Weighted pseudo almost periodic solutions to a class of semilinear integro-differential equations in Banach spaces

In this paper we prove the existence of weighted pseudo almost periodic mild solutions for the class of integro-differential equations in the form u′(t)=Au(t)+α∫−∞te−β(t−s)Au(s)ds+f(t,u(t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u'(t)=Au(t)+\alpha\int_{-\infty}^{t}e^{-\beta(t-s)}Au(s) \,ds+f(t,u(t)) $\end{document} where f(⋅,u(⋅))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(\cdot,u(\cdot))$\end{document} is a Stepanov-like weighted pseudo almost periodic function and A generates an immediately norm continuous C0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{0}$\end{document}-semigroup on a Banach space X. Also, we give a short proof to show that the vector-valued space of Stepanov-like weighted pseudo almost periodic functions is a Banach space.