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Some fixed point theorems for nonexpansive mappings in Hausdorff locally convex spaces Tan, Kok Keong
Abstract
Let X be a Hausdorff locally convex space, U be a base for closed absolutely convex Oneighborhoods in X , K C X be nonempty. For each U є U , we denote by P[subscript u] the gauge of U. Then T : K ↦ K is said to be nonexpansive w.r.t. U if and only if for each U є U, P[subscript u](T(x)  T(y)) ≤ P[subscript u](x  y) for all x, y є K; T: K ↦ K is said to be strictly contractive w.r.t. U if and. only if for each U є U, there is a constant λ[subscript u] with 0 ≤ λ[subscript u] < 1 such that P[subscript u](T(x)  T(y)) ≤λ[subscript u]P[subscript u](x  y) for all x, y є K . The concept of nonexpansive (respectively strictly contractive) mappings is originally defined on a metric space. The above definitions are generalizations if the topology on X is induced by a translation invariant metric, and in particular if X is a normed space. An analogue of the Banach contraction mapping principle is proved and some examples together with an implicit function theorem are shown as applications. Moreover several fixed point theorems for various kinds of nonexpansive mappings are obtained. The convergence of nets of nonexpansive mappings and that of fixed points are also studied. Finally on sets with 'complete normal structure', a common fixed point theorem is obtained for an arbitrary family of 'commuting' nonexpansive mappings while on sets with 'normal structure', a common fixed point, theorem for an arbitrary family of (not necessarily commuting) 'weakly periodic' nonexpansive mappings is obtained.
Item Metadata
Title  Some fixed point theorems for nonexpansive mappings in Hausdorff locally convex spaces 
Creator  Tan, Kok Keong 
Publisher  University of British Columbia 
Date Issued  1970 
Description 
Let X be a Hausdorff locally convex space, U be a
base for closed absolutely convex Oneighborhoods in X , K C X be
nonempty. For each U є U , we denote by P[subscript u] the gauge of U. Then
T : K ↦ K is said to be nonexpansive w.r.t. U if and only if for each
U є U, P[subscript u](T(x)  T(y)) ≤ P[subscript u](x  y) for all x, y є K; T: K ↦ K is
said to be strictly contractive w.r.t. U if and. only if for each U є U,
there is a constant λ[subscript u] with 0 ≤ λ[subscript u] < 1 such that
P[subscript u](T(x)  T(y)) ≤λ[subscript u]P[subscript u](x  y) for all x, y є K . The concept of nonexpansive (respectively strictly contractive) mappings is originally defined on a metric space. The above definitions are generalizations if the topology on X is induced by a translation invariant metric, and in particular if X is a normed space.
An analogue of the Banach contraction mapping principle is proved and some examples together with an implicit function theorem are shown as applications. Moreover several fixed point theorems for various kinds of nonexpansive mappings are obtained. The convergence of nets of nonexpansive mappings and that of fixed points are also studied.
Finally on sets with 'complete normal structure', a common fixed point theorem is obtained for an arbitrary family of 'commuting' nonexpansive mappings while on sets with 'normal structure', a common fixed point, theorem for an arbitrary family of (not necessarily commuting) 'weakly periodic' nonexpansive mappings is obtained.

Subject  Locally convex spaces 
Genre  Thesis/Dissertation 
Type  Text 
Language  eng 
Date Available  20110531 
Provider  Vancouver : University of British Columbia Library 
Rights  For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. 
DOI  10.14288/1.0080519 
URI  
Degree  Doctor of Philosophy  PhD 
Program  Mathematics 
Affiliation  Science, Faculty of; Mathematics, Department of 
Degree Grantor  University of British Columbia 
Campus  UBCV 
Scholarly Level  Graduate 
Aggregated Source Repository  DSpace 
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For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.