Existence and Uniqueness of Solution for a Class of Nonlinear Fractional Order Differential Equations

Abstract

We discuss the existence and uniqueness of solution to nonlinear fractional order ordinary differential equations (Dα-ρtDβ)x(t)=f(t,x(t),Dγx(t)), t∈(0,1) with boundary conditions x(0)=x0, x(1)=x1 or satisfying the initial conditions x(0)=0, x′(0)=1, where Dα denotes Caputo fractional derivative, ρ is constant, 1<α<2, and 0<β+γ≤α. Schauder's fixed-point theorem was used to establish the existence of the solution. Banach contraction principle was used to show the uniqueness of the solution under certain conditions on f.