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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.
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Title 
Some fixed point theorems for nonexpansive mappings in Hausdorff locally convex spaces

Creator  
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.

Genre  
Type  
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  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

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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.