A fixed point theorem for conical shells

We prove a fixed point theorem for mappings f defined on conical shells F in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb R^n$$\end{document}, where the image of f need not be a subset of F, nor even a subset of the cone that contains F. In this sense, our results extend, in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb R^n$$\end{document}, Krasnosel’skiĭ’s well-known fixed point result on cones in Banach spaces (Krasnosel’skiĭ, Soviet Math Dokl 1:1285–1288, 1960). Sufficiency for fixed points of f is dependent only on the behavior of f on the boundary of F. This behavior is related to notions of compressing or extending the conical shell F. We also discuss possible extensions of our theorem to infinite dimensional Banach spaces.