Existence of solutions for sequential fractional integro-differential equations and inclusions with nonlocal boundary conditions

Abstract

We investigate the existence of solutions for Caputo type sequential fractional integro-differential equations and inclusions subject to nonlocal boundary conditions involving Riemann–Liouville and Riemann–Stieltjes integrals. For the proofs of our main theorems we use the contraction mapping principle and the Krasnosel’skii fixed point theorem for the sum of two operators in the case of fractional equations, and the nonlinear alternative of Leray–Schauder type for Kakutani maps and the Covitz–Nadler fixed point theorem in the case of fractional inclusions. Some examples are presented to illustrate our results.