Integrable Solutions and Continuous Dependence of a Nonlinear Singular Integral Inclusion of Fractional Orders and Applications

Let E be a reflexive Banach space. In this article we study the existence of integrable solutions in the space of all lesbesgue integrable functions on E , L 1 ( [ 0, T] , E) , of the nonlinear singular integral inclusion of fractional orders beneath the assumption that the multi-valued function G has Lipschitz selection in E . The main tool applied in this work is the Banach contraction fixed point theorem. Moreover, the paper explores a qualitative property associated with these solutions for the given problem such as the continuous dependence of the solutions on the set of selections S 1 G (tau, x (tau)) . As an application, the existence of integrable solutions of the two nonlocal and weighted problems of the fractional differential ff erential inclusion is investigated. We additionally provide an example given as a numerical application to demonstrate the effectiveness ff ectiveness and value of our results.