Existence of Homoclinic Solutions for Difference Equations on Integers Via Variational Method

{ −∆(a(k)φp(∆u(k − 1))) + b(k)φp(u(k)) = λf(k, u(k)) + μg(k, u(k)) + h(u(k)), k ∈ Z, u(k) → 0 as |k| → ∞ (P f,g,h λ,μ ) where 1 < p < +∞, λ > 0, μ ≥ 0, φp(t) = |t|p−2t for all t ∈ R, a, b : Z → (0,∞), f, g : Z×R → R are two continuous functions in the second variables, h : R → R is a Lipschitz continuous function of order p− 1 with Lipschitzian constant L ≥ 0 such that h(0) = 0, ∆u(k) = u(k + 1) − u(k) is the forward difference operator. A solution u = u(k) of (P f,g,h λ,μ ) is homoclinic if lim|k|→∞ u(k) = 0. Also we define the following conditions:


Introduction
The aim of this paper is to establish the existence of homoclinic solutions for the following discrete boundary value problem −∆(a(k)φ p (∆u(k − 1))) + b(k)φ p (u(k)) = λf (k, u(k)) + µg(k, u(k)) + h(u(k)), k ∈ Z, u(k) → 0 as |k| → ∞ (P f,g,h λ,µ ) where 1 < p < +∞, λ > 0, µ ≥ 0, φ p (t) = |t| p−2 t for all t ∈ R, a, b : Z → (0, ∞), f, g : Z × R → R are two continuous functions in the second variables, h : R → R is a Lipschitz continuous function of order p − 1 with Lipschitzian constant L ≥ 0 such that h(0) = 0, ∆u(k) = u(k + 1) − u(k) is the forward difference operator. A solution u = u(k) of (P f,g,h λ,µ ) is homoclinic if lim |k|→∞ u(k) = 0. Also we define the following conditions: There are many papers about existence of solutions to boundary value problems for finite difference equations with p-Laplacian operator which branching out in many fields such as biologic, economic, farm and other areas. There are various methods such fixed point, variational methods, critical point theory, Morse theory and the mountain-pass theorem. For background and recent results, we refer the reader to [1,2,3,4,5,6,7,8,12,13,19,20,22,23,24,26,27,28,29,30,32,33]. For example, Henderson and Thompson investigated existence multiple solutions for second order discrete boundary value problems in [19]. Wong and Xie proved three symmetric solutions of lidstone boundary value problem for partial equation in [33]. In [12] Cabada and Tersian studied the existence of homoclinic solutions for semilinear p-Laplacian difference equations with periodic coefficients based on the Brezis-Nirenberg's mountain pass theorem. In [13] Candito and D'Aguì studied discrete nonlinear Neumann problems to find three solutions. In addition, Iannizzotto and Tersian in [20] via critical point theory investigated the existence of multiple homoclinic solutions for the discrete p-Laplacian problem −∆(φ p (∆u(k − 1)) + a(k)φ p (u(k)) = λf (k, u(k)), ∀k ∈ Z, u(k) → 0 as |k| → ∞ where p > 1 is a real number, φ p (t) = |t| p−2 t for all t ∈ R is a positive and coercive weight function and f : Z × R → R is a continuous function in the second variable. In [23] Kong studied the problem (P f,g,h λ,µ ), in the case µ = 0 and h ≡ 0 and proved the problem has infinitely many homoclinic solutions. Also, a variant of the fountain theorem was utilized. Kong in [22] studied the following higher order difference equation defined on Z with p-Laplacian where n > 1 is an integer, p > 1 is a real number, λ > 0 is a parameter, , and f : Z × R 3 → R is continuous in the second, third, and fourth variables. By using the critical point theory, sufficient conditions were obtained for the existence of infinitely many homoclinic solutions of the problem (1.2), based on the fountain theorem in combination with the variational technique. Stegliński in [28] determined a concrete interval of positive parameter λ, for prove the existence of homoclinic solutions for the problem (1.1), while in [30] dealt with the problem (P f,g,h λ,µ ), in the case µ = 0 and h ≡ 0, and using both the general variational principle of Ricceri and the direct method introduced by Faraci and Kristály proved the existence of infinitely many homoclinic solutions for a the problem where the nonlinear term f has an appropriate oscillatory behavior at zero. In [4], sufficient conditions for the existence of at least one homoclinic solution for a nonlinear second-order difference equation with p-Laplacian were presented. Inspired by the above results, in the present paper, we obtain the existence of at least three distinct nonnegative solutions for the problem (P f,g,h λ,µ ), in which two parameters are involved. Estimation of these two parameters λ and µ will be given. In particular, in Theorem 3.1 we establish the existence of at least three distinct nonnegative solutions for the problem (P f,g,h λ,µ ). Theorem 3.4 is a consequence of Theorem 3.1. The Examples 3.3 and 3.5 help us to illustrate our main results. In Example 3.3 the hypotheses of Theorem 3.1 are fulfilled and in Example 3.5 the condition of Theorem 3.4 are satisfied. As a special case of Theorem 3.4, we obtain Theorem 3.6 which under suitable conditions on f at zero and at infinity, ensures two positive solutions for the autonomous case of the problem. Finally, by the way of example, we point out Theorem 3.7, as simple consequence of Theorem 3.6.

Preliminaries
In this paper X denotes a finite dimensional real Banach space and as is shown in [20] (X, . ) is a reflexive Banach space and the embedding X ֒→ ℓ p is compact and I λ : X → R is a functional satisfying the following structure hypothesis: I λ (u) := Φ(u) − λΨ(u) for all u ∈ X where Φ, Ψ : X → R are two functions of class C 1 on X with Φ coercive, i.e. lim u →∞ Φ(u) = +∞, and λ is a positive real parameter.
In this framework a finite dimensional variant of [8,Theorem 3.3 ] (see also Corollary 3.1 and Remark 3.9 of [8]) is the following: Let X be a nonempty set and Φ, Ψ : X → R be two functions. For all r 1 , r 2 , r 3 , with r 2 > r 1 and r 2 > inf X Φ, and all r 3 > 0, we define Assume that there are three positive constants r 1 , r 2 , r 3 with r 1 < r 2 , such that Then, for each λ ∈] 1 β(r1,r2) , We refer the interested reader to the papers [11,14,15,16,17,18,21] in which Theorem 2.1 has been successfully employed to get the existence of at least three solutions for boundary value problems.
For Banach space X and the norm explained above the following inequality is obvious where (A 1 ) is satisfied. Let two functions f, g : Z × R → R be two continuous functions in the second variables and h : R → R be a Lipschitz continuous function of order p − 1 with Lipschitzian constant Corresponding to the function f, g and h, we introduce F, G : Z × R → R and H : R → R, respectively, as follows

Main results
Set If g is sign-changing, then G θ ≥ 0 and G η ≤ 0.

Remark 3.2.
If either f (k, t) = 0 for some k ∈ Z or g(k, t) = 0 for some k ∈ Z, or both hold true the solutions of Theorem 3.1 are not trivial. Now, we present the following example in which the hypotheses of Theorem 3.1 are satisfied.

by the definition of F and H its obvious
For positive constants θ 1 , θ 2 and η, put We, now deduce the following consequence of Theorem 3.1.

Theorem 3.4.
Assume that there exist three positive constants θ 1 , θ 4 , η and the integer k 0 which we put Then, for every and for every nonnegative function g : Z × R → R, there is δ ′ λ,G > 0 given by (3.8) such that for each µ ∈ [0, δ ′ λ,G ), the problem (P f,g,h λ,µ ) has at least three nonnegative solutions u 1 , u 2 , u 3 such that Proof. We put θ 2 = 1 p √ 2 θ 4 and θ 3 = θ 4 . So from (A 5 ) we get From (A 5 ) and taking into account θ 1 < η we have Hence, from (3.9), (3.10), and (A 5 ), the assumption (A 3 ) of Theorem 3.1 is satisfied, and since the critical points of function φ − λψ are the solutions of the problem (P f,g,h λ,µ ) we have the conclusion. Now, we present the following example in which the conditions of Theorem 3.1 are satisfied.

Example 3.5. Consider the problem
as |k| → ∞, where a(k) = 10 −k for every k ∈ Z and b(k) = | cos kπ| and also h(t) = 1 1 + t 2 − 1 for t ∈ R, so by the definitions of F and H it is obvious and such that F (θ 1 ) Finally, we point out the following simple consequence of Theorem 3.4 when µ = 0 and h ≡ 0. Then, for every λ > λ * * where