Nonuniform Exponential Dichotomy for Evolution Families on the Real Line

This paper presents some Perron-type results for the nonuniform exponential dichotomy of evolution families on the real line with nonuniform exponential growth. In this manuscript, we describe the admissibility of the pair of spaces (Lp(X),Lq(X))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\mathcal{L}^p(X), \mathcal{L}^q(X))}$$\end{document} to an evolution family, when (p,q)≠(1,∞).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(p,q) \neq (1,\infty).}$$\end{document} This notion is used to obtain a result for the nonuniform exponential dichotomy for an evolution family on the real line.