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

Volume 13, Issue 4, pp 180--186 http://dx.doi.org/10.22436/jnsa.013.04.02
Publication Date: January 08, 2020 Submission Date: August 26, 2019 Revision Date: October 31, 2019 Accteptance Date: November 27, 2019


A. M. A. El-Sayed - Faculty of Science, Alexandria University, Alexandria, Egypt. Sh. M. Al-Issa - Faculty of Science, Lebanes International University, Beirut, Lebanon. - Faculty of Science, The International University of Beirut, Saida, Lebanon.


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 \(L^1\)-Caratheodory 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.