Research Article | Open Access
Three-Point Boundary Value Problems for Conformable Fractional Differential Equations
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.
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 , -derivative , 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 ) and gives a new solution for some fractional differential equations. In this paper we present a boundary value problem involving this fractional derivative.
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.
We recall some definitions and results concerning conformable fractional derivative.
Definition 1 (see ). 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 .
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
Proof. Integrating (11) we obtainThus, 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 obtainImposing 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.
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.
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).
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.
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 .
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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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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