Ivimey, William (1972) Multivalued contraction mappings and fixed points in metric spaces. Masters thesis, Memorial University of Newfoundland.
[English]
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Abstract
The purpose of this thesis is to set forth some fixed point theorems in (complete) metric spaces for singlevalued and multivalued contraction mappings; with emphasis on multivalued 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 nonexpansive mappings are also given.  In Chapter II, in the main, we shall consider multivalued 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 multivalued 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 multivalued 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 singlevalued mappings will be given which are analogous to those for multivalued mappings.  In Chapter III, we consider sequences of singlevalued and multivalued contraction mappings and fixed points.
Item Type:  Thesis (Masters) 

URI:  http://research.library.mun.ca/id/eprint/7178 
Item ID:  7178 
Additional Information:  Bibliography: leaves 6568. 
Department(s):  Science, Faculty of > Mathematics and Statistics 
Date:  1972 
Date Type:  Submission 
Library of Congress Subject Heading:  Banach spaces; Distance geometry 
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