Existence and Approximate Solutions for Fractional Differential Equations with Nonlocal Conditions

In this paper the authors prove existence, uniqueness and approximation of the solutions for initial value problems of nonlinear fractional differential equations with nonlocal conditions, using the operator theoretic technique in a partially ordered metric space. The main results rely on the Dhage iteration principle embodied in the recent hybrid fixed point theorem of Dhage (2014) in a partially ordered normed linear space. The approximation of the solutions of the considered nonlinear fractional differential equations are obtained under weaker mixed partial continuity and partial Lipschitz conditions. Our hypotheses and result are also illustrated by a numerical example.