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 $\left({\mathcal{D}}^{\alpha }-\rho t{\mathcal{D}}^{\beta }\right)x\left(t\right)=f\left(t,x\left(t\right),{\mathcal{D}}^{\gamma }x\left(t\right)\right)$, $t\in (0,1)$ with boundary conditions $x\left(0\right)=x_0,x\left(1\right)=x_1$ or satisfying the initial conditions $x\left(0\right)=0,x^{\prime }\left(0\right)=1$, where ${\mathcal{D}}^{\alpha }$ denotes Caputo fractional derivative, $\rho$ is constant, and . 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 $f$.