Sequences of mappings and their fixed point.

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

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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.

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