Let X be a Hausdorff locally convex space, U be a
base for closed absolutely convex O-neighborhoods 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. / Science, Faculty of / Mathematics, Department of / Graduate
Identifer | oai:union.ndltd.org:UBC/oai:circle.library.ubc.ca:2429/34966 |
Date | January 1970 |
Creators | Tan, Kok Keong |
Publisher | University of British Columbia |
Source Sets | University of British Columbia |
Language | English |
Detected Language | English |
Type | Text, Thesis/Dissertation |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
Page generated in 0.0019 seconds