On a class of non-local discrete boundary value problem

PurposeIn this article, the authors discuss the existence and multiplicity of solutions for an anisotropic discrete boundary value problem in T-dimensional Hilbert space. The approach is based on variational methods especially on the three critical points theorem established by B. Ricceri.Design/methodology/approachThe approach is based on variational methods especially on the three critical points theorem established by B. Ricceri.FindingsThe authors study the existence of results for a discrete problem, with two boundary conditions type. Accurately, the authors have proved the existence of at least three solutions.Originality/valueAn other feature is that problem is with non-local term, which makes some difficulties in the proof of our results.


Introduction
The non-linear difference equations have been of great interest because of their important applications appearing in various fields of research, such as numerical analysis, non-linear differential equations, computer science, mechanical engineering, control systems, artificial or biological neural networks and social sciences, such as economics. To deal with these kind of problems, a various methods such as fixed points theorems, lower and upper solutions, Browder degree, variational approach and critical point theory have been applied by many different authors. For the recent progress in discrete problems, we refer the readers to valuable monograph by Agarwal [1] and the papers [2,3]. Let T ≥ 2 be a positive integer, ½a; b be the discrete interval fa; a þ 1; . . . ; bg with a and b are integers such that a < b.
In the present paper, we deal with the existence of solutions for the Neumann problem ÀKðI ðuÞÞðΔðw pðk−1Þ ðΔuðk À 1ÞÞÞ þ w pðkÞ ðuðkÞÞÞ ¼ λf ðk; uðkÞÞ þ μgðk; uðkÞÞ; k ∈ ½1; T; where ρ 0 ; ρ; L and h are constants associated to the effects of the changes in the length of strings during the vibrations. It is an extension of the classical D'Alembert's wave equation. A distinguish feature of the above equation is that it contains a non-local coefficient of the kinetic energy 1 2 j vu vx j 2 on [0, L], and hence the equation is no longer a pointwise identity. The study of these problems has received more attention. In [2,[5][6][7][8][9][10][11][12][13][14][15], a variety of different methods were applied to obtain the existence results to the discrete boundary value problem of the following type ÀðΔðw p ðΔuðk À 1ÞÞÞÞ ¼ λf ðk; uðkÞÞ; k ∈ ½1; T; where w p ðxÞ ¼ jxj p−2 .
For example, Jiang and Zhou in [16] employing a three critical point theorem, due to Ricceri, established the existence of at least three solutions for perturbed non-linear difference equations with discrete boundary conditions. Bonanno and Candito [11], employing critical point theorems in the setting of finite dimensional Banach spaces, investigated the multiplicity of solutions for non-linear difference equations involving the p -Laplacian. Cabada et al. in [2], based on three critical points theorems, investigated different sets of assumptions which guarantee the existence and multiplicity of solutions for difference equations involving the discrete p -Laplacian operator. Candito and Giovannelli [12], using variational methods, established the existence of at least three solutions for the problem above. Far from being exhaustive, further details can be found in [13,[17][18][19][20][21][22][23][24].
By taking into account the previous papers and inspired by [25], we study problems (1) and (2) and obtain the existence of three weak solutions by employing a kind of Ricceri's theorem [26]. As for the author's best knowledge, the present papers results are not covered in the related literature, and hence, it is original in its own right.
The structure of this paper is outlined as follows. In Section 2, some preliminary results and statement of main results are presented. In Section 3, the proof of the main results is given.

Preliminaries
Firstly, we recall some basic properties which will be used in the proof of the precise result.
Through the sequel, we say that the functional L ∈ W W if L : W → ℝ possesses the following property: ðu n Þ is a sequence in W converging weakly to u ∈ W and lim inf Lðu n Þ ≤ LðuÞ; then u n has a subsequence converging strongly to u. When W is finite dimensional, the weak convergence coincides with the strong one.
In order to prove our main results, we will use the following Ricceri's theorem.
ℝÞ is coercive and belongs to W W : The derivative of f admits a continuous inverse on W * ; J : W → ℝ a C 1 functional. Assume that f has a strict local minimum u 0 with fðu 0 Þ ¼ J ðu 0 Þ ¼ 0: Finally, setting assume that fðuÞ ; assume that α < β: , there exists R > 0 with the following property: for every λ ∈ ½a; b and every C 1 functional ψ : W → ℝ with compact derivative, there exists δ > 0 such that, for each μ ∈ ½0; δ; f 0 ðuÞ ¼ λJ 0 ðuÞ þ μψ 0 ðuÞ has at least three solutions in W whose norms are less than R: Denoting by F; G : ½1; T → ℝ the primitives of f and g; i.e., Solutions to (1) will be investigated in a space which is a T-dimensional Hilbert space, see [5], associated with the norm It can be verified that for all u ∈ W ; one has and We list also some inequalities that will be are used later.
We say that u ∈ W is a weak solution of problem (1)  KðsÞds: Let C 1 and C 2 be positive constants such that We make the following assumptions. ðF 1 Þ The following inequality holds Now, we provide an example of non-linear term which satisfies ðF 1 Þ: Example: Set f ðk; uÞ ¼ juj αðkÞ−1 − juj βðkÞ−2 for all ðk; uÞ ∈ ½1; T 3 ℝ where α and β are bounded functions such that 1 ≤ αðkÞ < βðkÞ; k ∈ ½1; T: Hence, Fðk; uÞ ¼ juj αðkÞ αðkÞ À juj βðkÞ βðkÞ : Non-local discrete boundary value problem There exists ξ u ∈ ½1; T which depends on u such that sup k∈½1; T Fðk; uÞ For juj large enough, we have, Similarly, sup k∈½1;T Fðk; uÞ Since αðkÞ < βðkÞ and p þ < α − then then for each compact interval ½a; b ⊂ ðα * ; β * Þ there exists a number δ > 0 with the following property: for every λ ∈ ½a; b and every g ∈ ℭ there exists μ * > 0 such that, for each μ ∈ ½0; μ * ; problem (1) has at least three weak solutions whose norms are less than δ: Now, suppose that we have: Fðk; uðkÞÞ > 0 : Solutions to (2) will be investigated in a space which is a T-dimensional Hilbert space, see [1], with the inner product Δuðk À 1ÞΔvðk À 1Þ; for all u; v ∈ W : Therefore, the associated norm is defined by Also, it is useful to introduce other norms on W, namely ; ∀u ∈ W and m ≥ 2 : It can be verified (see [15]) that We report our second main result. then for each compact interval Λ ⊂ ðθ * ; ∞Þ there exists a number δ > 0 with the following property: for every λ ∈ Λ and every g ∈ ℭ there exists μ * > 0 such that, for each μ ∈ ½0; μ * ; problem (2) has at least three weak solutions whose norms are less than δ: Non-local discrete boundary value problem which means that ðF 0 1 Þ is satisfied. For function g such that gðk; uÞ ¼ juj γðkÞ u if juj ≤ 1 and gðk; uÞ ¼ juj δðkÞ if juj ≥ 1 where p þ < γðkÞ < δðkÞ for all k ∈ ½1; T; and then f and g verify the hypothesis in Theorem 2.4.

Proof
Proof of Theorem 2.3. It is clear that since the functional w is continuously differentiable on a finite dimensional space its Gâteaux derivative is compact with juðkÞj pðkÞ−2 uðkÞvðkÞ ! for all u; v ∈ W : Since W is a Hilbert space and K is continuous and strictly increasing, it follows that f belongs to the class W W : which means that f is coercive. It is evident that u 0 ¼ 0 is the only global minimum of w and that wðu 0 Þ ¼ J ðu 0 Þ ¼ 0: In view of ðF 1 Þ; there exist 0 < ρ < 1 and R > 1 such that for all k ∈ ½1; T and jξj ≤ ρ ; for all k ∈ ½1; T and jξj ≥ R : From the fact that F is bounded on each subset of ½1; T 3 ℝ; we may choose r − > p þ and a suitable constant c 1 > 0 such that Fðx; ξÞ ≤ ðA þ εÞjξj p þ þ c 1 jξj r − for all ðx; ξÞ ∈ ½1; T 3 ℝ: AJMS Consequently, for kuk < 1 with r − > p þ ; then, using the inequality (5)  ) and β * ¼ ∞: Proof of Theorem 2.4. Let start by defining f as follows: which means that f is coercive. It is evident that u 0 ¼ 0 is the only global minimum of f and that fðu 0 Þ ¼ J ðu 0 Þ ¼ 0 In view of ðF 0 2 Þ; for ε > 0 there exist 0 < ρ < 1 such that Fðx; ξÞ ≤ εjξj p þ for all x ∈ ½1; T and jξj ≤ ρ : From ðF 0 3 Þ there exists R > 1 such that for all x ∈ ½1; T and jξj ≥ R : In view of the fact that F is bounded on each subset of ½1; T 3 ℝ; so we may choose that r − > p þ and for a suitable constant c 1 > 0 such that Fðx; ξÞ ≤ εjξj p þ þ c 1 jξj r − for all ðx; ξÞ ∈ ½1; T 3 ℝ: Consequently, for kuk < 1 with r − > p þ ; then, using the inequality (c) in Lemma 2.2 and the above estimation, we can write lim sup kuk→0 J ðuÞ wðuÞ ≤ C 2 εp þ : On the other hand, for each k u k> 1; it is well known that there exists h ∈ l 1 ð½1; TÞ such that Fðx; tÞ ≤ εjtj p − þ hðxÞ; for all ðx; tÞ ∈ ½1; T 3 ℝ: It yields there is C 1 > 0 such that J ðuÞ ≤ C 1 ðε k uk p − þ 1Þ: (13) Thus, combining (12) and (13), we get lim sup kuk→∞ J ðuÞ wðuÞ ≤ p þ C 2 ε: Since ε is arbitrary and β > 0; α ¼ 0; hence, all the assumptions of Theorem 2.3 are satisfied with θ * ¼ 1 β and the proof is complete. The following corollary is a direct application of Theorem 2.4. Then for each compact interval ½c; d ⊂ ½η; ∞Þ there exists a number ρ > 0 with the following property: for every λ ∈ ½c; d and g in ℭ there exists σ > 0 such that for μ ∈ ½0; σ the problem ÀΔ À jΔuðk À 1Þj pðk−1Þ−2 Δuðk À 1Þ Á ¼ λf ðuðkÞÞ þ μgðk; uðkÞÞ; k ∈ ½1; T: uð0Þ ¼ uðT þ 1Þ ¼ 0; has at least three weak solutions whose norms are less than ρ