A Nonlinear Implicit Fractional Equation with Caputo Derivative

Abstract

In this paper, we study a nonlinear implicit differential equation with initial conditions. The considered problem involves the fractional Caputo derivatives under some conditions on the order. We prove an existence and uniqueness analytic result by application of Banach principle. Then, another result that deals with the existence of at least one solution is delivered and some sufficient conditions for this result are established by means of the fixed point theorem of Schaefer. At the end, we discuss two examples to illustrate the applicability of the main results.