# Sequences of mappings and their fixed point.

Collins, Glenn Wilfred (1973) Sequences of mappings and their fixed point. Masters thesis, Memorial University of Newfoundland. [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. Download (3MB)

## Abstract

In Chapter I of this thesis, we attempt to give a comprehensive survey of most of the well known results related to fixed point theorems in metric spaces. The most famous, of course, is the Banach Contraction Principle which states: "A contraction mapping of a complete metric space into itself has a unique fixed point". Then, generalizations of this theorem in metric spaces are given. Results are also included for contractive and nonexpansive mappings. -- In Chapter II, we make a detailed study of the conditions under which the convergence of a sequence of contraction mappings to a mapping T of a metric space into itself implies the convergence of their fixed points to the fixed point of T. The solution given by Bonsall and its generalizations are first given. -- The converse problem as studied by Ng is also briefly considered. -- In the final section of the chapter, we investigate a few interesting results as a solution to the problem posed above for the following types of mappings introduced recently. -- f : X + X such that -- (i) d(f(x),f(y)) ≤ ad(x,f(x)) + bd(y,f(y)) -- (ii) d(f(x),f(y)) ≤ ad(x,f (y)) + bd(y,f(x)) -- (iii) d(f(x),f(y)) ≤ ad(x,f(x)) + bd(y,f(y)) + cd(x,y) -- (iv) d(f(x),f(y)) ≤ ad(x,f(y)) + bd(y,f(x)) + cd(x,y) -- (v) d(f(x),f(y)) ≤ ad(x,f(x)) + bd (y,f(y)) + cd(x,f(y)) + ed(y,f(x)) + gd(x,y) -- for all x,y εX where a,b,c,e and g are nonnegative real numbers.

Item Type: Thesis (Masters) http://research.library.mun.ca/id/eprint/10327 10327 Bibliography : leaves 51-54. Science, Faculty of > Mathematics and Statistics 1973 Submission Functional analysis; Metric spaces. View Item