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

We discuss the existence and uniqueness of solution to nonlinear fractional order ordinary differential equations (D-alpha - rho tD(beta))x(t) = f(t, x(t), D(gamma)x(t)), t is an element of (0, 1) with boundary conditions x(0) = x(0), x(1) = x(1) or satisfying the initial conditions x(0) = 0, x'(0) = 1, where D-alpha denotes Caputo fractional derivative, rho is constant, 1 < alpha < 2, and 0 < beta + gamma <= alpha. 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.