Fractional Differential Equations with Fractional Boundary Conditions

This work is dedicated to investigate the existence and uniqueness of solutions for nonlinear fractional differential equations with boundary conditions involving the Caputo fractional derivative in a Banach space. After introducing some basic preliminaries and the important concepts of fractional calculus, we considered two models of boundary value problems of Caputo fractional derivative. The first one is nonlinear fractional differential equation with nonlocal four-point fractional boundary conditions. The second equation is nonlinear impulsive boundary value problem of multi-orders fractional supplemented with nonlocal four-point fractional boundary conditions. The existence and uniqueness of solution are obtained via Banach’s fixed point theorem and Schauder’s fixed point theorem for the two models. In addition, both results are provided by the illustrative examples to support them.