Morse theory.
Definition 1 (see [22]). J is said to satisfy the Palais-Smale condition (PS condition for short), if any sequence u n ⊂ E for which J(u n ) is bounded and J ′ (u n ) ⟶ 0 as n ⟶ ∞ possesses a convergent subsequence in E.
In the Morse theory, functional J is always required to satisfy the deformation condition (D), which was introduced by Bartsch and Li [26]. Bartolo et al. [27] proved that J satisfies the deformation condition (D) if J satisfies the PS condition.
Assume that J ∈ C 1 (E, R), and let u 0 ∈ K be an isolated critical point of J with J(u 0 ) � c ∈ R, and U is a neighborhood of u 0 , containing the unique critical point, then we call the q-th critical group of J at u 0 , where H q (·, ·) stands for the q-th singular relative homology group with integer coefficients [22,23]. We say that u 0 is a homological nontrivial critical point of J, if at least one of its critical groups is nontrivial.
Let J ∈ C 1 (E, R) satisfy the PS condition and all critical values of J be greater than some α ∈ R, then the group is called the q-th critical group of J at infinity [26]. Assume #K < ∞ and J satisfies the PS condition. e Morse-type numbers of the pair (E, J α ) are defined by M q � M q (E, J α ) � u∈K dimC q (J, u), and the Betti numbers of the pair (E, J α ) are β q � dimC q (J, ∞). By the Morse theory [22,23], the following relations hold: By (14), it follows that M q ≥ β q for all q ∈ Z. Hence, when β q ≠ 0 for some q ∈ Z and J satisfies the PS condition, then J must have a critical point We need the following lemmas to prove the main results.
Lemma 1 (see [17]). Let E be a real Banach space and J ∈ C 1 (E, R) satisfy the (D) condition and be bounded from below. If J has a critical point that is homological nontrivial and is not the minimizer of J, then J has at least three critical points.
Obviously, we need to show that J has a homological nontrivial critical point, which is not the global minimizer in this lemma.
Lemma 2 (see [17,25]). Assume that J has a critical point Lemma 3 (see [26]). Let E be a Banach space and J ∈ C 1 (E, R) satisfy the PS condition. Suppose E splits as Lemma 4 (see [28]). Let Y ⊂ B ⊂ A ⊂ X be tophological spaces and q ∈ Z. If then

(22)
Theorem 1. Assume that the matrix P + Q is positive definite and f(k, t) satisfies (G 1 ) and (G 2 ). en, problem (1) has at least two nontrivial solutions in S.
To prove eorem 1, we need a series of lemmas.

Lemma 5. If the matrix P + Q is positive definite and (G 1 ) holds. en, J satisfies the PS condition.
Proof. For any sequence u n ⊂ S, with J(u n ) is bounded and J ′ (u n ) ⟶ 0 as n ⟶ + ∞, there exists a positive constant C ∈ R such that |J(u n )| ≤ C. By (G 1 ), we have en, for any n ∈ Z, Since 1 < θ < 2, thus u n is bounded in S and the Bolzano-Weierstrass theorem implies that u n has a convergent subsequence. □ Lemma 6. If the matrix P + Q is positive definite and (G 1 ) holds. en, J is coercive on S, that is, J(u) ⟶ + ∞ as ‖u‖ ⟶ + ∞.
Proof. If the matrix P + Q is positive definite, then λ i > 0, ∀i ∈ Z(1, T). Since (G 1 ) is satisfied, it implies that there exist constants a 1 > 0 and a 2 > 0 such that then we have e proof is complete. Let S(λ l ) denote the eigenspace of λ l for some l ∈ Z(1, T − 1), V denote the subspace of S spanned by the eigenfunctions corresponding to the eigenvalues λ 1 , . . . , λ l− 1 , and W denote the subspace of S spanned by the eigenfunctions corresponding to the eigenvalues λ l+1 , . . . , λ T , then we are given an orthogonal decomposition (29) □ Lemma 7. Assume that (G 2 ) holds. en, J has a local linking at the origin with respect to S � (V ⊕ S(λ l )) ⊕ W and For each u ∈ W, 0 < ‖u‖ ≤ δ implies |u(k)| ≤ δ ∀k ∈ Z(1, T). We have the following estimates: is completes the proof.

□
Proof. Proof of eorem 1. By Lemmas 5 and 6, J satisfies the PS condition and is coercive, hence J is bounded from below. Combining Lemma 7 with Lemma 2, the trivial solution u � 0 is homological nontrivial and is not a minimizer. We apply Lemma 1 to conclude that problem (1) has at least two nontrivial solutions in S.
Example 1. Let T � 2, and we consider the boundary value problem (1) with for all k ∈ Z(1, 2). en, Fix θ � 3/2, p(k) � 1, and q(k) � 0 ∀k ∈ Z(1, 2), then the matrix P + Q, is positive definite and admits two distinct eigenvalues given by thus the condition (G 1 ) holds. Let δ � 1, we see that then, the condition (G 2 ) holds. Clearly, the conditions of eorem 1 are satisfied, and problem (1) admits at least two nontrivial solutions in S.
In fact, if |u(k)| ≥ 1 for k ∈ Z(1, 2), then problem (1) is Proof. We now show that we can find at least one nontrivial critical point of functional J.
Since the matrix P + Q is negative definite, in this case, λ i < 0 for every i ∈ Z(1, T) and the inequality (20) holds. If u ∈ S satisfies ‖u‖ ≥ M, by (G 1 ), we have We notice that if − J is continuity and coercive, then J has at least a local maximum u 1 ∈ S by the above inequality. is means that C q (J, u 1 ) � δ q,T Z. If u 1 � 0, then C q (J, 0) � δ q,T Z, noticing Lemma 7, this contradicts Lemma 2. Hence, 0 is different from u 1 . □ Theorem 3. Assume that matrix P + Q is negative semidefinite, and Complexity 5 en, problem (1) has no nontrivial solution.
Proof. If we assume that (1) has a nontrivial solution, then J(u) has a nonzero critical point u. Since J ′ (u) � 0 and the matrix P + Q is negative semidefinite, one has is contradicts with (G 3 ). e proof is complete. If the matrix P + Q is just nonsingular, we may suppose that its first positive eigenvalue is λ l > 0. We denote its eigenvalues as follows: en, we have □ Theorem 4. If the matrix P + Q is just nonsingular and 2 ≤ l < T, we assume that (G 1 ) and the following conditions hold: en, problem (1) has at least one nontrivial solution.
On the other hand, we have for every u ∈ S(λ l ) ⊕ W. Hence, J is bounded from below on S(λ l ) ⊕ W. It can be easily seen that the functional J satisfies all the assumptions of Lemma 3; hence, we have C l− 1 (J, ∞) ≇ 0. (51) We note that l − 1 ≠ l − 1 + ](0), and then J has a critical point u 1 ≠ 0 such that C l− 1 (J, u 1 ) ≇ 0. us, J has at least one nontrivial solution u 1 .

Superlinear Case
In this section, we will study the case where f(k, t) is superlinear in t at infinity for each k ∈ Z(1, T). We give the following assumption: (i) (G 6 ) ere exist β ∈ (2, ∞)and M 1 > 0 such that tf(k, t) ≥ βF(k, t) > 0, for |t| ≥ M 1 , ∀k ∈ Z(1, T). (52) We apply a similar method in [18] to look for nontrivial critical points of J in S.