Recent Developments and Applications on Discrete Fractional Equations and Related TopicsView this Special Issue
Solvability for Discrete Fractional Boundary Value Problems with a -Laplacian Operator
This paper is concerned with the solvability for a discrete fractional -Laplacian boundary value problem. Some existence and uniqueness results are obtained by means of the Banach contraction mapping principle. Additionally, two representative examples are presented to illustrate the effectiveness of the main results.
For any number and each interval of , we denote and throughout this paper. It is also worth noting that, in what follows, we appeal to the convention that the empty sum is taken to be .
In this paper, we will consider the existence and uniqueness of solutions for the following discrete fractional boundary value problem involving a -Laplacian operator where , , , , is the forward difference operator with stepsize 1, denotes the discrete Caputo fractional difference of order is a continuous function, and is the -Laplacian operator; that is, , . Obviously, is invertible and its inverse operator is , where is a constant such that .
The theory of fractional differential equations has become a new important branch of mathematics (see, e.g., [1–8]). At the same time, boundary value problems for fractional differential equations have received considerable attention [9–18]. It is well known that discrete analogues of differential equations can be very useful in applications [19, 20], especially for using computer to simulate the behavior of solutions for certain dynamic equations. Compared to continuous case, significantly less is known about the discrete fractional calculus. However, within the recent years, a lot of papers have appeared on discrete fractional calculus and discrete fractional boundary value problems; see [21–37]. For example, in , Atıcı and Eloe explored a discrete fractional conjugate boundary value problem with the Riemann-Liouville fractional difference. To the best of our knowledge, this is pioneering work on discussing boundary value problems in discrete fractional calculus. After that, Goodrich studied discrete fractional boundary value problems involving the Riemann-Liouville fractional difference intensively and obtained a series of excellent results; see [26–31]. In [33, 34], Bastos et al. considered the discrete fractional calculus of variations and established several necessary optimality conditions for fractional difference variational problems. Abdeljawad introduced the conception of Caputo fractional difference and developed some useful properties of it in . Ferreira in  initially investigated the existence and uniqueness of solutions for some discrete fractional boundary value problems of order less than one by the Banach fixed point theorem.
Very recently, some authors have focused their attention on the existence of solutions for fractional boundary value problems with the -Laplacian operator in continuous case [38–44]. However, as far as we know, few papers can be found in the literature for the discrete fractional boundary value problems with the -Laplacian operator .
Inspired by the aforementioned results, we will investigate the discrete fractional -Laplacian boundary value problem (1) and establish some sufficient conditions for the existence and uniqueness of solutions to it by using the Banach contraction mapping principle.
The remainder of this paper is organized as follows. Section 2 preliminarily provides some necessary basic knowledge for the theory of discrete fractional calculus. In Section 3, the existence and uniqueness results for the solution to problem (1) will be established with the help of the contraction mapping principle. Finally, in Section 4, two concrete examples are provided to illustrate the possible applications of the established analytical results.
For the convenience of the reader, we begin by presenting here some necessary basic definitions and lemmas on discrete fractional calculus theory.
Definition 1 (see ). For any and , the falling factorial function is defined as provided that the right-hand side is well defined. We appeal to the convention that ifis a pole of the Gamma function and is not a pole, then .
Definition 2 (see ). The th fractional sum of a function , for , is defined by
Definition 3 (see ). The th Caputo fractional difference of a function , for , , is defined by where is the smallest integer greater than or equal to and is the th forward difference operator. If , then .
Lemma 4 (see ). Assume that and is defined on . Then where , , and is the smallest integer greater than or equal to.
Now, we state and prove the following lemma, which provides a representation for the solution to (1) if the solution exists.
Lemma 5. Let , and let . Then the following problem has a unique solution where .
Proof. The definition of the discrete Caputo fractional difference, together with condition , implies that . So from (6), we have
Hence, in view of Lemma 4, we can get
where , .
Furthermore, we have Then by conditions , , we can get Substituting the values ofandin (11), we get (8). This completes the proof.
Finally, we list below the following basic properties of the-Laplacian operator which will be used in the sequel.(1) If, and, then (2) If,, then
3. Main Results
In this section, we will use the Banach contraction mapping principle to prove the existence and uniqueness for the solution to problem (1).
Let denote the Banach space of all functions from into endowed with the norm defined by .
For the sake of convenience to the following discussion, we set where is as given in Lemma 5. Also, for any , we denote for . Obviously, .
Now, we state the main results as follows.
Theorem 6. Suppose ,,, and the following condition holds: there exist positive numbers and with such that Then the problem (1) has a unique solution.
Proof. For any, by (21), we can get that
Due to and , we know that. By (14) and (22), for any,, we have
Next, for any and for each , together with the fact that , we obtain where. From (20), we get that, which implies that is a contraction mapping. By means of the Banach contraction mapping principle, we get that has a unique fixed point in ; that is. the problem (1) has a unique solution. This completes the proof.
With a similar proof to that of Theorem 6, we can get the following theorem.
Theorem 7. Suppose , , , and the following condition holds: there exist constantsandwith such that Then the problem (1) has a unique solution.
Theorem 8. Suppose,,, and the following condition holds: there exists a nonnegative function and such that and there exists a positive constantsuch that Then the problem (1) has a unique solution provided that
Proof. By (28), we can get that, for, In view of and , we can get . From (15) and (29), for any , we have Hence, for any , by , we have where . In view of (30), is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle. This completes the proof.
In this section, we will illustrate the possible application of the above established analytical results with the following two concrete examples.
Example 1. Consider the discrete fractional boundary value problem hereis a real number.
Conclusion. Problem (34) has a unique nonnegative solution.
Proof. Corresponding to problem (1), , , , , , , and ,.
Choosing and , by direct calculation, we can verify that It is easy to verify that for , . Therefore, by Theorem 6, the boundary value problem (34) has a unique solution. Furthermore, from the nonnegativeness of and the expression of , we also get that the unique solution of (34) is nonnegative.
Example 2. Consider the nonlinear discrete fractional boundary value problem where is a real number.
Conclusion. Problem (37) has a unique solution.
Proof. The problem (37) can be regarded as problem (1), where , , , , ,, and , . Taking , , then . Let ; we have
Moreover, we can verify that
for , .
Therefore, by Theorem 8, problem (37) has a unique solution.
This work was supported by the Longdong University Grant XYZK-1010 and XYZK-1007.
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