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Solvability for a Coupled System of Fractional Integrodifferential Equations with -Point Boundary Conditions on the Half-Line
The aim of this paper is to study the solvability for a coupled system of fractional integrodifferential equations with multipoint fractional boundary value problems on the half-line. An example is given to demonstrate the validity of our assumptions.
The theory of derivatives and integrals of fractional order has undergone rapid development over the years and played a very important role in modern applied mathematical models of real processes arising in phenomena studied in physics, mechanics, engineering, and so on [1–3]. Recently, the existence of solutions for coupled systems involving fractional differential equations is one of the theoretical fields investigated by many authors [4–13].
Very recently, Wang et al.  studied the existence of solutions for the following coupled system of nonlinear fractional differential equations by using Schauder’s fixed point theorem: where , , and and denote Riemann-Liouville fractional derivatives of order and order , respectively; also , are such that and .
Motivated by , in this paper, we consider a coupled system of nonlinear fractional integrodifferential equations on an unbounded domain and more general boundary conditions: where , , , are real numbers, , and denotes Riemann-Liouville fractional derivative. It is clear that boundary value problem (2) includes problem (1) as special case.
Integrodifferential equations have become important in recent years as mathematical models of phenomena in both the physical and social sciences. In particular, some physical phenomena involving certain type of memory effects are represented by integrodifferential equations [14–18].
However, to the best of our knowledge, no work has been reported on the existence results for coupled system of nonlinear fractional integrodifferential equations on an unbounded domain.
The paper is organized as follows. In Section 2, we recall some basic definitions, notations, and preliminary facts. Section 3 is devoted to the existence results for system of nonlinear fractional integrodifferential equations on an unbounded domain. In Section 4, an example is given to demonstrate the applicability of our results.
Definition 1. The Riemann-Liouville fractional integral of order of a function is given by provided that the right-hand side is pointwise defined.
Definition 2. The Riemann-Liouville fractional derivative of order of a continuous function is given by where , provided that the right-hand side is pointwise defined.
Remark 3. The following properties are well known:
Lemma 4. For , the equation is valid if and only if where is the smallest integer greater than or equal to .
Lemma 5. Assume that ; then, where is the smallest integer greater than or equal to .
For any we can define the space equipped with the norm Clearly, is a Banach space . For we define then, is a Banach space.
3. Main Results
In this section, we prove the existence results for the boundary value problem (2). For convenience we use the following notation: By replacing with , respectively, we can define .
Lemma 6. Let and ; then, the unique solution of is given by where is Green’s function given by with
Proof. By Lemma 5, the solution of (12) can be writen as Using the boundary conditions (13), we find that and Now considering the second boundary condition, we have Therefore, the unique solution of the boundary value problem (12)–(14) is where , , and are defined by (16), (17), and (18), respectively. The proof is complete.
Now, we introduce the following function: where
Remark 7. From the definition of and , for any , we have where
Let an operator be defined by From the definition of operator , the problem (2) has a solution if and only if the operator has a fixed point.
Theorem 8. Assume the following. (H1)There exist nonnegative functions such that
where is the beta-function.(H2)There exist nonnegative functions such that
where is the beta-function.
Then, the system (2) has a solution.
and define a ball
At the first step, we prove that the operator transforms the ball into itself. For any we have
In a similar way, we can get
Hence, and this shows that .
Next, we show that is completely continuous. First, Let as in . From (32) we have Then, by the Lebsegue dominated convergence theorem and continuity of , we obtain as . Therefore, by Remark 7, we have as . Similar process can be repeated for ; thus, operator is continuous.
Now, we show that is equicontinuous operator. Let and ; without loss of generality, we may assume that . Since and , for any and , we have In view of (37), by the similar process used in , we can easily prove that operator is equicontinuous. Similar process can be repeated for ; thus, is equicontinuous. On the other hand, is uniformly bounded as . Therefore, is completely continuous operator. Hence, by Schauder fixed point theorem the boundary value problem (2) has at least one solution in .
4. An Example
Consider the following boundary value problem on unbounded domain: Here , , , , , , , , , , , , and . We have For by direct calculation we obtain , , , and Thus all the conditions of Theorem 8 are satisfied and the problem (38) has at least one solution.
In the current paper, we have studied the existence results for a coupled system of nonlinear fractional integrodifferential equations with -point fractional boundary conditions on an unbounded domain. The result obtained in this paper is based on Schauder’s fixed point theorem. In order to show the validity of the assumptions made in our result, we also include an illustrative example.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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