Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 785738, 12 pages
http://dx.doi.org/10.1155/2015/785738

Research Article

On a Time-Fractional Integrodifferential Equation via Three-Point Boundary Value Conditions

Dumitru Baleanu,1,2 Shahram Rezapour,³ Sina Etemad,³ and Ahmed Alsaedi⁴

¹Department of Mathematics, Cankaya University, Ogretmenler Caddesi 14, Balgat, 06530 Ankara, Turkey
²Institute of Space Sciences, Magurele, Bucharest, Romania
³Department of Mathematics, Azarbaijan Shahid Madani University, Azarshahr, Tabriz, Iran
⁴Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 15 July 2014; Revised 15 October 2014; Accepted 20 October 2014

Academic Editor: Guido Maione

Copyright © 2015 Dumitru Baleanu 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.

The existence and the uniqueness theorems play a crucial role prior to finding the numerical solutions of the fractional differential equations describing the models corresponding to the real world applications. In this paper, we study the existence of solutions for a time-fractional integrodifferential equation via three-point boundary value conditions.

1. Introduction

There is no doubt that the fractional calculus has various important applications in many fields as mathematics (see the following monographs [1, 2] and the references therein), physics [3], and economics [4], as well as in many other branches of science and engineering [3, 5, 6]. Recently the numerical methods were applied intensively to solve complicated fractional differential equations (FDE) which describe the real world applications. As an example, we mention that several physical processes exhibit fractional order behavior varying with time and/or space [7]. Thus, a class of phenomena can be obtained by analyzing this most general model. A fundamental step is to prove the existence and uniqueness of the proposed fractional nonlinear differential equations for these models. There are many works on fractional differential equations and inclusions. The existence of solutions for integrodifferential equations of fractional order with nonlocal three-point fractional boundary conditions was developed in [8] while the importance of antiperiodic type integral boundary conditions was discussed in [9]. The existence of solutions for fractional differential inclusions in the presence of the separated boundary conditions in Banach space was developed in [10]. We recall that the existence and multiplicity of positive solutions for singular fractional boundary value problems were analyzed in [11]. The existence and multiplicity of positive solutions for singular fractional boundary value problems were the topic debated in [12]. Making use of the fixed point results on cones, the existence and uniqueness of positive solutions for some nonlinear fractional differential equations were developed in [13]. Further results on positive solutions of a boundary value problem for nonlinear fractional differential equations were reported in [14] and new results on the existence results on nonlinear fractional differential equations were reported in [15]. For other related results we suggest for the readers [16, 17] as well as the references therein. We recall that the existence of solutions for some fractional partial differential equations was investigated in [18, 19] and the references therein.

Let be a natural number, , , and . The Riemann-Liouville time-fractional order integral of the function is defined by for all whenever the integral exists (more details regarding the basic definitions of the fractional calculus can be seen in [1]). Also, the Caputo derivative of time-fractional of order for the function is defined by (see for more details [1]). It has been proved that the general solution of the time-fractional differential equation is given by , where are real constants and (see [20]). Also, for each and we have , where are real constants and ([20]). Let be a bounded set in a metric space . Then the Kuratowski measure of noncompactness is defined by (see [21])

Let be a bounded and continuous operator on a Banach space . Then is called a condensing map whenever for all bounded sets , where denotes the Kuratowski measure of noncompactness (see [21]).

In this paper, we study the existence of solutions for a time-fractional integrodifferential equation via three-point boundary value conditions under certain conditions. We mention that the investigated equation generalized a huge class of classical ordinary differential equations which can be found in applications in engineering and physics. The boundary conditions used in this paper are as general as possible and for particular cases of the parameters we recover several cases of fractional nonlinear differential equations. In this way, we use the next Lemma and Sadovskii’s fixed point theorem for condensing maps (see [22]).

Lemma 1 (see [23]). Let be a Banach space, , , a -contraction, and a compact operator. Then is a -set contraction and so a condensing map.

Theorem 2 (see [22]). Let be a convex, bounded, and closed subset of a Banach space and a condensing map. Then has a fixed point.

2. Main Results

Suppose that , , and endowed via the norm

where denotes the standard Caputo time-fractional derivative. Let , , , , and , and and are continuous functions. We investigate the nonlinear time-fractional integrodifferential equation as follows:

via the three-point boundary value conditions , , and . In this way, we give first next result.

Lemma 3. An element in is a solution for the problem via the three-point boundary value conditions if and only if is a solution for the time-fractional integral equation

where

whenever ,

whenever ,

whenever ,

whenever ,

whenever , and

whenever .

Proof. Let be a solution for the time-fractional integrodifferential equation via the boundary value conditions. Put . Choose such that . Thus, we get

and () = + . Now by using the boundary value conditions, we obtain ,

Hence,

Thus, is a solution for the time-fractional integral equation. It is obvious that is a solution for the time-fractional integrodifferential equation whenever is a solution for the time-fractional integral equation. This completes the proof.

By considering the proof of last result, one can get that the solution of the time-fractional integrodifferential equation is in the form

where , , , , and with .

Theorem 4. Let , , , , and be a nondecreasing bounded function. Suppose that and are continuous functions such that and for all and . Put

If , then the time-fractional integrodifferential equation has at least one solution.

Proof. First, put

where

Now, choose and consider the closed, convex, and bounded subset of . Define the operator by

for all . Now, decompose by , where

for all . Now, we show that . Let , , and . Then, we have

and similarly

Thus, we get

If , then we have

and similarly

Hence,

and so

This implies that . Now, we show that is continuous. Let and be a sequence with . Then, we have

Thus, is continuous. On the other hand, is a -contraction because

for all . Now, we show that is a compact map. We showed that is uniformly bounded. Now, we show that maps the bounded sets into equicontinuous sets. Let such that . Then, we have

for all . Hence, as . By using the Arzela-Ascoli theorem, we get that the operator is completely continuous and so is compact on . Now by using Lemma 1, the operator is a condensing map on and so has a fixed point by using Theorem 2. Now by using Lemma 3, it is easy to see that the fixed point of is a solution for the time-fractional integrodifferential equation .

Example 5. Let , , , , , , , and . Now, consider the time-fractional integrodifferential equation

via the boundary conditions , , and . Define the maps by and by . Define and for all . Then, we have . Define by for all . It is easy to check that

and for all and . One can calculate that and . Now by using Theorem 4, the time-fractional integrodifferential equation has a solution.

3. Conclusion

A time-fractional integrodifferential equation via three-point boundary value conditions was investigated and the existence of the solution was proved for three-point boundary value conditions. One example was studied in detail. For particular forms of the functions and and of the investigated equation and for various values of , , , , , and we can reobtain the forms of several nonlinear time-fractional differential equations describing the complex phenomena which arise in science and engineering.

Conflict of Interests

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

Acknowledgment

Research of the authors was supported by Azarbaijan Shahid Madani University.

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