# Fixed point theorems in uniform spaces

Russell, Wayne Cyril (1970) Fixed point theorems in uniform 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.

• [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

A mapping F of a metric space X into itself is said to satisfy a Lipschitz condition with Lipschitz constant K if d(F(x), F(y)) ≤ K d(x, y) , (x, y εX). If this condition is satisfied with a Lipschitz constant K such that 0 ≤ K < 1 then F is called a contraction mapping. If we let K = 1 the mapping is called non-expansive, and if K = 1 and we have a strict inequality it is called contractive. -- In this thesis we give a survey of the various definitions offered for non-expansive, contractive and contraction mappings in uniform spaces. In particular we study the following definition of a U-contractive mapping given by Casesnoves) [3 ]. DEFINITION: If (E, U) is a complete uniform space and F a map of E into itself such that g = (F, F) is the extension of F to the product space E x E, then F is said to be U-contractive, provided the following conditions are satisfied. -- (a) V ε U , g(V) C V -- (b) V V, V W ε U, k ε N, V p > 0 , V n ≥ k -- gn(V)0gn+1 (V) 0 ... 0 gn+p (V) c W. -- We consider also sequences of contraction mappings in metric and uniform spaces. In metric spaces we prove a theorem for a sequence of contraction mapping of a complete ε - chainable metric space. In uniform spaces we prove the following theorem and then show how it may be used to prove other results for sequences of mappings in uniform spaces. -- THEOREM: Let (E, U) be a complete uniform space and Fk a U-contractive mapping from E into itself, with fixed points Uk (k = 1, 2, ... ). Suppose lim -- [special characters omitted] -- Fk(x) = F(x) for every x ε E, where F is a U-contractive mapping from E into itself. Then lim -- [special characters omitted] -- Uk = U, where U is a fixed point of F.

Item Type: Thesis (Masters) http://research.library.mun.ca/id/eprint/7645 7645 Bibliography: leaves 43-46. Science, Faculty of > Mathematics and Statistics 1970 Submission Metric spaces; Fixed point theory View Item