Iterated contraction mappings and fixed point theorems in metric spaces

Hynes, Frank (1972) Iterated contraction mappings and fixed point theorems in metric spaces. Masters thesis, Memorial University of Newfoundland.

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Abstract

The main object of this thesis is to study fixed point theorems for iterated contraction mappings in metric and generalized metric spaces. -- In the beginning, we have discussed Banach’s Contraction Principle, "A contraction mapping of a complete metric space into itself has a unique fixed point”, together with its various generalizations in metric spaces. -- Then the iterated contraction mapping, “d(Tx,TTx) ≤ kd(x,Tx), for all x, Tx E (X,d), T : X – X, and for some constant k, 0 ≤ k < 1", has been considered. We have given the iterated contraction mapping principle, "An iterated contraction mapping, which is continuous at the limit of its sequence of iterates, of a complete metric space into itself has a fixed point", by following the procedure of the Banach Contraction Principle. Then generalizations of the iterated contraction principle have been given in metric spaces. Iterated contractive mappings, "d(Tx,TTx) < d(x,Tx), for all x, Tx E (X,d), x = Tx, T : X – X” and iterated nonexpansive mappings, “d(Tx,TTx) ≤ d(x,Tx), for all x, Tx E (X,d), T : X – X”, have also been considered briefly in metric spaces. -- In the end, we have given the results on iterated contraction mappings in generalized metric spaces.

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