Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay

Abstract

Abstract In this work, we investigate the existence and uniqueness of solutions for retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay. To achieve this goal, we select an appropriate phase space that satisfies the axiomatic theory developed by Hale and Kato for studying differential equations with infinite delay. Then, we reformulate the problem of existence of solutions as the problem of finding fixed points of an operator. To this end, we apply the Karakostas fixed point theorem, which is an extension of the well-known Krasnoselskii fixed point theorem. Under certain conditions, we establish the uniqueness of solutions for our problem. Finally, we analyze the prolongation of solutions and demonstrate that, given certain conditions, these solutions are globally defined. Our research has important implications for the study of delayed dynamical systems, including models of population dynamics, physiological processes, and engineering systems.