Abstract
There are many published papers about fractional integrodifferential equations and system of fractional differential equations. The goal of this paper is to show that we can investigate more complicated ones by using an appropriate basic theory. In this way, we prove the existence and uniqueness of solution for a -dimensional system of multiterm fractional integrodifferential equations with antiperiodic boundary conditions by applying some standard fixed point results. An illustrative example is also presented.
1. Introduction
Fractional differential equations have recently been studied by many researchers for a variety of problems (see, e.g., [1–33] and the references therein). Antiperiodic boundary value problems occur in the mathematical modeling of a variety of physical processes (see, e.g., [3–6, 29, 30] and the references therein). On the other hand, the study of a coupled system of fractional order is also very significant because this kind of system can often occur in applications (see, e.g., [7, 15, 24, 28, 29] and the references therein). We are going to investigate a complicated case in this work. Let and . In this paper, we study the existence and uniqueness of solution for the -dimensional system of multiterm fractional integrodifferential equations with antiperiodic boundary conditions , , and for , where denotes the Caputo fractional derivative, , , for , , and , are continuous functions for all . Hereafter, we will use vector notations. Define the space endowed with the norm . In fact, and the product space endowed with the norm are Banach spaces. The Riemann-Liouville fractional integral of order is defined by ( and ), provided the integral exists. The Caputo derivative of order for a function is defined by for and [25]. Recently, Wang et al. proved the following result [30].
Lemma 1. For each , the unique solution of the boundary value problem is given by , where is Green's function defined as
One can find the next result in [34].
Theorem 2. Let be a Banach space and a completely continuous operator. Suppose that the set is bounded. Then has a fixed point in .
We will use the last two results for solving the problem (1).
2. Main Results
Now, we are ready to state and prove our main results. For each , put and , where for all . Define the operator by where and for , where Thus, for each , we have
Theorem 3. The operator is completely continuous.
Proof. First, we show that the operator is continuous. Let and for and let be a sequence in such that . Then, we have for . Since for , the sequences , , and converge uniformly on and also , , and converge uniformly on for . Since by using the above inequalities and the continuity of (), we get Thus, is continuous in . Let be a bounded subset of . Choose positive constants such that for all and . Thus, for each we have for all . Hence, for all and so . This implies that the operator is uniformly bounded. Now, we show that is an equicontinuous set. Let . Then, we have for all . As , the right-hand side of the above inequalities tends to zero. Thus, by using the Arzela-Ascoli theorem one can conclude that the operator is completely continuous. This completes the proof.
Theorem 4. Assume that there exist positive constants , , , , and () such that and for all , , and . Then problem (1) has at least one solution.
Proof. First, we show that for some is bounded. Let . Then, for each we have Hence, for all . Thus, we get Hence, for . This implies that and so . Therefore, the set is bounded. Now by using Theorem 2, the operator has at least one fixed point. This implies that the problem (1) has at least one solution.
Theorem 5. Suppose that there exist nonnegative constants , , , and for such that for all , , and . Then the problem (1) has a unique solution.
Proof. Let for , , and We show that . Let . Then for . Hence, for . Hence, . Now, for each , , and we get for . This implies that for and so Since , is a contraction and so, by using the Banach contraction principle, has a unique fixed point. Now, one can easily get that the problem (1) has a unique solution.
3. Example
Here, we provide an example to illustrate one of our results. It is considerable that there are some examples which provide nonuniqueness of solutions for some fractional differential equations (see, e.g., [32]).
Example 1. Consider the following 3-dimensional system:
where , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and , with , , , , , , , , and . Then, we have
Here, , , ,
Thus, by using Theorem 4, 3-dimensional system (30) has at least one solution.
4. Conclusions
Fractional integrodifferential equations, system of fractional differential equations, and their applications represent a topic of high interest in the area of fractional calculus and its applications in various fields of science and engineering. Antiperiodic boundary value problems occur in the mathematical modeling of a variety of physical processes. The goal of this paper is to investigate a complicated case by using an appropriate basic theory. In this way, we prove the existence and uniqueness of solution for a new -dimensional system of multiterm fractional integrodifferential equations with antiperiodic boundary conditions.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
Research of the second and third authors was supported by Azarbaijan Shahid Madani University. Also, the authors express their gratitude to the referee for the helpful suggestions which improved the final version of this paper.