Bounded Mild Solutions for Semilinear Integro Differential Equations in Banach Spaces

In this paper we study the structure of various classes of spaces of vector-valued functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{M}(\mathbb{R};X)}$$\end{document} ranging between periodic functions and bounded continuous functions. Some of these functions are introduced here for the first time. We propose a general operator theoretical approach to study a class of semilinear integro-differential equations. The results obtained are new and they recover, extend or improve variety of recent works.