Journal of Function Spaces
Volume 2015 (2015), Article ID 706383, 6 pages
http://dx.doi.org/10.1155/2015/706383

Research Article

Three-Point Boundary Value Problems for Conformable Fractional Differential Equations

H. Batarfi,¹ Jorge Losada,² Juan J. Nieto,1,2 and W. Shammakh¹

¹Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
²Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain

Received 30 October 2014; Revised 2 March 2015; Accepted 6 March 2015

Academic Editor: Aurelian Gheondea

Copyright © 2015 H. Batarfi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study a fractional differential equation using a recent novel concept of fractional derivative with initial and three-point boundary conditions. We first obtain Green's function for the linear problem and then we study the nonlinear differential equation.

1. Introduction

In this paper, we study a class of differential equations supplemented with three-point boundary conditions. Precisely, we consider the following problem:

where is the conformable fractional derivative of order , is the ordinary derivative, is a known continuous function, and are real numbers, , and .

Fractional calculus and fractional differential equations are relevant areas of research. There are several concepts of fractional derivatives, some classical, such as Riemann-Liouville or Caputo definitions, and some novel, such as conformable fractional derivative [1], -derivative [2], or a new definition [3, 4]. The relation between these definitions and their potential applications needs further study.

The conformable fractional derivative aims at extending the usual derivative satisfying some natural properties (see [1]) and gives a new solution for some fractional differential equations. In this paper we present a boundary value problem involving this fractional derivative.

Sequential fractional differential equations have been considered for other types of fractional derivatives, see, for example, [5, 6].

The paper is organized as follows. In Section 2 we recall some concepts relative to the conformable fractional calculus. In Section 3 we solve the corresponding linear problem and obtain Green's function. In Section 4 we study the nonlinear problem and finally, in Section 5, we present an example to illustrate the applicability of our results.

2. Preliminaries

We recall some definitions and results concerning conformable fractional derivative.

Definition 1 (see [1]). Given a function , the conformable fractional derivative of order of at is defined by

for all . If is -differentiable in some interval with , then we define

whenever the limit of the right hand side exists.

We remark that if is differentiable, then

Reciprocally, if exists, then for we have

Hence, . Of course, for this is not valid and it would be useful to deal with equations and solutions with singularities.

Definition 2. Let and let be an -differentiable function at ; the fractional conformable derivative of order at is given by

where is the smallest integer greater than or equal to . For we proceed in a similar way as in Definition 1.

Definition 3. Given , the fractional integral of order at is given by

Definition 4. Given , the fractional integral of order is given by

where denotes the operator (usual integration) of order .

Remark 5. Some authors (see [7, 8]) have argued that conformable fractional derivative is not a truly fractional operator. This question seems today to still be open and perhaps it is a philosophical issue. However, in any case, the study of boundary value problems involving this new derivative has, in our opinion, a point of interest and deserves to be researched in more detail.

In [1, Theorem 3.1], authors have proved that for and a given continuous function, for . In this paper, we consider , so we need the following results.

Lemma 6. Given and a continuous function defined in the domain of , one has that for .

Proof. Since is continuous, then is twice differentiable. In view of [1, Remark 2.1] we have

Thus, statement of Lemma 6 has been proved.

Lemma 7. Given and an -differentiable function, one has that if and only if , where , .

Proof. This fact follows easily in view of the mean value theorem for conformable fractional differentiable functions (see [1, Theorem 2.4]).

3. Linear Boundary Problem

In order to study boundary value problem (1)-(2), we consider now the linear equation

where and .

Lemma 8. Consider

Then, the unique solution of (11) subject to the boundary conditions (2) is given by

where

Proof. Integrating (11) we obtain

Thus, in view of Lemmas 6 and 7, every solution of (15) is a solution for (11).
Let . Equation (15) can be rewritten as

Integrating from to , we obtain

Imposing boundary conditions (2), we conclude that , , and

Substituting these values of , , and in (17), we finally obtain (13) and that expression gives the unique solution.

We now obtain Green’s function corresponding to the fractional differential equations (11) of order with subject to boundary conditions (2).

By changing the order of integration, we note that

Hence, solution (17) with takes the form

where

Now, using the boundary condition , we get

Therefore, we finally conclude the following:

so, we deduce the following result.

Theorem 9. The unique solution of (11) subject to boundary conditions (2) is given by

where

with

Remark 10. In other words, corresponding Green's function for the homogeneous problem (11) satisfying the boundary conditions (2) is given by (25).

Remark 11. Note that is independent of , but the solution depends, of course, on .

4. Nonlinear Problem

Let be the Banach space of all continuous functions defined in endowed with the usual supremum norm defined by .

For the sake of convenience, we set

with

where is given by (14).

In view of Lemma 8, we transform boundary value problem (1)-(2) into

where is defined by

Observe that problem (1)-(2) has solutions if the operator (30) has fixed points.

Theorem 12. Let be a jointly continuous function satisfying the condition

where is the Lipschitz constant. Then, boundary value problem (1)-(2) has a unique solution if , where is given by (27).

Proof. First, for defined by (30), we show that , where is the closed ball of radius in ; that is, . Now we consider and we choose

For , we have

Now, for , and for each , we obtain

As , we conclude that is a contraction. Thus, the statement of the theorem follows by the classical Banach fixed point theorem. This concludes the proof.

Now we recall a known result due to Krasnoselskii (see [9, Theorem ]) which we will use to prove existence of at least one solution to (1)-(2).

Theorem 13. Let be a closed, convex, and nonempty subset of a Banach space . Let and be operators such that

(i) whenever , ,(ii) is compact and continuous,(iii) is a contraction mapping.

Then there exists such that .

Theorem 14. Let be a jointly continuous function satisfying the following conditions:

(H1) for all , , ,(H2) for all with .

Then, boundary value problem (1)-(2) has at least one solution on if

Proof. Letting , we fix

and we consider as in Theorem 12.
Define the operators and as

For , , it follows by (36) that

Thus, . In view of condition (35), we have that is a contraction mapping.
Now we show that is compact and continuous. The continuity of implies that the operator is continuous. In addition, is uniformly bounded on as

Setting , we define , and consequently we have that, for ,

which is independent of and tends to zero as . This shows that is relatively compact on . Hence, by the Ascoli-Arzelá Theorem, is compact on . Thus, all the hypotheses of Theorem 13 are satisfied and the conclusion of Theorem 13 implies that the boundary value problem (1)-(2) has at least one solution on , with satisfying (36). This completes the proof.

5. Example

Consider the boundary value problem over the interval , given by:

Here, , , , . Clearly,

For , it follows, by Theorem 12, that boundary value problem (41) has a unique solution.

Remark 15. Authors of [5] obtained a similar result considering fractional Caputo derivative instead of fractional conformable derivative in (41).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their gratitude to the referees for their very valuable comments and suggestions. This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under Grant no. 88-130-35-HiCi. The authors, therefore, acknowledge technical and financial support of KAU.

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