Abstract
Existence and uniqueness of solutions for order fractional differential equations with three-point fractional boundary and integral conditions involving the nonlinearity depending on the fractional derivatives of the unknown function are discussed. The results are obtained by using fixed point theorems. Two examples are given to illustrate the results.
1. Introduction
Recently, the theory on existence and uniqueness of solutions of linear and nonlinear fractional differential equations has attracted the attention of many authors; see, for example, [1–19] and references therein. Many of the physical systems can better be described by integral boundary conditions. Integral boundary conditions are encountered in various applications such as population dynamics, blood flow models, chemical engineering, and cellular systems. Moreover, boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two-point, three-point, multipoint, and nonlocal boundary value problems as special cases. The existing literature mainly deals with first order and second order boundary value problems and there are a few papers on third order problems.
El-Shahed [14] studied existence and nonexistence of positive solution of nonlinear fractional two-point boundary value problem: where denotes the Caputo derivative of fractional order , is a positive parameter, and is continuous function.
In [8], Ahmad and Ntouyas studied a boundary value problem of nonlinear fractional differential equations of order with antiperiodic type integral boundary conditions: where denotes the Caputo derivative of fractional order , denotes th derivative of , are given continuous functions, and (). The same problem for fractional differential inclusions is considered in [9].
Ahmad and Nieto [7] studied existence and uniqueness results for the following general three-point fractional boundary value problem involving a nonlinear fractional differential equation of order :
However, very little work has been done on the case when the nonlinearity depends on the fractional derivative of the unknown function. Su and Zhang [17] and Rehman et al. [16] studied the existence and uniqueness of solutions for following nonlinear two-point and three-point fractional boundary value problem when the nonlinearity depends on the fractional derivative of the unknown function.
In this paper, we investigate the existence (and uniqueness) of solution for nonlinear fractional differential equations of order when the nonlinearity depends on the fractional derivatives of the unknown function with the three-point and integral boundary conditions where denotes the Caputo fractional derivative of order , are continuous functions, and , for .
2. Preliminaries
Let us recall some basic definitions [20–22].
Definition 1. The Riemann Liouville fractional integral of order for continuous function is defined as provided the integral exists.
Definition 2. For -times absolutely continuous function , the Caputo derivative fractional order is defined as where denotes the integral part of the real number .
Lemma 3. Let . Then, the differential equation has solutions where and , .
Caputo fractional derivative of order for is given as
Assume that and For convenience, we set
Lemma 4. For any , the unique solution of the fractional boundary value problem is given by
Proof. By Lemma 3, for , the general solution of the equation can be written as where are arbitrary constants. Moreover, by the formula (9), and order derivatives are as follows: Using boundary conditions (13), we get the following algebraic system of equations, for , Solving the above system of equations for , we get the following: Inserting into (15), we get the desired representation for the solution of (12)-(13).
Remark 5. The Green function of the BVP (4)-(5) is defined by where
Remark 6. For , , , and , the BVP (4)-(5) can be written as follows: In this case, the Green function can be written as follows: where Moreover, the case is investigated in [10]. In this case,
3. Existence and Uniqueness Results
In this section we state and prove an existence and uniqueness result for the fractional BVP (4)-(5) by using the Banach fixed-point theorem. We study our problem in the space equipped with the norm where is the sup norm in .
The following notations, formulae, and estimations will be used throughout the paper:
Theorem 7. Assume the following.The function is jointly continuous.There exists a function with such that
for each .The function is jointly continuous and there exists such that
for each .
If
then the problem (4)-(5) has a unique solution on .
Proof. In order to transform the BVP (4)-(5) into a fixed point problem, we consider the operator which is defined by and take its th and th fractional derivative to get Clearly, due to , and being jointly continuous, the expressions (32)-(33) are well defined. It is obvious that the fixed point of the operator is a solution of the problem (4)-(5). To show existence and uniqueness of the solution (12)-(13), we use the Banach fixed point theorem. To this end, we show that is contraction: On the other hand, Similarly, Here, in estimations (34)–(36), we used the Hölder inequality: From (34)–(36), it follows that Consequently, by (31), is a contraction mapping. As a consequence of the Banach fixed point theorem, we deduce that has a fixed point which is a solution of the problem (4)-(5).
Remark 8. In the assumptions (H2), if is a positive constant, then the condition (31) can be replaced by
4. Existence Results
To prove the existence of solutions for BVP (4)-(5), we recall the following known nonlinear alternative.
Theorem 9 (nonlinear alternative). Let be a Banach space; let be a closed, convex subset of ; let be an open subset of and . Suppose that is a continuous and compact map. Then, either (a) has a fixed point in or (b) there exist an (the boundary of ) and with .
Theorem 10. Assume thatfunctions , are jointly continuous;there exist nondecreasing functions and functions , with such that , for all and ;there exists a constant such that Then the problem (4)-(5) has at least one solution on .
Proof. Let .
Step 1. We show that the operator defined by (32) maps into bounded set.
For each , we have
By the Hölder inequality, we have
In a similar manner,
Thus,
Step 2. The families , , and are equicontinuous.
Because of the continuity of and assumption (H5) we have
Therefore, is equicontinuous. Similarly, we may prove that and are equicontinuous.
Hence, by the Arzela-Ascoli theorem, the sets , , and are relatively compact in . Therefore, is a relatively compact subset of . Consequently, the operator is compact.
Step 3. has a fixed point in .
We let for . Then, for each ,
In other words,
According to the assumption (H6), we know that there exists such that and
In other words, for all , we have . Since the operator is continuous and compact, from Theorem 9(a), we can deduce that has a fixed point in .
Remark 11. Notice that analogues of Theorems 7 and 10 for the case were considered in [9]. Thus, our results are a generalization of [9] in the special case when (fractional) differential inclusion is replaced by (fractional) differential equation.
Remark 12. Since the number can be negative, the function . That is why in Theorems 7 and 10 it is assumed that , .
5. Examples
Example 1. Consider the following boundary value problem of fractional differential equation: Here, Since , , , and , with simple calculations, we show that Furthermore, Thus, all the assumptions of Theorem 7 are satisfied. Hence, the problem (50) has a unique solution on .
Example 2. Consider the following boundary value problem of fractional differential equation: where is given by We have Thus, Moreover, From the condition we find that Thus, all the conditions of Theorem 10 are satisfied. So, there exists at least one solution of problem (54) on .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors would like to thank the reviewers for their constructive suggestions and helpful comments.