Recent Abstracts on Fixed Point Theory

Abstracts on Fixed Point Theory for 1997

Kirk W.A., Martinez Yanez C., Sik Shin S., Asymptotically Nonexpansive Mappings
Abstract: Suppose $X$ is a reflexive Banach space which has the property that each of its bounded closed convex sets has the fixed point property for nonexpansive mappings. It is known that in such spaces each bounded closed convex set also has the fixed point property for mappings which are eventually nonexpansive as well. It is shown here that this result extends to asymptotically nonexpansive mappings if at least one of the iterates is nonexpansive. It is also shown that if a Banach space has the super-fixed point property for nonexpansive mappings then it has the approximate fixed point property for pointwise asymptotically nonexpansive mappings. This leads to an entirely new approach in proving fixed point theorems for asymptotically nonexpansive mappings via a banach space ultrapower techniques..
For more on this abstract please contact the authors via email at william-kirk@uiowa.edu
Posted: February 25,1997.

Kirk W.A., Hyperconvexity of R-Trees
Abstract: It is shown that for a metric space $(M,d)$ the following are equivalent. (i)M is a complete R-tree. (ii) M is hyperconvex and has unique metric segments.
For more on this abstract please contact the authors via email at william-kirk@uiowa.edu
Posted: February 25,1997.

Priess-Crampa S., Ribenboim P. , Applications of Ultrametric Spaces to Logic Programming
Abstract: In this paper, the authors use a fixed point theorem in generalized ultrametric spaces to obtain a criterion for a program to fix a valuation, having thereby a Herbrand Model.
For more on this abstract please contact the authors via email at priess@rz.mathematik.uni-muenchen.de
Posted: March 4,1997.

Jachymski J.R. , Some Consequences of Fundamental Ordering Principles in Metric Fixed Point Theory
Abstract: We present some results of metric fixed point theory, which can be derived from the following fixed point theorems involving a partial ordering: Zermelo's theorem, the Knaster-Tarski theorem and the Tarski-Kantorovitch theorem. Using ideas of Fuchssteiner and Manka we also establish another generalization of Zermelo theorem, which enables to give constructive proofs (without a help of the Axiom of Chaoice) of fixed point theorems due to Caristi and Khamsi-Kreinovich.
For more on this abstract please contact the authors via email at jachym@ck-sg.p.lodz.pl
Posted: November 19,1997.

Jachymski J.R. , Caristi Fixed Point Theorem and Selection of Set-Valued Contractions
Abstract: We show (without the Axiom of Choice) that the Zermelo Theorem implies directly a restriction of the Caristi fixed point theorem to continuous functions. Under the Axiom of Choice, this restriction is proved to be equivalent to Caristi Theorem. We also discuss Kirk's problem on an extension of the Caristi theorem and we establish two selection theorems for set-valued contractions.
For more on this abstract please contact the authors via email at jachym@ck-sg.p.lodz.pl
Posted: November 19,1997.

Jachymski J.R. , Fixed Point Theorems in Metric and Uniform spaces via the Knaster-Tarski Principle
For more on this abstract please contact the authors via email at jachym@ck-sg.p.lodz.pl
Posted: November 19,1997.

Jachymski J.R. , Some Consequences of the Tarski-Kantorovitch ordering theorem in metric fixed point theory
Abstract: We show that the Tarski-Kantorovitch principle for continuous maps on a partially ordered set yields some fixed point theorems for contractive maps on a uniform space. Our proofs do not depend on the Axiom of Choice.
For more on this abstract please contact the authors via email at jachym@ck-sg.p.lodz.pl
Posted: November 19,1997.

B. Sims, H. K. Xu and George Xian-Zhi Yuan, The Homotopic Invariance for Fixed Points of Set-valued Nonexpansive Mappings.
Abstract: The homotopic invariance of fixed points of set-valued contractions and nonexpansive mappings is studied at the first time. As application, nonlinear alternative principles are given. A Leray-Schauder alternative and an antipodal theorem for set-valued nonexpansive mappings are also included.
For more on this abstract please contact the authors via email at xzy@maths.uq.edu.au
Posted: December 21,1997.

W. Art Kirk, B. Sims and George Xian-Zhi Yuan., The Knaster - Kuratowski and Mazurkiewicz Theory in Hyperconvex Metric Spaces and Some of Its Applications
Abstract: In this paper, we establish the characterization of the Knaster-Kuratowski and Mazurkiewicz principle in hyperconvex metric spaces. The characterization for a family of subsets with the finite intersection property in hyperconvex metric spaces is then given. As applications, we give hyperconvex versions of Fan's celebrated minimax principle and Fan's best approximation theorem for set-valued mappings, which in turn are applied to give the Browder-Fan fixed point theorem and the Schauder-Tychonoff fixed point theorem in hyperconvex metric spaces for set-valued mappings. Finally the existence theorems for saddle points, intersection theorems and Nash equilibria are given. Our results unify and improve corresponding results in the literature as special cases.
For more on this abstract please contact the authors via email at xzy@maths.uq.edu.au
Posted: December 21,1997.

Q. Luo, George Xian-Zhi Yuan and T. X. Sun, The existence of essential components for coincident points of set-valued mappings.
Abstract: In this paper, the concept of essential components for coincident points of set-valued mappings is first introduced and then we prove the existence of essential components of coincident points for upper semicontinuous set-valued mappings in the setting of Banach spaces.
For more on this abstract please contact the authors via email at xzy@maths.uq.edu.au
Posted: December 23,1997.

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