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EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A DISCRETE FRACTIONAL BOUNDARY VALUE PROBLEM

Abstract: This present work discusses existence and uniqueness of solutions for the following discrete fractional antiperiodic boundary value problem of the form C 0 ∆ α kx(k) = f (k + α− 1, x(k + α− 1)) , for k ∈ [0, l + 2]N0 = {0, 1, ..., l + 2}, with boundary conditions x(α − 3) = −x(α+ l), ∆x(α−3) = −∆x(α+ l), ∆2x(α−3) = −∆2x(α+ l), where f : [α− 2, α+l]Nα−2×R → R is continuous and C 0 ∆ α k is the Caputo fractional difference operator with order 2 < α ≤ 3. Finally, the main results are illustrated by suitable examples.


Introduction
The theory of fractional calculus is one of the branches of study which involves integrals and derivatives of an arbitrary order. In the past few decades, fractional calculus is one of the most novel types of tools applied to problems in engineering, science and technology (see [16], [17]). It also played a key role for the improvement of the specialized research in mathematical modeling of several phenomena such as enzyme kinetics, nonlinear oscillation of earthquake, blood flow problems, aerodynamics, cancer modeling being among the popular ones, control theory, electrical circuits, fluid-dynamic traffic model, regular variation in thermodynamics, polymer rheology, arterial and heart disease modeling, etc (see [13], [14]). In recent years, the theory of discrete fractional calculus has been developed by a very few researchers (see [8], [9], [11], [15]) and they obtained many important properties and interesting fruitful results.
Anti-periodic boundary value problems (classical and fractional), recently have received great attention as they occur in the mathematical modeling of a diversity of physical processes (see [2], [3], [4], [7]). As a matter of fact, many numerical problems converge faster when anti-periodic boundary conditions are used instead of periodic boundary conditions. However, the concept of non local anti-periodic boundary conditions has not been addressed yet. In the last two decades, many researchers have focused their attention on the study of stability of solutions, existence and uniqueness, positive and multiplicity of solutions of boundary value problems (BVP's) for nonlinear fractional differential and difference equations by using fixed point techniques such as Krasnoselskii fixed point theorem, Brower fixed point theorem and contraction mapping principle (see [1], [3], [5], [6], [11], [12], [15]).
Although the discrete boundary value problems with fractional order 0 < α ≤ 1 and 1 < α ≤ 2 have been studied by researchers, very little is known in the literature about discrete fractional equation with order 2 < α ≤ 3. Motivated by the aforesaid work, in this present work, we investigate the existence and uniqueness of solutions for the following discrete fractional order antiperiodic boundary value problem of the form The plan of this paper is as follows. Some definitions and theorems needed to prove the main results are provided in Section 2. In Section 3, a form of solutions of boundary value problem (1)-(2) is obtained. In Section 4, existence and uniqueness of a solution to equation (1)-(2) are established by using the contraction mapping principle and Brouwer fixed point theorem. In Section 5, suitable examples to illustrate the main results are presented and the paper ends with a conclusion.

Solution for antiperiodic BVP
In order to discuss the main results, now we state and prove important theorems which will aid to obtain a form of the solution of (1)-(2), provided the solution exists. where is a solution of (7). From Theorem 3, we obtain a general solution for (7) as for some c 0 , c 1 , c 2 ∈ R. Using Theorem 4, we obtain , the value of c 0 , c 1 and c 2 as follows: , and Substituting the values of c 0 , c 1 and c 2 into (9), we obtain (8). This shows that if (7) is a solution, then it can be represented by (8).
Conversely, every function of the form (8) is a solution of (7). The proof is completed.
Theorem 6. If α and k are any numbers then .

Existence and uniqueness of solution
In this section, we prove that under certain conditions, discrete fractional antiperiodic boundary value problem (1)-(2) has at least one solution. We observe that boundary value problem (1)-(2) may be recast as an equivalent summation equation. It follows from Theorem 5 that x(k) is a solution of (1)-(2) if and only if x(k) is a fixed point of the operator T : is Lipschitz in x. That is, there exists L > 0 such that |f (k, x) − f (k, y)| ≤ L |x − y| whenever x, y ∈ R. Then it follows that discrete fractional antiperiodic boundary value problem (1)-(2) has a unique solution provided that the condition holds.
Above inequality leads to the conclusion When the condition (13) holds, boundary value problem (1) -(2) has a unique solution, which completes the proof of the theorem.
Theorem 9. Assume that there exists a constant M > 0 such that f (k, x) satisfies the inequality where β = 3 α(α − 1) . Then it follows that discrete fractional antiperiodic boundary value problem (1) -(2) has at least one solution Proof. Consider the Banach space B := x ∈ R ℓ+4 : x ≤ M . Let T be the operator defined in (12). It is clear that T is a continuous operator. Thus, the main purpose is to prove that T : B → B. That is, whenever x ≤ M , it follows that T x ≤ M . Once this is proved, we use the Brouwer fixed point theorem to obtain the conclusion. Suppose that inequality (14) holds for given f . For convenience, we assume which is a strictly positive constant. Then we obtain . By an application of Theorem 6 and Corollary 7, we get .
From the above inequality, we have Finally, by the definition of Φ given in (15), we get that (16) implies that Thus, from (17) we deduce that T : B → B. Consequently, it follows from Brouwer fixed point theorem that there exists a fixed point of the map T , say T x 0 = x 0 with x 0 ∈ B. So this x 0 is a solution of (1) - (2). Moreover, x 0 satisfies the bound |x 0 (k)| ≤ M , for each k ∈ [α − 3, α + ℓ] N α−3 . Proof of the theorem is completed.

Examples
This section provides two examples to illustrate Theorems 8 and 9. We start with an example illustrating Theorem 8, subsequently by an example illustrating Theorem 9.
Therefore, from Theorem 8 we conclude that boundary value problem (18) -(19) has a unique solution. For the values of α ∈ (2, 3] (with an increment of 0.1), the corresponding values of Λ are computed and tabulated in Table 1.
Taking the values of α on the horizontal axis, the computed values of Λ are plotted in Figure 1. The resulting curve is nonlinear and shows monotonically increasing trend in α ∈ (2, 3] (with an increment of 0.025).

Conclusions
Arguments in this paper are based on a contraction mapping theorem and Brouwer fixed point theorem for Banach spaces. Conditions for the existence and uniqueness of solutions for the fractional order difference equations in the presence of antiperiodic boundary conditions are obtained. Results obtained in the main results are illustrated with suitable examples supported with numerical calculations.