Abstract and Applied Analysis

Volume 2013 (2013), Article ID 176180, 8 pages

http://dx.doi.org/10.1155/2013/176180

Research Article

Existence Results for a Class of Fractional Differential Equations with Periodic Boundary Value Conditions and with Delay

Hadi Karami,¹ Azizollah Babakhani,¹ and Dumitru Baleanu2,3

¹Department of Mathematics, Faculty of Basic Science, Babol University of Technology, Babol 47148-71167, Iran

²Department of Mathematics and Computer Science, Cankaya University, Turkey

³Institute of Space Sciences, P.O. Box, MG-23, 76900 Magurele-Bucharest, Romania

Received 11 June 2013; Accepted 15 August 2013

Academic Editor: Ali H. Bhrawy

Copyright © 2013 Hadi Karami 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 discuss the existence and uniqueness of solution for two types of fractional order ordinary and delay differential equations. Fixed point theorems are the main tool used here to establish the existence and uniqueness results. First we use Banach contraction principle to prove the uniqueness of solution and then Krasnoselskii's fixed point theorem to show the existence of the solution under certain conditions in a Banach space.

1. Introduction

In mathematics delay differential equations are a type of differential equation in which the derivative of unknown function at a certain time is given in terms of the values of the function at previous times.

While physical events such as acceleration and deceleration take little time compared to the times needed to travel most distances, times involved in biological processes such as gestation and maturation can be substantial when compared to the data-collection times in most population studies. Therefore, it is often imperative to explicitly incorporate these process times into mathematical models of population dynamics. These process times are often called delay times, and the models that incorporate such delay times are referred as delay differential equation models [1, 2].

Recently theory of fractional differential equations attracted many scientists and mathematicians to work on them [3–12]. For the existence of solutions for fractional differential equations, one can see [13–30] and references therein. The results have been obtained by using fixed point theorems like Picard’s, Schauder fixed-point theorem, and Banach contraction mapping principle. About the development of existence theorems for fractional functional differential equations, many publications exist [31–35]. Many applications of fractional calculus amount to replacing the time derivative in a given evolution equation by a derivative of fractional order. The results of several studies clearly stated that the fractional derivatives seem to arise generally and universally from important mathematical reasons. Recently, interesting attempts have been made to give physical meaning to the initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives which could be found in [8, 9, 36, 37].

Recently Benchoohra et al. [28] studied existence of solutions for a class of fractional differential equations with infinite delay; namely,

where is the standard Riemann-Liouville fractional derivative and satisfies some assumptions.

First, in this paper we consider nonlinear delayed fractional differential equations:

associated with boundary conditions

where is the standard Riemann-Liouville fractional derivative and is a continuous function. Here represents the properitoneal state from time up to time which is defined by ,  . We proved the uniqueness of existence solutions for (2) with periodic boundary condition (3) under some further conditions.

For investigating to establish an existence theorem, we also consider a class of nonlinear delayed fractional differential equations of the form

with periodic boundary condition

The paper has been organized as follows. In Section 2 we give basic definitions and preliminary. Unique solution of (2)-(3) under some conditions is proved in Section 3. The existence solution of (4)-(5) under some assumptions has been presented in Section 4.

2. Preliminaries

For the convenience of the readers, we firstly present the necessary definitions from the fractional calculus theory and functional analysis. These definitions and results can be found in the literature [3, 7, 38].

Let be the Banach space of all continuous real functions defined on with the norm

Let , be the space of all functions such that which is a Banach space when endowed with the norm

Definition 1. For a function defined on an interval , the Riemann-Liouville fractional integral of of order is defined by

and Riemann-Liouville fractional derivative of of order defined by

provided that the right-hand side of the pervious equation is pointwise defined on .

We denote as and as . Further and are referred as and , respectively.

Definition 2. A two-parameter function of the Mittag-Leffler type is defined by

Definition 3. The beta function is usually defined by

and we have also the following expression for the beta function:

Theorem 4 (Arzela-Ascoli’s theorem). A subset of is compact if and only if it is closed, bounded, and equicontinuous.

Theorem 5 (Banach’s fixed point theorem). Consider a metric space , where . Suppose that is complete and is a contraction on . Then has precisely one fixed point.

Theorem 6 (Krasnoselskii’s fixed point theorem). Let be a nonempty closed convex subset of a Banach space . Suppose that , and map into such that

(1) for any we have ,(2) is a contraction,(3) is continuous and is contained in a compact set.

Then there exists such that .

3. Uniqueness of Solution

In this section we prove (2) with boundary condition (3) and another condition on has a unique solution. Before proving, we need to introduce some notations that will be provided in the following.

Let . Consider the operator defined by

where

is given in Section 4. Let be the function defined by

For each with we denote the function defined by

If satisfies the integral equation,

we can decompose as ,  , which implies for every , and function satisfies

Set . is Banach space with the norm . Let be defined by

Note that as operator has a fixed point, equivalently has a fixed point and so instead we try to prove that has a fixed point.

Theorem 7. Assume that there exists a constant such that for each and all . Then the problem (2)-(3) has a unique solution in provided that

Proof. We prove that is a contraction map. For each and for we have

using the definition of we get

Moreover,

Indeed we have

Note that

Hence we have

Using (22) and (23) we get

This completes the proof.

4. Existence of Solution

In this section, by using Krasnoselskii’s theorem, we discuss the existence solution of (4) under some assumptions on and further conditions. Before proving this theorem, we prove the following lemma which will be used in the next theorem.

Lemma 8. Consider the following nonlinear fractional differential equation of the form

with periodic boundary conditions

where is a continuous function. Then the periodic boundary value problem (28)-(29) is equivalent to an integral equation given by , where

Proof. We consider the following fractional differential equation:

with

where . Laplace transform of (31) yields

from which

and the inverse Laplace transform gives the solution

Hence we have

Therefore,

which leads to

since we have

Then the solution of the problem (28)-(29) is given by

This completes the proof.

Now we prove our main result using Lemma 8 and two more assumptions which follow next.

(H1) We assume that can be written as , where , are Lipschitz continuous. Moreover assume that the function , satisfies the following relations:

(H2) Let denote collection of the space of all function such that . Define the set , where satisfies

Furthermore we assume that

where is a Lipschitz constant of .

Theorem 9. If the assumptions (H1) and (H2) satisfied, then the problem (4) with periodic boundary value condition (5) has at least one solution.

Proof. (i) Note that by Lemma 8, (4)-(5) is equivalent to integral equation (17). Define by

For we have,

(ii) We will prove that is a contraction:

Then is a contraction.

(iii) Finally we prove that is continuous and is contained in a compact set. To prove the continuity of let us consider a sequence converging to . Taking the norm of we have

Hence whenever we have . This proves the continuity of .

On the other hand for and we have

Then we have

Hence we deduce that if then .

Then is equicontinuous. Moreover we show that is a bounded set in . Indeed we have

Then by Arzela-Ascoli’s theorem we conclude that is compact. By using Krasnoselskii’s theorem there exists such that is a fixed point of . This completes the proof.

5. Conclusions

We considered two types of nonlinear delay fractional differential equations (FDE) with periodic boundary conditions involving Remann-Liouville fractional derivative possessing with a lower terminal at 0. In order to obtain the results in this paper, we have shown the existence and the uniqueness of solution for a class of nonlinear delayed FDE by Banach contraction principle. Then using Krasnoselskii's fixed point theorem we established an existence theorem for a different type of the equation that we have proven its uniqueness theorem.

Acknowledgment

The authors would like thank to the anonymous referees for their valuable comments and suggestions.

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