On the Solvability of Discrete Nonlinear Two-Point Boundary Value Problems

where T ≥ 2 is a positive integer andΔu k u k 1 −u k is the forward difference operator. Throughout this paper, we denote by Z a, b the discrete interval {a, a 1, . . . , b}, where a and b are integers and a < b. We consider in 1.1 two different boundary conditions: a Dirichlet boundary condition u 0 0 and a Neumann boundary condition Δu T 0 . In the literature, the boundary condition considered in this paper is called a mixed boundary condition. We also consider the function space


Introduction
In this paper, we study the following nonlinear discrete boundary value problem: where T ≥ 2 is a positive integer and Δu k u k 1 −u k is the forward difference operator.Throughout this paper, we denote by Z a, b the discrete interval {a, a 1, . . ., b}, where a and b are integers and a < b.
We consider in 1.1 two different boundary conditions: a Dirichlet boundary condition u 0 0 and a Neumann boundary condition Δu T 0 .In the literature, the boundary condition considered in this paper is called a mixed boundary condition.
We also consider the function space where W is a T-dimensional Hilbert space 1, 2 with the inner product The associated norm is defined by For the data f and a, we assume the following.
H 2 a k, • : R → R for all k ∈ Z 0, T and their exists a mapping A : Z 0, T × R → R which satisfies a k, ξ ∂/∂ξ A k, ξ , for all k ∈ Z 0, T and A k, 0 0, for all k ∈ Z 0, T .
The theory of difference equations occupies now a central position in applicable analysis.We just refer to the recent results of Agarwal et The problem 1.5 is referred as the "semipositone" problem in the literature, which was introduced by Castro and Shivaji 2 .Semipositone problems arise in bulking of mechanical systems, design of suspension bridges, chemical reactions, astrophysics, combustion, and management of natural resources.
The studies regarding problems like 1.1 or 1.5 can be placed at the interface of certain mathematical fields such as nonlinear partial differential equations and numerical analysis.On the other hand, they are strongly motivated by their applicability in mathematical physics as mentioned above.

International Journal of Mathematics and Mathematical Sciences 3
In 11 , Jiang and Zhou studied the following problem: where T is a fixed positive integer, f : Z 1, T × R → R is a continuous function.Jiang and Zhou proved an existence of nontrivial solutions for 1.7 by using strongly monotone operator principle and critical point theory.
In this paper, we consider the same boundary conditions as in 11 but the main operator is more general than the one in 11 and involves variable exponent.
Problem 1.1 is a discrete variant of the variable exponent anisotropic problem where Our goal in this paper is to use a minimization method in order to establish some existence results of solutions of 1.1 .The idea of the proof is to transfer the problem of the existence of solutions for 1.1 into the problem of existence of a minimizer for some associated energy functional.This method was successfully used by Bonanno et al. 27 for the study of an eigenvalue nonhomogeneous Neumann problem, where, under an appropriate oscillating behavior of the nonlinear term, they proved the existence of a determined open interval of positive parameters for which the problem considered admits infinitely many weak solutions that strongly converges to zero, in an appropriate Orlicz-Sobolev space.Let us point out that, to our best knowledge, discrete problems like 1.1 involving anisotropic exponents have been discussed for the first time by Mihȃilescu et where T ≥ 2 is a positive integer and the functions p : Z 0, T → 2, ∞ and q : Z 1, T → 2, ∞ are bounded while λ is a positive constant.In 4 , Koné and Ouaro proved, by using minimization method, existence and uniqueness of weak solutions for the following problem: where T ≥ 2 is a positive integer.
The function a k − 1, Δu k − 1 which appears in the left-hand side of problem 1.1 is more general than the one which appears in 1.9 .Indeed, as examples of functions which satisfy the assumptions H 2 -H 5 , we can give the following.
, for all k ∈ Z 0, T and ξ ∈ R.

1.11
The function a k − 1, Δu k − 1 has the same properties as in 4 , but the boundary conditions are different.For this reason, Guiro et al. defined a new norm in the Hilbert space considered in order to get, by using minimization methods, existence of a unique weak solution which is also a classical solution since the Hilbert space associated is of finite dimension .Indeed, they used the following norm: 1.12 which is associated to the Hilbert space In order to get the coercivity of the energy functional, the authors of 5 assumed the following on the exponent: The assumption above allowed them to exploit the convexity property of the map x → x p − /2 .Problem 1.11 is a discrete variant of the following problem: The remaining part of this paper is organized as follows.Section 2 is devoted to mathematical preliminaries.The main existence and uniqueness result is stated and proved in Section 3. In Section 4, we discuss some extensions, and, finally, in Section 5, we apply our theoretical results to an example.

Preliminaries
From now, we will use the following notations: Moreover, it is useful to introduce other norms on W, namely, We have the following inequalities see 6, 8 which are used in the proof of Lemma 2.1: In the sequel, we will use the following auxiliary result.
for all u ∈ W with u > 1.
We have the following result.

Existence and Uniqueness of Weak Solution
In this section, we study the existence and uniqueness of weak solution of 1.1 .
Note that, since W is a finite dimensional space, the weak solutions coincide with the classical solutions of the problem 1.1 .
We have the following result.
Theorem 3.2.Assume that (H 1 )-(H 5 ) hold.Then, there exists a unique weak solution of 1.1 .The energy functional corresponding to problem 1.1 is defined by We first present some basic properties of J.
Proposition 3.3.The functional J is well defined on W and is of class C 1 W, R with the derivative given by International Journal of Mathematics and Mathematical Sciences 7 The proof of Proposition 3.3 can be found in 5 .
We now define the functional I by We need the following lemma for the proof of Theorem 3.2.
Lemma 3.4.The functional I is weakly lower semicontinuous.
Proof.A is convex with respect to the second variable according to H 2 .Thus, it is enough to show that I is lower semicontinuous.For this, we fix u ∈ W and > 0. Since I is convex, we deduce that, for any v ∈ W,

3.5
We define H and B by

3.6
By using Schwartz inequality, we get

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The same calculus gives Finally, we have We conclude that I is weakly lower semicontinuous.
We also have the following result.
Proposition 3.5.The functional J is bounded from below, coercive, and weakly lower semicontinuous.
Proof .By Lemma 3.4, J is weakly lower semicontinuous.We will only prove the coerciveness of the energy functional since the boundedness from below of J is a consequence of coerciveness.The other proofs can be found in 5 .By H 4 , we deduce that

3.10
To prove the coercivity of J, we may assume that u > 1, and we get from the above inequality and Lemma 2.1, the following:
We can now give the proof of Theorem 3.2.
Proof of Theorem 3.2.By Proposition 3.5, J has a minimizer which is a weak solution of 1.1 .
In order to end the proof of Theorem 3.2, we will prove the uniqueness of the weak solution.
Let u 1 and u 2 be two weak solutions of problem 1.1 , then we have

3.13
Adding the two equalities of 3.13 , we obtain Using H 3 , we deduce from 3.14 that Therefore, by using discrete's Wirtinger inequality, we get

Extension 1
In this section, we show that the existence result obtained for 1.1 can be extended to more general discrete boundary value problem of the form where T ≥ 2 is a positive integer, and we assume that By a weak solution of problem 4.1 , we understand a function u ∈ W such that, for any We have the following result.Proof .For u ∈ W, is such that J ∈ C 1 W; R and is weakly lower semicontinuous, and we have This implies that the weak solutions of problem 4.1 coincide with the critical points of J.We then have to prove that J is bounded below and coercive in order to complete the proof. As Using Proposition 3.5, we deduce that J is bounded below and coercive.
Let u 1 and u 2 be two weak solutions of problem 4.1 , then we have

4.7
Adding these two equalities, we obtain

4.8
We deduce that

Extension 2
In this section, we show that the existence result obtained for 1.1 can be extended to more general discrete boundary value problem of the form where T ≥ 2 is a positive integer, λ ∈ R , and f : Z 1, T × R → R is a continuous function with respect to the second variable for all k, z ∈ Z 1, T × R.
For every k ∈ Z 1, T and every t ∈ R, we put F k, t t 0 f k, τ dτ.By a weak solution of problem 4.11 , we understand a function u ∈ W such that

4.12
We assume the following.Therefore, for u ∈ W, International Journal of Mathematics and Mathematical Sciences 13 is such that J ∈ C 1 W; R and is weakly lower semicontinuous, and we have This implies that the weak solutions of problem 4.11 coincide with the critical points of J.We then have to prove that J is bounded below and coercive in order to complete the proof.
Then, for u ∈ W such that u > 1, where we put λ * C β with C a positive constant.Furthermore, by the fact that 1 < β − < p − , it turns out that Therefore, J is coercive.

Extension 3
We consider the problem where T ≥ 2.

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We suppose the following.f k, u k v k , ∀v ∈ W.

4.18
We have the following result.Proof.We consider

4.21
For all u ∈ W such that u > 1, we have F k, u k .

4.22
Therefore, similar to the proof of Theorem 4.3, Theorem 4.5 follows immediately.
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H 10 1 k 1 a k − 1 , 1 T k 1
There exist two positive constants C 5 and C 6 such that f k, t ≤ C 5 C 6 |t| β−1 , for all k, t ∈ Z 1, T × R, where 1 < β < p − .Definition 4.4.A weak solution of problem 4.17 is a function u ∈ W such that T Δu k − 1 Δv k −
al. see 8 , in a second time by Koné and Ouaro 4 , and in a third time by Guiro et al. 5 .In eigenvalues for the problem