Measure of noncompactness, densifying mappings and fixed points

Abstract

The purpose of this thesis is to set forth some fixed point theorems in metric and Banach spaces for densifying mappings and k-set-contractions. -- In Chapter I, we discuss the Banach Contraction Principle and give an extensive coverage of its generalizations. Results are also given for contractive and nonexpansive mappings, and for mappings of an iterative type. -- In Chapter II, we introduce the concepts of α-measure (and X-measure) of noncompactness; some geometric properties are given. -- In Chapter III and IV we make a detailed study of some of the recent fixed point results in metric and Banach spaces respectively. -- Particular attention is given to the works of Furi, Vignoli, Nussbaum, and Petryshyn. -- In Chapter V, we give applications of α-measure to problems in optimization theory.