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

Hadi Karami,Azizollah Babakhani,Dumitru Baleanu

doi:10.1155/2013/176180

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.

Highlights

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
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

Summary

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. Benchoohra et al [28] studied existence of solutions for a class of fractional differential equations with infinite delay; namely, Dαu (t) = u (t) + f (t, ut) , for each t ∈ [0, b] , 0 < α < 1, u (t) = φ (t) , t ∈ [−∞, 0] , (1). For investigating to establish an existence theorem, we consider a class of nonlinear delayed fractional differential equations of the form.

Preliminaries

Uniqueness of Solution

Existence of Solution

Conclusions