Integro-differential equations with delays: A perturbation approach

Abstract: This paper focuses on the study of integro-differential equations with delays, presenting a novel perturbation approach. The primary objective is to introduce the concepts of classical and mild solutions for these equations and establish their existence and uniqueness, under suitable assumptions. Furthermore, we provide a variation of constants formula that characterizes these solutions. To illustrate the applicability of the proposed methodology, we present an example of integro-differential Volterra equations with a nonlocal kernel. In addition to the aforementioned contributions, a secondary goal of this paper is to address an issue concerning the statement and proof of a fundamental theorem presented in a previous work \cite{Zaza}. Specifically, we aim to rectify the statement and provide a corrected proof for Theorem 2.6 in \cite{Zaza}. By doing so, we enhance the accuracy and reliability of the existing literature in this field.