Multivalued Exponentiation Analysis. Part I: Maclaurin Exponentials

The exponentiation theory of linear continuous operators on Banach spaces can be extended in manifold ways to a multivalued context. In this paper we explore the Maclaurin exponentiation technique which is based on the use of a suitable power series. More precisely, we discuss about the existence and characterization of the Painlevé–Kuratowski limit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[{\rm Exp}\;F](x)= \lim_{n\to\infty}\sum_{p=0}^n \frac{1}{p!}F^p(x)$$\end{document}under different assumptions on the multivalued map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F\!:X\rightrightarrows X$\end{document}. In Part II of this work we study the so-called recursive exponentiation method which uses as ingredient the set of trajectories associated to a discrete time evolution system governed by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F$\end{document}.