Russell, Wayne Cyril (1970) Fixed point theorems in uniform spaces. Masters thesis, Memorial University of Newfoundland.
[English]
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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 nonexpansive, 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 nonexpansive, contractive and contraction mappings in uniform spaces. In particular we study the following definition of a Ucontractive 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 Ucontractive, 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 Ucontractive 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 Ucontractive mapping from E into itself. Then lim  [special characters omitted]  Uk = U, where U is a fixed point of F.
Item Type:  Thesis (Masters) 

URI:  http://research.library.mun.ca/id/eprint/7645 
Item ID:  7645 
Additional Information:  Bibliography: leaves 4346. 
Department(s):  Science, Faculty of > Mathematics and Statistics 
Date:  1970 
Date Type:  Submission 
Library of Congress Subject Heading:  Metric spaces; Fixed point theory 
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