An extension of picard-lindelöff theorem to fractional differential equations^*

Abstract

The method of contraction mapping defined on a complete metric space is used to prove the existence and uniqueness of solutions of the system of differential equations: together with the initial condition y(xo)=yo, where y(α is the Riemann-Liouville derivative of non integer order a of a real valued vector function y(x), under usual continuity and Lipschitz conditions on the function f. The classical Picard-Lindeloff theorem for ordinary differential equations of integer order is a special case of the main result when α = 1. Some examples are given. Finally, consequences for the linear case are obtained.