Existence of integrable solutions for integro-differential inclusions of fractional order; coupled system approach

In this article, we establish the existence of solutions for a functional integral equation of fractional order. The study upholds the case when the set-valued function has L1-Carathèodory selections, we reformulate the functional integral inclusion according to these selections via a classical fixed point theorem of Schauder and present theorem for the existence of integrable solutions. As an application, the existence of solutions of nonlinear functional integro-differential inclusion with an initial value, and the initial value problem for the arbitrary-order differential inclusion will be studied.