Existence of Solutions for a Class of Coupled Fractional Differential Systems with Nonlocal Boundary Conditions

Abstract

Applying Schauder fixed point theorem and Leray-Schauder nonlinear alternative theory, this paper is concerned with the existence of solutions to coupled fractional differential systems with fractional integral boundary value conditions. Meanwhile, two examples are worked out to illustrate the application of the main results.

1. Introduction

Fractional differential equations have a wide range of applications in many science and engineering, such as in physics, chemistry, biology, and electrodynamics. We refer the reader to see [1–5]. The main reason is that fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. Therefore, this topic has attracted much attention of scientists and engineers. More and more good results are obtained. See [6–10] and references therein.

In recent years, fractional differential equations with the nonlinear terms involving fractional derivative of unknown functions have been investigated by some authors. See [11–16] and references therein. For example, in [13], Su studied the following nonlinear coupled fractional differential systems: where , , , , and are given continuous functions. is the standard Riemann-Liouville fractional derivative. Applying Schauder fixed point theorem, the existence of solution was studied.

On the other hand, integral boundary conditions have various applications in applied fields such as chemical engineering, underground water flow, blood flow problems, thermoelasticity, population dynamics, and finite element method approaches with the minimization of constitutive error. In consequence, the integral boundary value problem of fractional differential equations is gaining much importance and attention. See [17–21] and references therein. For instance, in [19], Zhao and Liu studied the following coupled fractional differential systems with integral boundary conditions:where , , . is the Caputo fractional derivate and are given continuous functions. By using the monotone method and the theory of fixed point index on cone, they investigated the existence and uniqueness of solution for this coupled system.

In [21], Ahmad et al. investigated the following coupled fractional differential system with nonlocal and integral boundary value conditions: where , , . , are given continuous functions and are real number. Applying Banach contraction mapping principle and Leray-Schauder nonlinear alternative theory, the existence and uniqueness of solution were studied.

To the best of our knowledge, there are fewer results for coupled fractional differential systems with nonlocal and fractional integral boundary value conditions. Motivated by the above-mentioned references, we consider the existence of solutions of the following systems: where is Riemann-Liouville fractional integral, , , , , . , and are given continuous functionals. Applying Schauder fixed point theorem and Leray-Schauder nonlinear alternative theory, some existence results of solutions to this systems are obtained. Finally, two examples are worked out to illustrate the application of our results.

The main features of this paper are as follows. () The coupled fractional differential systems with nonlocal and fractional integral boundary value conditions are first studied. () The nonlinear term here involves fractional derivative of unknown functions. () Nonlocal conditions such as and and fractional integral boundary conditions such as and are more extensive and superior to local conditions.

The rest of this paper is organized as follows. Section 2 introduces some basic definitions and lemmas. In Section 3, the main results are presented. Finally, in Section 4, some examples are given to illustrate the effectiveness of the main results.

2. Preliminary Results

In this section, we first introduce some definitions and lemmas for fractional calculus. For details, please refer to [1, 22].

Definition 1. The Caputo fractional derivative of order of a function is given by where denotes the integer part of the real number .

Definition 2. The Riemann-Liouville fractional integral of order of a function is given by where is the gamma function.

Lemma 3. Let . Then the fractional differential equation has solution where

Lemma 4. Let Then where

Lemma 5. Let , and . Then

Lemma 6. Suppose are given continuous functionals and . Then the boundary value problem is equivalent to systemwhere ,

Proof. In view of Lemmas 3 and 4, the solution of is where From , one can get By means of Lemma 5 and (14), we have Since , one has which implies Thus, Substituting and to (14), we obtain that (11) holds. Similarly, one can prove that (12) holds.

Let be endowed with the norm . Then we have the following conclusions.

Lemma 7. is a Banach space.
Let be endowed with . Then is a Banach space. Let . Clearly is also a Banach space.
Defining an operator by where are described as in Lemma 6.

Lemma 8. Suppose Then is a solution of problem (4) if and only if is a fixed point of the operator .

Proof. By Lemma 6, the necessity is obvious. Now we show sufficiency.
Suppose is a fixed point of the operator . This together with (19) indicatesNotice that , , where is the smallest integer greater than or equal to . Therefore, Similarly, one has By direct computation, we can easily get . Therefore, is a solution of (4).

Lemma 9 ([23] (Leray-Schauder nonlinear alternative)). Let be a Banach space, be a bounded open subset of , and be a completely continuous operator. Then, either there exists such that or there exists a fixed point

3. Main Results

For convenience, some notations and assumptions are stated as follows: are continuous functions and there exist constants , and , such that are continuous functionals, , and there exist constants such that, for all , are continuous functions, and there exist and nondecreasing functions , such thatThere exists such that , where are described as in .

Theorem 10. Assume that hold. Then FDE (4) has at least one solution.

Proof. LetwhereNow we prove that . In fact, for any , it follows from Definition 2 and thatOn the other hand, by Definition 1, we can obtain Repeating a process similar to that of (31), it is easy to see It follows from (32) and (33) that By virtue of (31) and (34), we have Similarly, we can get Therefore, which means that .
From the continuity of the functions , and , it is not difficult to see that is continuous.
Next we show that are equicontinuous. For this sake, let For any , one has On the other hand, we have Notice that This together with (39)-(40) guarantees that Similarly, we can getThis together with (39), (42), and (43) implies that are equicontinuous. By virtue of the Arzela-Ascoli theorem, we can infer that the operator is completely continuous. Hence, applying Schauder fixed theorem, FDE (4) has at least one solution in .

Theorem 11. Assume that hold. Then FDE (4) has at least one solution.

Proof. Firstly, we show that is completely continuous operator. From the continuity of the functions , and , it follows that the operator is continuous. Let , where is described as in ; we now prove that is relatively compact.
For any , by similar computation as (31) and (34), we obtain thatwhich imply Similarly, we have Combining (45) and (46), one can get By (47) and we know . Using similar computation as in (39), (40), (42), and (43), we can obtain are equicontinuous. Thus, is completely continuous by Arzela-Ascoli theorem.
We now claim that has at least one solution in .
Suppose there exists such that for some This together with (44) implies thatBy virtue of (44)–(47), it is easy to see This is a contradiction with . Therefore, it follows from Lemma 9 that the operator has a fixed point . Thus FDE (4) has at least one solution in .

Remark 12. Comparing with , we know that condition is more extensive than . However, the assumptions of Theorem 10 are easier to verify than that of Theorem 11.

4. Examples

Example 1. Consider the following coupled system:where , , , , , , , , , .
(50) can be regarded as the form of (4), where It is not difficult to see Choose , , , , , , , , , , , , , . Obviously, . Therefore, and are satisfied. Immediately, we can conclude that FDE (4) has at least one solution by Theorem 10.

Example 2. Consider the following problem: where , , , , , , , , , .
(53) can be regarded as the form of (4), where It is not difficult to see Choose , , , , , , , , , , , , , . Now we need to verify . By direct calculation, it is easy to see Thus, we can get for , which means that holds. Consequently, FDE (4) has at least one solution by Theorem 11.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This project is supported by NNSF of China (11671237) and NSF of Shandong Province (ZR2013AM005).