Multi-valued contraction mappings and fixed points in metric spaces

Ivimey, William (1972) Multi-valued contraction mappings and fixed points in metric spaces. Masters thesis, Memorial University of Newfoundland.

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Abstract

The purpose of this thesis is to set forth some fixed point theorems in (complete) metric spaces for single-valued and multi-valued contraction mappings; with emphasis on multi-valued contraction mappings. -- In Chapter I, we discuss Banach's Contraction Mapping Principle and present some fixed point theorems in metric and complete metric spaces which extend and generalize Banach's result. Some results on contractive and non-expansive mappings are also given. -- In Chapter II, in the main, we shall consider multi-valued contraction mappings. In this respect, we have the following main definition and theorem due to NadlerJr[21]. -- Definition. Let (X,d) be a complete metric space, let CB(X) denote the nonempty closed and bounded subsets of X, and let H be the Hausdorff metric for CB(X). A function F : X → CB(X) is said to be a multi-valued contraction mapping if and only if there is a real number α, 0 ≤ α < 1, such that H(F(x), F(y)) ≤ αd(x,y), for all x,y ε X. -- Theorem. If (X,d) is a complete metric space and F : X → CB(X) is a multi-valued contraction mapping, then F has a fixed point (i.e., there is an x₀ ε X such that x₀ ε F(x₀)). -- Other results due to Nadler Jr. will also be given for metric spaces and generalized metric spaces, and these results will be extended to the following mappings F : X → CB(X) such that -- (A) H(F(x), F(y)) ≤ α[D(x, F(x)) + D(y, F(y))] for all x,y ε X, 0 ≤ α < 1/2; -- (B) H(F(x), F(y)) ≤ α[D(x, F(x)) + D(y, F(y)) + d(x,y)] for all x,y ε X, 0 ≤ α < 1/3. -- Some fixed point theorems for single-valued mappings will be given which are analogous to those for multi-valued mappings. -- In Chapter III, we consider sequences of single-valued and multi-valued contraction mappings and fixed points.

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