Existence and multiplicity of solutions for a Dirichlet problem involving the discrete p(x)-Laplacian operator

In the present paper, using the three critical points theorem and variational method, we study the existence and multiplicity of solutions for a Dirichlet problem involving the discrete p(x)−Laplacian operator.


INTRODUCTION
In this paper, we study the existence and multiplicity of solutions of the discrete boundary value problem where f (k, t) = |t| q(k)−2 t − |t| s(k)−2 t for all t ∈ R + and k ∈ Z[1, T ], T ≥ 2 is a positive integer, λ is a positive constant and ∆u(k) = u(k + 1) − u(k) is the forward difference operator.Here and hereafter, Z [a, b] denotes the discrete interval {a, a + 1, ..., b} with a and b are integers such that a < b.Moreover, we assume that functions p : Z [0, T ] → [2, ∞) and q, s : Z [1, T ] → [2, ∞) are bounded and we denote It is well known that in various fields of research, such as computer science, mechanical engineering, control systems, artificial or biological neural networks, economics and many others, the mathematical modeling of important questions leads naturally to the consideration of nonlinear difference equations.For this reason, in recent years, many authors have widely developed various methods and techniques, such as fixed points theorems or upper and lower solutions methods, to study discrete problems [7].In order to support the above ideas we refer the reader to [2,3,4,8,10,14,17,23,24] and the reference there in.
In [16], the author studied an elliptic equation with nonstandard growth conditions and the Neumann boundary condition.He established the existence of at least three solutions by using as the main tool a variational principle due to Ricceri.
In [17], the authors studied the problem where T ≥ 2 is a positive integer, the functions p : are bounded and λ is a positive constant.By using critical point theory, they showed the existence of a continuous spectrum of eigenvalues for the problem (1.2).
In [15], the authors dealt with the following problem which is a generalization of (1.2) where T ≥ 2 is a positive integer, ∆u(k) = u(k + 1) − u(k) is the forward difference operator and a (k, ξ) = |ξ| p(k)−2 ξ such that k ∈ Z [0, T ] and ξ ∈ R. By using critical point theory, the authors proved the existence and uniqueness of weak solutions for the problem (1.3).For the case p(x) = p = const, the authors investigated the following problem where [9]).By using critical point theory, they showed the existence of multiple solutions for the problem (1.4).This paper is organized as follows.In Section 2, we present some necessary preliminary results.In Section 3, using three critical points theorem and the variational method we show the existence and multiplicity of solutions of problem (P).

PRELIMINARIES
Let W be the function space (see [3]).The associated norm is defined by On the other hand, it is useful to introduce other norms on W , namely In [10] it is verified that (2.1) Therefore, since u(0) = u(T + 1) = 0, a straightforward computation gives and by using the discrete Hölder inequality, we have that max Lemma 2.1. [17] (a) There exist two positive constants C 1 and C 2 such that (b) There exists a positive constant C 3 such that (c) For any m ≥ 2 there exists a positive constant c m such that Theorem 2.1. [6]Let X be a separable and reflexive real Banach space; Φ : X → R a continuously Gâteaux differentiable and sequentially weakly lower semi-continuous functional whose Gâteaux derivative admits a continuous inverse on X * ;Ψ : X → R a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact.Assume that, (i) lim . EJQTDE, 2011 No. 67, p. 3 Then there exist an open interval Λ ⊂ (0, ∞) and a positive real number ρ such that for each λ ∈ Λ the equation Φ ′ (u) + λΨ ′ (u) = 0 has at least three solutions in X whose norms are lees than ρ.

AUXILIARY RESULTS
For any λ > 0 the energy functional corresponding to problem (P) is defined as and From the standard arguments J λ ∈ C 1 (W, R) and its derivative is given by for any u, v ∈ W .
Definition 3.1.We say that u ∈ W is a weak solution of (P) if where ϕ ∈ W . Hence, the critical points of functional J λ are the weak solutions for problem (P).
Proof of Theorem 3.1.
(i) First, we point out that Thus, we have Using the above inequality, we obtain Now, consider the case u > 1 for u ∈ W .Then, using Lemma 2.1 (c) and relation (2.1), we get that For the case u > 1 for u ∈ W , using Lemma 2.1 (a), the relation (2.1) and (3.3), the inequality (3.2) imply that and hence, Theorem 2.1 (i) is verified.
In the following, we show that Theorem 2.1 (ii) and Theorem 2.1 (iii) hold. Define It is clear that H(k, t) and H t (k, t) are continuous for all k ∈ Z [1, T ] and satisfies the following equality EJQTDE, 2011 No. 67, p. 5 Thus, H t (k, t) ≥ 0 for all t ≥ 1 and all k ∈ Z [1, T ] and H t (k, t) ≤ 0 for all t ≤ 1 and all k ∈ Z [1, T ].It follows that H (k, t) is increasing for t ∈ (1, ∞), and decreasing for t ∈ (0, 1).Moreover, lim t→∞ Using the above information, we get that there exists δ > 1 such that Let a, b be two real numbers such that 0 < a < min 1, We also define r = . Clearly, r ∈ (0, 1).A simple computation implies Thus, we obtain Φ (u 0 ) < r < Φ (u 1 ) .
On the other hand, we have > 0. Now, we investigate the case Φ (u) ≤ r < 1 for u ∈ W .Then, using Lemma 2.1 (b) and for , we have ≤ r < 1.
EJQTDE, 2011 No. 67, p. 6 Thus, using Remark 2.1, for any u ∈ W with Φ (u) ≤ r we obtain that where r < 1 p + .The above inequality means that Therefore, we obtain , Consequently, all the assumptions of Theorem 2.1 are satisfied.Therefore, we conclude that there exists an open interval Λ ⊂ (0, ∞) and constant ρ > 0 such that for any λ ∈ Λ, the equation Φ ′ (u) + λΨ ′ (u) = 0 has at least three solutions in W whose norms are less than ρ.The proof of Theorem 3.1 is complete.
Proof.It is obvious that f (k, t) = |t| q(k)−2 t − |t| s(k)−2 t ∈ C R 2 , R , for every k ∈ Z [1, T ].Since s − > q + , we get Hence, the condition (g 2 ) holds for f (k, t) = |t| q(k)−2 t − |t| s(k)−2 t.For u ∈ W with u > 1 by Lemma 3.1, we get Since J λ is coercive, continuous ( therefore, weakly lower semi-continuous since W is finite dimensional) and Gâteaux differentiable on W , using the relation between critical points of J λ and problem (P), we deduce Theorem 3.2.holds.

Remark 3 . 1 .Lemma 3 . 1 .
Applying ([5], Theorem 2.1) in the proof of Theorem 3.1, an upper bound of the interval of parameters λ for which (P) has at least three weak solutions is obtained.To be precise, in the conclusion of Theorem 3.H (k, b)   for each h > 1 and b as in the proof of Theorem 3.1 (namely, b > δ is such that H (k, b) > 0).EJQTDE, 2011 No. 67, p. 7In the sequel, we investigate the solutions of the problem (P) under the following conditions, (g 1 ) : g ∈ C R 2 , R , (g 2 ) :there exists η > 0 such that tg(k, t) ≤ 0 for |t| ≥ η for all k ∈ Z [1, T ].Assume that (g 1 ) and (g 2 ) hold.Then,T k=1 G(k, u(k)) ≤ B, where G(k, u(k)) = u(k) 0 g(k, t)dt and B > 0 is a constant.Proof.From (g 1 ) and (g 2 ), for u ∈ W we have , t)|dt = B