Generalized Dichotomies and Hyers–Ulam Stability

We consider a semilinear and nonautonomous differential equation 1x′=A(t)x+f(t,x)t≥0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} x'=A(t)x+f(t,x) \quad t\ge 0, \end{aligned}$$\end{document}acting on an arbitrary Banach space X. Provided that the linear part x′=A(t)x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x'=A(t)x$$\end{document} exhibits a very general form of dichotomic behaviour and that the nonlinear term f is Lipschitz in the second variable (with a suitable Lipshitz constant), we prove that (1) admits two different forms of a generalized Hyers–Ulam stability. Moreover, we obtain the converse result which shows that under suitable additional assumptions, the presence of these two forms of a generalized Hyers–Ulam stability for the linear equation x′=A(t)x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x'=A(t)x$$\end{document} implies that it exhibits this general dichotomic behaviour.