A Cauchy Problem for Functional Inclusions with Causal Operators in Banach Spaces

Abstract

In this paper, we consider the Cauchy problem for functional inclusions containing the sum of n causal single-valued operators and a multivalued causal operator in Banach spaces. The peculiarity of single-valued operators is that, starting from the second term, each operator is represented by an integral with a variable upper limit of the previous term. Such functional inclusions generalize the Cauchy problem for semilinear differential equations and inclusions of arbitrary order n, as well as a Cauchy-type problem in the case of inclusions and equations of fractional order not exceeding n. To solve the problem, we will apply the theory of topological degree for multivalued condensing mappings. To prove the existence of a solution, we will construct a resolving multivalued operator in the space of continuous functions corresponding to the problem. Based on the properties of the resolving operator, we will prove a theorem on the existence of solutions to this problem.