Existence of infinitely many solutions for a class of difference equations with boundary value conditions involving p(k)-Laplacian operator

The existence of infinitely many solutions was investigated for an anisotropic discrete non-linear problem involving p(k)-Laplacian operator with Dirichlet boundary value condition. The technical approach is based on a local minimum theorem for differentiable functionals in finite dimensional space.


Introduction
One of the reasons for the huge development of the theory of difference equations is the inclusion of a great number of applications in different fields of research, such as mechanical engineering, control systems, economics, social sciences, computer science, physics, artificial or biological neural networks, cybernetics and ecology. There seems to be increasing interest in the existence of results to boundary value problems for finite difference equations with p(k)-Laplacian operator because of their applications in many fields. Results on this topic are usually achieved using various fixed point theorems in cone; see Avci (2016), Avci and Pankov (2015), and Liu and Ge (2003) and references therein for details. Another tool in the study of non-linear difference equations is upper and lower solution techniques; see, for instance, Chu and Jiang (2005), Henderson and Thompson (2002) and references therein. It is well known that critical point theory is an important tool to deal with the problems for differential equations. More, recently, in Bonanno andD'Aguì (2010), Chu and Jiang (2005), Candito and Giovannelli (2008), Khaleghi Moghadam and Avci (2017), Khaleghi Moghadam, Heidarkhani, andHenderson (2014), Khaleghi Moghadam and Henderson (2017), andKhaleghi Moghadam, Li, andTersian (2018) by starting from the seminal paper Agarwal, Perera, and O'Regan (2005), the existence and multiplicity of solutions for non-linear discrete boundary value problems have been investigated by adopting variational methods.

PUBLIC INTEREST STATEMENT
Theory of difference equations is the inclusion of a great number of applications in different fields of research, such as mechanical engineering, control systems, economics, social sciences, computer science, physics, artificial or biological neural networks, cybernetics and ecology. There is increasing interest in the existence of infinitly many solutionsto boundary value problems for finite difference equations with p(k)-Laplacian operator and their applications in many fields.
The main goal of the present paper is to establish the existence of infinitely many solutions for the following discrete anisotropic problem for any k ∈ [1, T], where T is a fixed positive integer, [1, T] is the discrete interval {1, ..., T}, f :[1, T] × ℝ → ℝ is a continuous function, > 0 is a parameter and w:[0, T] → [1, ∞) is a fix function and Δu(k) = u(k + 1) − u(k) is the forward difference operator and the function p:[0, T + 1] → [2, ∞) is bounded and the function q:[0, T + 1] → [1, ∞) is bounded, we denote for short We want to remark that problem (1.1) is the discrete variant of the variable exponent anisotropic problem where Ω ⊂ ℝ N , N ≥ 3 is a bounded domain with smooth boundary, f ∈ C(Ω × ℝ, ℝ) is a given function that satisfies certain properties and p i (x), w i (x) ≥ 1 and q(x) ≥ 1 are continuous functions on Ω with 2 ≤ p i (x) for each x ∈ Ω and every i ∈ {1, 2, ⋯ , N}, > 0 is real number.
In this article, in the framework of variational methods, we look for the existence of infinitely many solutions to problem (1.1) based on a recent local minimum theorem obtained (Theorem 2.1) which is given in finite dimensional spaces in Bonanno and Candito (2014) due to Bonanno, Candito and D'Aguì. We ensure exact intervals of the parameter , in which the problem (1.1) admits infinitely solutions.
In this article, after presenting a main tools theorem (Theorem 2.1) and an applicable lemma (Lemma 2.1), we present a lemma ( Lemma 2.2) which is fundamental to our aims where lies in a well-defined half-line. Bearing in mind a fundamental lemma, we obtain our results where the existence of an unbounded sequence of solutions (Theorem 3.1) converges to infinity depending on the non-linear term having suitable behaviours at infinity. Moreover, we also emphasize that by strong maximum principle, if f is non-negative and f (k, 0) = 0 for all k ∈ [1, T], our results guarantee infinitely many positive solutions (Remark 3.3).
Further, as an example, we point out a special case of our main results with respect to Theorem 3.1, in the following theorem.  T+1] q(k), Then, for any the problem admits an unbounded sequence of solutions u n such that The local minimum theorem (Theorem 2.1) due to Bonanno, Candito and D'Aguì (2014) is also successfully employed to the existence of infinitely solutions for two-point boundary value problems in   Salari, Caristi, Barilla, and Puglisi (2000).
The remainder of this paper is arranged as follows. In Section 2, we recall the main tools (Theorem 2.1) and give some basic knowledge. In Section 3, we state and prove our main results of the paper that contains several theorems and corollaries, and prove a special case of our main result (Theorem 1.1) and illustrate the results by giving concrete examples as applications to (1.1).

Preliminaries
Our main tool is the following infinitely many critical points theorem. Assume that: (H) Let (X, ‖ ⋅ ‖) be a real finite dimensional Banach space and let Φ, Ψ:X ⟶ ℝ be two continuously Gateaux differentiable functionals with Φ coercive and such that Put for all r > 0, Bonanno, Candito, & DAgu'i, 2014). The following property holds: Assume that ∞ < +∞ and for each ∈]0, 1 ∞ [ the function I = Φ − Ψ is unbounded from below. Then, there is a sequence {u n } of critical points (local minima) of I such that lim n→+∞ Φ(u n ) = +∞.
Remark 2.2 Theorem 2.1 is the finite dimensional version of [Bonanno, 2012, Theorem 7.4] (see also [Ricceri, 2000, Theorem 2.3] and observations in Remark 3.1). Since W is finite-dimensional, we can also define the following equivalent norm on W Now, let :W → ℝ be given by the formula In the sequel, we will use the following inequalities. To study the problem (1.1), we consider the functional I , :W → ℝ defined by We want to remark that since problem (1.1) is settled in a finite-dimensional Hilbert space W, it is not difficult to verify that the functional I satisfies the regularity properties. Therefore, I is of class C 1 on W (see, e.g., Jiang and Zhou, (2008)) with the derivative for all u, v ∈ W.
It is clear that the critical points of I and the solutions of the problem (1.1) are exactly equal.
Now we give two lemmas and the following notation. Put and Lemma 2.4 If 0 < B ∞ , then I is unbounded from below for each ∈] A p − B ∞ , +∞[.
Proof Fix l such that B ∞ > l > A p − and let d n be a sequence of positive numbers, with lim n→∞ d n = +∞, such that for each n ∈ ℕ large enough. Set Clearly, w n ∈ W. Bearing in mind p − ≤ p(k) ≤ p + , we obtain from lim sup n→∞ Therefore, (2.7) (k, u). Thus, one has that is, lim n→+∞ I (w n ) = −∞.

First, put
We state our main result as follows. Proof Our aim is to apply Theorem 2.1 to our problem. To this end, first, we observe that due to 0 ≤ B ∞ < ∞, the interval Λ is non-empty, so fix in Λ.
To settle the variational framework of problem (1.1), take X = W, and put Φ, Ψ as defined in (2.6), for every u ∈ W. Again, because W is finite dimensional, an easy computation ensures that Φ and Ψ are of class C 1 on W with the derivatives
Standard arguments show that I : = Φ − Ψ ∈ C 1 (W, ℝ) as well as that critical points of I are exactly the solutions of the problem (1.1).