Qualitative Analysis of Differential, Difference Equations, and Dynamic Equations on Time ScalesView this Special Issue
Research Article | Open Access
Uniqueness and Existence of Solution for a System of Fractional -Difference Equations
We prove the existence and uniqueness of solution for a system of fractional differential equations. Our results are based on the nonlinear alternative of Leray-Schauder type and Banach’s fixed-point theorem.
This paper is mainly concerned with the uniqueness and existence of solution for a system of fractional -difference equations given by where , is the fractional -derivatives of the Caputo type, , , , , , and are arbitrary real constants, and are given continuous functions.
In the last few years, fractional differential equations (in short FDEs) have been studied extensively. The motivation for those works stems from both the development of the theory of fractional calculus itself and the applications of such constructions in various sciences such as physics, mechanics, chemistry, and engineering. For an extensive collection of such results, we refer the readers to the monographs by Kilbas et al. , Miller and Ross , Oldham and Spanier , Podlubny , and Samko et al. .
Some basic theory for the initial value problems of fractional differential equations involving Riemann-Liouville differential operator has been discussed by Lakshmikantham and Vatsala ([6–8]), Babakhani and Daftardar-Gejji ([9–11]), Bai , and so on. Also, there are some papers which deal with the existence and multiplicity of solutions (or positive solution) for nonlinear FDE of BVPs by using techniques of nonlinear analysis (fixed-point theorems, Leray-Shauder theory, topological degree theory, etc.)—see ([13–18]) and the references therein. The study of a coupled system of fractional order is also very significant because this kind of system can often occur in applications. The reader is referred to the papers ([19–22]) and the references cited therein.
The pioneer work on -difference calculus or quantum calculus dates back to Jackson’s papers ([23, 24]), while a systematic treatment of the subject can be found in [25, 26]. For some recent existence results on -difference equations, see [27–29] and the references therein.
There has also been a growing interest on the subject of discrete fractional equations on the time scale . Some interesting results on the topic can be found in a series of papers [30–38]. Fractional -difference equations have recently attracted the attention of several researchers. For some earlier work on the topic, we refer to [39, 40], whereas some recent work on the existence theory of fractional -difference equations can be found in [41–45]. However, the study of boundary value problems of fractional -difference equations is at its infancy and much of the work on the topic is yet to be done.
From the above works, we can see a fact, although the fractional boundary value problems have been investigated by some authors. To the best of our knowledge, there have been few papers which deal with problem (1) for nonlinear fractional differential equation. Motivated by all the works above, in this paper we discuss problem (1). Using nonlinear alternative of Leray-Schauder type, we will give the existence and uniqueness of solution for a system of fractional differential equations with Riemann-Liouville integral boundary conditions of different order (1).
The paper is organized as follows. In Section 2, we give some preliminary results that will be used in the proof of the main results. In Section 3, we establish the uniqueness and existence of a solution for the nonlinear fractional differential equation boundary value problem (1). In last section, we give two examples to illustrate our work.
2. Preliminaries and Lemmas
In this section, we cite some definitions and fundamental results of the -calculus as well as of the fractional -calculus ([46, 47]). We also give a lemma that will be used in obtaining the main results of the paper.
The -derivative of a function is here defined by and -derivatives of higher order by The -integral of a function defined in the interval is given by If and is defined in the interval , its integral from to is defined by Similarly, as done for derivatives, an operator can be defined, namely, by The fundamental theorem of calculus applies to these operators and ; that is, and if is continuous at , then Basic properties of the two operators can be found in the book that is mentioned in . We now point out three formulas that will be used later ( denotes the derivative with respect to variable ) :
Remark 1. We note that if and , then .
Definition 2 (see ). Let and let be a function defined on . The fractional -integral of the Riemann-Liouville type is and
Definition 3 (see ). The fractional -derivative of the Riemann-Liouville type of order is defined by and where is the smallest integer greater than or equal to .
Definition 4 (see ). The fractional -derivative of the Caputo type of order is defined by where is the smallest integer greater than or equal to .
Lemma 5. Let and let be a function defined on . Then the next formulas hold:(1),(2).
Lemma 6 (see ). Let and . Then the following equality holds:
Lemma 8. Let and ; then the unique solution of the linear fractional boundary value problem
is given by
The following lemma is fundamental in the proofs of our main result.
Lemma 9 (see ; nonlinear alternative of Leray-Schauder type). Let be a Banach space with closed and convex. Assume that is a relatively open subset of with and is continuous, compact (i.e., is a relatively compact subset of ) map. Then either(i) has a fixed point in or(ii)there exist and with .
3. Main Results
In this section, we will discuss the uniqueness and existence of solutions for boundary value problem (1).
First of all, we define the Banach space endowed with the norm . For , let ; then is a Banach space.
For convenience, we set and let . Note
Lemma 10. Suppose that and are continuous; then is a solution of BVP (1) if and only if is a solution of the integral equations (24).
Let ; define an operator as where then, by Lemma 10, the fixed point of operator coincides with the solution of system (1).
In the first result, we prove uniqueness of solution of the boundary value problem (1) via Banach’s contraction principle.
Theorem 11. Assume that are continuous functions and the following conditions hold:(H1)there exist two -integrable functions that satisfy In addition, assume that where where and are given by (19) and (22), respectively. Then system (1) has a unique solution.
Proof. Let us set , ,
For , we obtain Then, for , , we have In view of (31), we obtain From the last estimate we deduce that .
By a similar way as done above we have and .
Therefore, we obtain From the last estimate we can choose ; then, for every , we have .
In order to show that is a contraction, let , and, for any , we get which, in view of and (31), implies that
Similarly, we have .
Thus, we have
Since , , therefore, the operator is a contraction. Hence, by Banach’s contraction principle, the operator has a unique fixed point, which is the unique solution of the system (1). This completes the proof.
The second result is based on the nonlinear alternative of Leray-Schauder type (Lemma 9).
Theorem 12. Assume that are continuous functions and the following conditions hold:(H2)there exist four functions , , and two nondecreasing functions , such that where .(H3)There exists a constant such that where Then system (1) has at least one solution on .
Proof. Consider the operator defined by (25). The proof consists of several steps. As a first step, it will be shown that maps bounded sets into bounded sets in . For a positive number , let be bounded set in ; then, for , we have
As before, it can be shown that Similarly, we have Thus, maps bounded sets into bounded sets in .
Next, we show that maps bounded sets into equicontinuous sets of . Let with and , where is a bounded set of . Then taking into consideration the inequality , for , we obtain Clearly, the right-hand side of the above inequality tends to zero independently of as . Thus, it follows by the Arzelá-Ascoli theorem that is completely continuous. Similarly, is completely continuous. Therefore, is completely continuous.
Let us set . Note that the operator is continuous and completely continuous. From the choice of , assume that there is such that , for some . By (H3), we obtain which is a contradiction. In consequence, by the nonlinear alternative of Leray-Schauder type (Lemma 9), we deduce that has a fixed point which is a solution of the system (1). The proof is complete.
Example 1. Consider the following fractional -difference nonlocal boundary value problem: