Fixed point theorems in uniform spaces

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.