Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-22T17:28:36.293Z Has data issue: false hasContentIssue false
  • English
  • Français

AMENABLE SEMIGROUPS OF NONLINEAR OPERATORS IN UNIFORMLY CONVEX BANACH SPACES

Part of: Nonlinear operators and their properties

Published online by Cambridge University Press:  28 November 2018

KHADIME SALAME Open the ORCID record for KHADIME SALAME [Opens in a new window]
KHADIME SALAME*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada email [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In 1965, Browder proved the existence of a common fixed point for commuting families of nonexpansive mappings acting on nonempty bounded closed convex subsets of uniformly convex Banach spaces. The purpose of this paper is to extend this result to left amenable semigroups of nonexpansive mappings.

Keywords

amenabilityasymptotic centrecommon fixed pointnonexpansive mappinguniformly convex space

MSC classification

Primary: 47H20: Semigroups of nonlinear operators
Secondary: 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc. 47H10: Fixed-point theorems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 2 , April 2019 , pp. 284 - 292
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

Agarwal, R. P., O’Regan, D. and Sahu, D. R., Fixed Point Theory for Lipschitzian-type Mappings with Applications (Springer, New York, 2009).Google Scholar
Alspach, D., ‘A fixed point free nonexpansive map’, Proc. Amer. Math. Soc. 82 (1981), 423–424.Google Scholar
Berglund, J. F., Junghenn, H. D. and Milnes, P., Analysis on Semigroups (John Wiley & Sons, New York, 1989).Google Scholar
Browder, F. E., ‘Nonexpansive nonlinear operators in a Banach space’, Proc. Natl Acad. Sci. USA 54 (1965), 1041–1044.Google Scholar
Day, M. M., ‘Amenable semigroups’, Illinois J. Math. 1 (1957), 509–544.Google Scholar
DeMarr, R., ‘Common fixed points for commuting contraction mappings’, Pacific J. Math. 13 (1963), 1139–1141.Google Scholar
Granirer, E., ‘On amenable semigroups with a finite dimensional set of invariant means II’, Illinois J. Math. 7 (1963), 49–58.Google Scholar
Kirk, W. A. and Goebel, K., Topics in Metric Fixed Point Theory (Cambridge University Press, Cambridge, 1990).Google Scholar
Mitchell, T., ‘Fixed point of reversible semigroups of non-expansive mappings’, Kodai Math. Sem. Rep. 22 (1970), 322–323.Google Scholar
Salame, K., ‘Nonlinear common fixed point properties of semitopological semigroups in uniformly convex spaces’, J. Fixed Point Theory Appl. 19 (2017), 1041–1057.Google Scholar
Takahashi, W., ‘Fixed point theorem for amenable semigroups of nonexpansive mappings’, Kodai Math. Rep. 21 (1969), 383–386.Google Scholar
Takahashi, W., Nonlinear Functional Analysis: Fixed Point Theory and its Applications (Kindai-Kagakusha, Tokyo, 1988), (in Japanese).Google Scholar
Takahashi, W., Nonlinear Functional Analysis: Fixed Point Theory and its Applications (Yokohama Publishers, Yokohama, 2000).Google Scholar