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

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 in $\Omega =\{y:(-\infty ,b]\to ℝ:y{|}_{(-\infty ,0]}\in ℬ\}$ such that $y{|}_{[0,\mathrm{b}]}$ is continuous and $ℬ$ is a phase space.