Fixed point theorems for single and multi-valued mappings

Abstract

The purpose of this thesis is to set forth some fixed point theorems in metric and Banach spaces for single-valued and multi-valued mappings of a contractive type. -- In Chapter I, we discuss the Banach Contraction Principle and its generalization in metric spaces, including the major known results for contraction and contractive mappings. We also consider recent developments in the study of fixed points for multi-valued mappings of this type. -- Chapter II is devoted to fixed point theorems for nonexpansive mappings and for mappings characterized by the property that they do not increase the “measure of non-compactness” of bounded non-precompact sets. Again we mention results for both the single and multi-valued case. -- In Chapter III, we focus our attention on those fixed point theorems that have been obtained by imposing a convexity condition on the mapping. We also provide some generalizations of these results as well as a theorem for commutative families of mappings. Recent extensions of the convexity concept and related results for multi-valued mappings are also given.