Abstract
We discuss the existence and uniqueness of solutions for boundary value problems involving multiterm fractional integral boundary conditions. Our study relies on standard fixed point theorems. Illustrative examples are also presented.
1. Introduction
In this paper, we study the existence and uniqueness of solutions for the following fractional differential equation: subject to nonlocal fractional integral boundary conditions where denotes the Caputo fractional derivative of order , is a continuous function, , , for all , and is the Riemann-Liouville fractional integral of order .
The significance of studying problem (1)-(2) is that condition (2) is very general and includes many conditions as special cases. In particular, if , for all , , then condition (2) reduces to Note that condition (3) does not contain values of unknown function at the left-side and right-side of boundary points and , respectively.
In recent years, the boundary value problems of fractional order differential equations have emerged as an important area of research, since these problems have applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, biology, economics, control theory, signal and image processing, polymer rheology, regular variation in thermodynamics, biophysics, aerodynamics, viscoelasticity and damping, electrodynamics of complex medium, wave propagation, and blood flow phenomena (see [1–5]). Many researchers have studied the existence theory for nonlinear fractional differential equations with a variety of boundary conditions; for instance, see the papers [6–16] and the references therein.
The main objective of this paper is to develop some existence and uniqueness results for the boundary value problem (1)-(2) by using standard fixed point theorems. The paper is organized as follows. In Section 2, we recall some preliminary facts that we need in the sequel, and Section 3 contains our main results. Finally, illustrative examples are presented in Section 4.
2. Preliminaries
In this section, we introduce some notations and definitions of fractional calculus [2, 3] and present preliminary results needed in our proofs later.
Definition 1. For an at least -times differentiable function , the Caputo derivative of fractional order is defined as where denotes the integer part of the real number .
Definition 2. The Riemann-Liouville fractional integral of order is defined as provided the integral exists.
Lemma 3. For , the general solution of the fractional differential equation is given by where , .
In view of Lemma 3, it follows that for some , .
For convenience, we set
Lemma 4. Let , , , for , , and . Then, the problem has a unique solution given by
Proof. Using Lemma 3, (9) can be expressed as an equivalent integral equation: Taking the Riemann-Liouville fractional integral of order for (12), we have From the first condition of (10) and the equation (13) with , it follows that According to the above process, the second conditions of (10) and (13) with imply that Solving the system of linear equations for constants and , we have Substituting constants and into (12), we obtain (11) as required.
3. Main Results
Let denote the Banach space of all continuous functions from to endowed with the norm defined by . As in Lemma 4, we define an operator by It should be noticed that problem (1)-(2) has solutions if and only if the operator has fixed points.
We are in a position to establish our main results. In the following subsections, we prove existence as well as existence and uniqueness results for the BVP (1)-(2) by using a variety of fixed point theorems.
3.1. Existence and Uniqueness Result via Banach’s Fixed Point Theorem
Theorem 5. Assume thatthere exists a constant such that , for each and .If where is defined by then problem (1)-(2) has a unique solution in .
Proof. We transform problem (1)-(2) into a fixed point problem, , where the operator is defined as in (17). Observe that the fixed points of the operator are solutions of problem (1)-(2). Applying the Banach contraction mapping principle, we will show that has a unique fixed point.
We let and choose
where a constant is defined by
Now, we show that , where . For any , we have
which implies that .
Next, we let . Then, for , we have
which implies that . As , is a contraction. Therefore, we deduce by Banach's contraction mapping principle that has a fixed point which is the unique solution of problem (1)-(2). The proof is completed.
3.2. Existence Result via Krasnoselskii’s Fixed Point Theorem
Lemma 6 (Krasnoselskii’s fixed point theorem, [17]). Let be a closed, bounded, convex, and nonempty subset of a Banach space . Let and be the operators such that (a) whenever ; (b) is compact and continuous; and (c) is a contraction mapping. Then, there exists such that .
Theorem 7. Let be a continuous function satisfying . Assume that, for all , and .Then, the boundary value problem (1)-(2) has at least one solution on provided
Proof. Setting and choosing
( and are defined in (19) and (21), resp.), we consider . We define the operators and on by
For any , we have
This shows that . It is easy to see using (24) that is a contraction mapping.
Continuity of implies that the operator is continuous. Also, is uniformly bounded on as
Now, we prove the compactness of the operator .
We define , and consequently we have
which is independent of and tends to zero as . Thus, is equicontinuous. So is relatively compact on . Hence, by the Arzelá-Ascoli theorem, is compact on . Thus, all the assumptions of Lemma 6 are satisfied. So the conclusion of Lemma 6 implies that the boundary value problem (1)-(2) has at least one solution on .
3.3. Existence Result via Leray-Schauder’s Nonlinear Alternative
Theorem 8 (nonlinear alternative for single valued maps, [18]). Let be a Banach space, a closed, convex subset of , an open subset of , and . Suppose that is a continuous, compact (i.e., is a relatively compact subset of ) map. Then, either(i) has a fixed point in , or(ii)there is (the boundary of in ) and with .
Theorem 9. Assume thatthere exist a continuous nondecreasing function and a function such that there exists a constant such that where and are defined in (19) and (21), respectively.Then, the boundary value problem (1)-(2) has at least one solution on .
Proof. Let the operator be defined by (17). Firstly, we will show that maps bounded sets (balls) into bounded sets in . For a number , let be a bounded ball in . Then, for we have
and, consequently,
where and are defined by (19) and (21), respectively.
Next, we will show that maps bounded sets into equicontinuous sets of . Let with and . Then, we have
As , the right-hand side of the above inequality tends to zero independently of . Therefore, by the Arzelá-Ascoli theorem, the operator is completely continuous.
Let be a solution. Then, for , following the similar computations as in the first step, we have
which leads to
In view of , there exists such that . Let us set
We see that the operator is continuous and completely continuous. From the choice of , there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder type, we deduce that has a fixed point which is a solution of problem (1)-(2). This completes the proof.
3.4. Existence Result via Leray-Schauder Degree
Theorem 10. Let be a continuous function. Assume thatthere exist constants , where is given by (19) and such that Then, the boundary value problem (1)-(2) has at least one solution on .
Proof. Let us define an operator as in (17) and consider the fixed point problem In view of the fixed point problem (39), we are going to prove the existence of at least one solution satisfying (39). Define a ball with a constant radius given by Hence, it is sufficient to show that satisfies We define As shown in Theorem 9, we deduce that the operator is continuous, uniformly bounded, and equicontinuous. Then, by the Arzelá-Ascoli theorem, a continuous map defined by is completely continuous. If (41) holds, then the following Leray-Schauder degree is well defined, and by the homotopy invariance of topological degree, it follows that where denotes the unit operator. By the nonzero property of Leray-Schauder degree, for at least one . In order to prove (41), we assume that for some and for all . Then, Computing directly for , we have Let ; then (41) holds. This completes the proof.
4. Examples
In this section, we present some examples to illustrate our results.
Example 1. Consider the following boundary value problem with multiterm fractional integral:
Here, , , , , , , , , , , , , , , , , , , , , , and . Since , then, is satisfied with . We can show that Hence, by Theorem 5, the boundary value problem (46) has a unique solution on .
Example 2. Consider the following boundary value problem with multiterm fractional integral:
Here, , , , , , , , , , , , , , , , , , , , , , , , , and . It is easy to verify that Clearly, Choosing and , we can show that which implies that . Hence, by Theorem 9, the boundary value problem (48) has at least one solution on .
Example 3. Consider the following boundary value problem with multiterm fractional integral:
Here, , , , , , , , , , , , , , , , , , , , , , , , , , , , and . We can show that Since then, is satisfied with and such that Hence, by Theorem 10, the boundary value problem (52) has at least one solution on .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The research of J. Tariboon and A. Singubol is supported by King Mongkut's University of Technology North Bangkok, Thailand. Sotiris K. Ntouyas is a Member of Nonlinear Analysis and Applied Mathematics- (NAAM-) Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.