Variational Approaches for Lagrangian Discrete Nonlinear Systems

In this paper, we study the multiple solutions for Lagrangian systems of discrete second-order boundary value systems involving the discrete p-Laplacian operator. The technical approaches are based on a local minimum theorem for differentiable functionals in a finite dimensional space and variational methods due to Bonanno. The existence of at least one solution, as well as three solutions for the given system are discussed and some examples and remarks have also been given to illustrate the main results.


Introduction
In recent years, equations involving the discrete p-Laplacian operator, subjected to classical conditions, have been studied by many authors using various techniques.The variational method appears to be a very fruitful one.In this direction, we mention Refs.[1][2][3][4][5][6][7][8].In [3][4][5], using a variational approach, the authors obtained the existence of periodic solutions for systems involving a general discrete φ-Laplacian operator.In addition, in recent years, boundary value problems with discrete p(•)-Laplacian have been studied (we refer the reader to [1,7]).Existence results for the discrete p(.)-Laplacian equations subjected to a general potential type boundary condition are obtained in [1] using Szulkin's critical point theory [1].By mountain pass type arguments and the Karush-Kuhn-Tucker theorem, in [9], the existence of at least two positive solutions in the case of Dirichlet boundary conditions are established.
In all the aforementioned papers, discrete boundary value problems involving a variety of operators and boundary conditions are studied in a variational framework.The solutions are seen as critical points of a convenient energy functional, defined on a function space.In general, such function spaces have a finite dimension, which makes things easier (in comparison with the variational methods for differential equations).
There seems to be increasing interest in the existence of solutions to boundary value problems for finite difference equations with the p-Laplacian operator.This is as a result of their applications in many fields.Recently, difference equations have attracted the interest of many researchers since they provide a natural description of several discrete models.Such discrete models are often investigated in various fields of science and technology such as computer science, economics, neural networks, ecology, cybernetics, optimal control and population dynamics.These studies cover many branches of difference equations, such as stability, attractiveness, periodicity, oscillation, and boundary value problems (see [1][2][3][4][5][6][7][8] and the references therein).This article, using variational methods, aims at studying the existence of multiple solutions for the Lagrangian discrete boundary-value system of second-order difference equations involving the discrete p-Laplacian operator having the following form: where φ p : R −→ R N , for all p ∈ (1, ∞) is a homeomorphism given by the and ∆ p is the discrete p-Laplace operator In this case, T > 1 is a fixed positive integer, [1, T] Z is a discrete interval {1, • • • , T}, and the potential where Inspired by the above works, in this article, we discuss the existence of multiple solutions for the second-order discrete Lagrangian boundary value system with a real parameter.The main tool used in ensuring the existence of multiple non-trivial solutions to the system in Equation ( 1) is a version of Ricceri's variational principle [6].We establish the existence of a precise interval Λ such that for every λ ∈ Λ, the system in Equation (1) admits one nontrivial solution, which is in the space W and is introduced below.
In detail, using the local minimum theorem (see Theorem 1) the existence of at least one nontrivial solution of Equation ( 1) is proven.Under suitable conditions and Theorem 2, we get the existence of at least three solutions.To prove the main result, we introduce some suitable hypotheses.In Theorem 3, we establish the existence of at least one nontrivial weak solution for the system in Equation (1).In Theorem 4, we prove the existence of at least three solutions of the system in Equation (1).

Preliminaries and Basic Notations
Our main tool to investigate the existence of multiple solutions for the system in Equation ( 1) is a smooth version of Theorems 1 and 2, consequences the existence result of the a local minimum ( [10], Theorem B) is used).Theorem 1 ([7], Theorem 2.1).Let X be a finite-dimensional Banach space, and J, Ψ : X −→ R be lower semi-continuous functional, with J a coercive and J(0) = Ψ(0) = 0. Further, set J − λΨ.Then, for each r > inf u∈X Ψ(u) and each λ satisfying Then, for each the restriction of I λ to J −1 ((−∞, r]) has a global minimum.
Theorem 2 ([10], Theorem B).Let X be a reflexive real Banach space, J : X → R a continuously Gâteaux differentiable functional and sequentially weakly lower semicontinuous whose Gâteaux derivative admits a continues inverse on X * , Ψ : X → R a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact.Assume that there exist r > 0 and x ∈ X with r < J(x) such that for each r > inf X J, where J −1 ((−∞, r)) w is the closure of J −1 ((−∞, r)).Then, for each , the functional I is coercive and the equation J (u) − λΨ (u) = 0, has at least three critical points in X.
For a given positive integer T, we define the T-dimensional Banach space where W is equipped with the norm Then, one knows Furthermore, via [11], one gets Form Equation (3), we get In fact, the norm of the space W is but we use the notation of Equation ( 2) since they are equal (see [11]).
Remark 1.Since W is a finite dimensional Banach space, it is reflexive Banach space with the norm is given in the relation in Equation (2).Now, to show that the inclusion i : W −→ C is a compact operator, for this end, we suppose that u n is a sequence in W and since W is finite and from (u m ) has a bounded subsequence (u m k ) in W and since W ⊆ C, thus C the subsequent (u m k ) is also in C thus, the operator i is compact.Since Then, from [11], there exists a positive constant α such that Now, define J, Ψ : W −→ R as follows: and define the functional I : W −→ R as follows: It is clear that the global minimum (local minimum) of I λ are exactly the solutions of the problem in Equation (1).

Existence of at Least One Nontrivial Solution
In the following, by using the conditions of Theorem 1, we prove that the system in Equation ( 1) has at least one nontrivial weak solution.
(H 1 ) Suppose that H : R N −→ R is a strictly monotone Lipschitz continuous function of order p with p > 1 and Lipschitazian constant L satisfying 0 < L < 1 2 , i.e., and Theorem 3. Assume that (H 1 ) holds and suppose that there is a positive real vector − → and positive constant α with 0 < α < 1 and T p−1 > pL(T − 1) 2p−1 , such that the following condition is satisfied.
Then, the system in Equation (1) has at least one nontrivial solution.
Proof.To apply Theorem 1 to our problem, let us prove that the functionals J and Ψ satisfy the required conditions in Theorem 1. From 2, we can get that the functionals J and Ψ are Gâteaux differentiable function.
Since J and Ψ are continuous and since (every continuous real valued function on W is lower semi-continuous), they are lower semi-continuous, and since W is finite dimensional, they are weakly lower semi-continuous, thus it follows that the functional J − λΨ is lower semi-continues in W. Now, we want to show that the functional J is coercive on W, taking into account the relations in Equations ( 4) and ( 5) and supposing that for any sequence ( − → u m ) ∈ W such that Suppose that Now, we want to show that zero is not local minimum for the functional I λ ; to this end, we claim that the mapping λ −→ I λ ( − → u ) is negative.From the definition of Λ, we observe that Now, suppose that τ > 0 such that Moreover, since − → w ∈ W defined as Moreover, since − → w ∈ J −1 ((0, r)), we have since λτ < 1, it follows that the functional I λ ( − → u ) is negative and thus zero is not a local minimum for the functional I λ ( − → u ).
Therefore, the assertion of Theorem 1 follows and the existence of one solution − → u ∈ Ψ −1 ((−∞, r)) to our problem is established.
for every λ ∈ (0, 5.768) the system in Equation (11) has at least one nontrivial solution by Theorem 3.

Existence of Three Solutions
In this section, our goal is to obtain the existence of three distinct weak solutions for the problem in Equation (1).The following result is obtained by applying Theorem 2. We introduce the suitable hypothesis for calculating of the critical points of the system in Equation ( 1) and give some auxiliary lemmas used in the proof of the main results.Lemma 1.The functional J is sequentially weakly lower semi-continuous.
Proof.From the continuity of H, we observe that the functional J Gâteaux differentiable whose Gâteaux derivative of the point − → u ∈ W is the functional J ( − → u ) ∈ W * given by for every − → v ∈ W We can assert that J is sequentially weakly lower semi-continuous.As a matter of fact, owing to Since the inner product is sequentially weakly lower semi-continuous in Banach space, we have lim inf Lemma 2. The functional Ψ is a compact operator.
Proof.We want to prove that the Gâteaux derivative of Ψ is compact operator.Indeed, it is enough to show that Ψ is strongly continuous on W. Let ( − → u m ) be a bounded sequence in W. Since W is reflexive and since the embedding (W, 3), the functional Ψ belongs to W. By Equation ( 3), the following inequality holds Using the Lebesgue dominated convergence theorem, we conclude that J ( − → u m ) converge to J (u) in W * , thus J is compact operator.such that the following conditions are satisfied.
Then, for each , the system in Equation (1) at least three nontrivial weak solutions.
Proof.For each − → u ∈ W, the functionals J, Ψ : W −→ R are given by Equations ( 6) and ( 7).Now, we set the functional To apply Theorem 2 to our problem, let us prove that the functionals J,Ψ satisfy the required conditions in Theorem 2.
It follow from Lemma 1 that the functional J is sequentially weakly upper semi-continuous.Using Lemma 2, we get that the functional Ψ is a compact operator.
Taking (H 5 ) into account and from (H 4 ) there exists ε > 0 such that lim sup Then, there exists a positive constant θ ε such that To prove the other conditions of Theorem 2, for each r > 0 and for any − → u ∈ W. In fact, taking into account that J −1 ((−∞, r)) w = J −1 ((−∞, r]) and by the definition of r, it follows that By considering the above computations and since − → 0 ∈ J −1 (−∞, r) and and It is clear that − → v ∈ W and ).
Remark 5. We observe that, in our results, no asymptotic conditions on F is needed and only algebraic conditions on F are imposed to guarantee the existence of solution.Moreover, in the conclusions of the above results, one of the three solutions may be trivial since the values of F(t, 0, • • • , 0) for t ∈ [1, T] Z are not determined.Remark 6. Scalar case.As an application of Theorems 3 and 4, we consider the following problem: where φ : R −→ R is a homeomorphism such that φ(0) = 0, ∆ denotes the forward difference operator defined by ∆u All amputations of 3 and 4 are satisfied for the scalar case.
We here present the following consequence of Theorems 3 and 4.
Author Contributions: All the authors contributed equally.