The Existence and Uniqueness of Solutions for a Class of Nonlinear Fractional Differential Equations with Infinite Delay

Azizollah Babakhani,Dumitru Baleanu,Ravi P Agarwal

doi:10.1155/2013/592964

Abstract

We prove the existence and uniqueness of solutions for two classes of infinite delay nonlinear fractional order differential equations involving Riemann-Liouville fractional derivatives. The analysis is based on the alternative of the Leray-Schauder fixed-point theorem, the Banach fixed-point theorem, and the Arzela-Ascoli theorem inΩ={y:(−∞,b]→ℝ:y|(−∞,0]∈ℬ}such thaty|[0,b]is continuous andℬis a phase space.

Highlights

Fractional derivatives and integrals have been vastly used in different fields, facing a huge development especially during the last few decades
The existence and the uniqueness of solutions for the nonlinear fractional differential equations with infinite delay comprising standard Riemann-Liouville derivatives have been discussed in the phase space
Further generalizations can be developed to some other class of fractional differential equations such as L(D)y(t) = f(t, yt), where L(D) = Dαn − ∑nj=−11 pj(t)Dαn−j, 0 < α1 < ⋅ ⋅ ⋅ < αn < 1, pj(t) = ∑Nk=j0 ajktk, and Nj is nonnegative integer

Summary

Introduction

Fractional derivatives and integrals have been vastly used in different fields, facing a huge development especially during the last few decades (see, e.g., [1,2,3,4,5,6,7,8,9] and the references therein). Varieties of schemes for numerical solutions of fractional differential equations are reported (see, e.g., [6, 21,22,23] and the references therein). We notice that some investigations have been done on the existence and uniqueness of solutions for fractional differential equations with delay (see, e.g., [24, 25] and the references therein). Having all the aforementioned facts in mind, in this paper we study the existence and uniqueness of solutions for a class of delayed fractional differential equations, namely,. We consider the following nonlinear fractional differential equation:. The notion of the phase space B plays an important role in the study of both qualitative and quantitative theories for functional differential equations.

Preliminaries

Existence and Uniqueness

Conclusions