Classes of maps with continuous selections and homotopy extension

This paper presents a variety of homotopy principles for multivalued maps with continuous selections based on null-homotopic, extendability, essentiality and the Urysohn function. In particular we first present two fixed point results for a multimap phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} where (i). phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} has a continuous single-valued selection f, (ii). f is homotopic to a constant map, and (iii). a "null-homotopic compact self map" property holds. Next some examples of the null-homotopic property are given via the Arens-Eells theorem and the Schauder fixed point theorem. The final part of the paper discusses essential maps in a very general setting and in this setting we establish that if two multivalued maps (with continuous selections) are homotopic and one is an essential map then the other map has a fixed point.