Multiple Solutions for the Discrete p-Laplacian Boundary Value Problems

for all s ∈ R. HereΔ is the forward difference operator defined byΔu(t) = u(t+1)−u(t) for all t ∈ Z, whilef : [1, T]×R → R is a continuous function. λ is a nonnegative parameter. Boundary value problems for difference equations have been extensively studied; see the monographs [1–5]. Difference equations represent the discrete counterpart of ordinary equations, and the classical theory of difference equations employs numerical analysis and features from the linear and nonlinear operator theory, such as fixed point methods; for an exhaustive description of the subject, we refer the reader to the monographs of Agarwal [1], Kelley and Peterson [6], and Lakshmikantham and Trigiante [3]. We remark that, usually, most methods yield existence results for solutions of a difference equation. The issue of multiplicity of solutions can be investigated through variational methods. Recently, many results have been established by applying variational methods. For example, we may recall here the works of Cabada et al. [7, 8], Cai and Yu [9], Guo and Yu [10], and Deng et al. [11]. In all the aforementioned papers, the variational methods are applied to difference equations on discrete intervals and the variational approach represents an important advance as it allows proving multiplicity results as well. In [7], the authors consider (1) in this form

Boundary value problems for difference equations have been extensively studied; see the monographs [1][2][3][4][5].Difference equations represent the discrete counterpart of ordinary equations, and the classical theory of difference equations employs numerical analysis and features from the linear and nonlinear operator theory, such as fixed point methods; for an exhaustive description of the subject, we refer the reader to the monographs of Agarwal [1], Kelley and Peterson [6], and Lakshmikantham and Trigiante [3].We remark that, usually, most methods yield existence results for solutions of a difference equation.
The issue of multiplicity of solutions can be investigated through variational methods.Recently, many results have been established by applying variational methods.For example, we may recall here the works of Cabada et al. [7,8], Cai and Yu [9], Guo and Yu [10], and Deng et al. [11].In all the aforementioned papers, the variational methods are applied to difference equations on discrete intervals and the variational approach represents an important advance as it allows proving multiplicity results as well.
In [7], the authors consider (1) in this form and give conclusion that there exists  > 0 such that (2) admits at least three solutions under the following assumptions: In this paper, under convenient assumptions on the reaction term , we not only give the result that (1) admits at least three distinct solutions but also find the two open intervals which  lies in and make estimation of the norm of .
The rest of the paper has the following structure.In Section 2 we introduce the variational framework for problem (1) and transfer the existence of solutions of boundary value problem (1) into the existence of critical points of the corresponding functional.Employing the critical point theorem of Bonanno [12], we state our main results and give proofs of the main results in Section 3. Finally, we exhibit a simple example to demonstrate the applicability of our results.

Variational Framework
This section is devoted to the formulation of a variational framework for (1).We are going to define a suitable Banach space (, ‖ ⋅ ‖) and an energy functional  ∈ C 1 (), such that critical points of  in  are exact solutions of (1).
We define the real vector space and, for every  ∈ , we denote So (, ‖ ⋅ ‖) is a reflexive Banach space and dim() = .We also put, for every  ∈ , By classical results, the norms ‖ ⋅ ‖ and ‖ ⋅ ‖ ∞ are equivalent on ; then there exists a constant  > 0 such that For later use, here we give the estimation of constant .
From Lemma 2, we easily get the following lemma which yields a variational formulation for (1).Lemma 3.For all  > 0, every critical point  ∈  of () is a solution of (1).
Proof.Fix  > 0 and assume that  ∈  is a critical point of ; then From Lemma 2, this is equivalent to this is exactly (1).So,  solves (1); that is,  is in fact a solution of (1).

Main Results
In order to present our main results, we introduce some notations at first.Denote  = max (,)∈[1,]×[−,] (, ); then write and, for each ℎ > 1, Now, we state the following convenient assumptions on the function .
We need the following assumptions.
To give proofs of our results, we need the following three critical point theorems, established by Bonanno [12].
For the reader's convenience, we recall the definition of weak closure.
Suppose that  ⊂ .We denote   as the weak closure of , that is,  ∈   , if there exists a sequence {  } ⊂  such that (  ) → () for every  ∈  * .
Then for each With the aid of Lemma 11, we devote ourselves to giving proof of Theorem 5.
Proof of Theorem 5. Let space  be the Banach space .It follows from ( 8) that Φ is a nonnegative Gâteaux differentiable and weakly lower semicontinuous functional, whose Gâteaux derivative admits a continuous inverse on  * , and let  :  → R be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact.Moreover, Φ(0) = (0) = 0.
Similar to the proof of Theorem 5, we can give proof of Theorem 9; here we omit it.
Finally, we exhibit a simple example to demonstrate the applicability of our results.
Equation (29) admits at least three solutions in  and, moreover, there exist an open interval Λ 4 ⊆ [0, b] and a positive real number  such that, for each  ∈ Λ 4 , (29) admits at least three solutions in  whose norms in  are less than .