Some fixed point theorems in metric spaces

Meade, Barbara Ann Moores (1977) Some fixed point theorems in metric spaces. Masters thesis, Memorial University of Newfoundland.

[English] PDF (Migrated (PDF/A Conversion) from original format: (application/pdf)) - Accepted Version Available under License - The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. Download (16MB) [English] PDF - Accepted Version Available under License - The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. (Original Version)

Abstract

The main object of this thesis is to study contractive type mappings on a complete metric space. These mappings are generalizations of the well known Banach contraction and have the property that each such mapping has a unique fixed point. -- In the the beginning we give some definitions of mappings of this type and theorems which give the conditions necessary to guarantee, for different definitions, the existence of a unique fixed point. We also consider common fixed points of pairs of mappings which satisfy a contractive type condition between the pair. -- In Chapter II, special consideration is given to mappings T which satisfy a condition involving ᶲ : R⁺ → R⁺ such that d(Tx,Ty) < ᶲ(d(x,y)). This idea has been extended to a pair of maps so that ᶲ : (R⁺)⁵ → R⁺ and ᶲ acts upon the five terms: d(x,y), d(x,Tx), d(y,Ty), d(x,Ty), and d(y,Tx). Also mappings which satisfy a commuting condition are considered. -- In the end sequences of mappings are considered. These mappings or their limiting mappings satisfy contractive type conditions and theorems are given which contain the conditions necessary for the sequence of fixed points to converge to the fixed point of the limiting mapping.

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