Initial-value problems for nonlinear hybrid implicit Caputo fractional differential equations

Abstract

We study the existence and uniqueness of solutions of a nonlinear hybrid implicit Caputo fractional differential equation ................................... ................................... .................................. where.................is the Caputo fractional derivative of order ................... In the process we convert the given fractional differential equation into an equivalent integral equation. Then we construct an appropriate mapping and employe the Banach’s fixed point theorem to show the existence and uniqueness of solutions of this equation. We also use the generalization of Gronwall’s inequality to show the estimate of the solution. Keywords: Implicit fractional differential equations, Caputo fractional derivatives, fixed point theorems, existence, uniqueness.