Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T16:34:49.919Z Has data issue: false hasContentIssue false
  • English
  • Français

Fixed Point Theorem and Nonlinear Ergodic Theorem for Nonexpansive Semigroups Without Convexity

Published online by Cambridge University Press:  20 November 2018

Wataru Takahashi
Wataru Takahashi*
Affiliation:
Department of Information Science, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo 152, Japan
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.

We first prove a nonlinear ergodic theorem for nonexpansive semigroups without convexity in a Hilbert space. Further we prove a fixed point theorem for non-expansive semigroups without convexity which generalizes simultaneously fixed point theorems for left amenable semigroups and left reversible semigroups.

Keywords

Primary: 47A3547H09Fixed pointnonexpansive mappinginvariant meannonlinear ergodic theorem
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 4 , 01 August 1992 , pp. 880 - 887
Copyright
Copyright © Canadian Mathematical Society 1992

References

1. Baillon, J.B., Un théorème de type ergodic pour les contractions non linéares dans un espace de Hilbert, C.R. Acad. Sci. Paris Sér. A-B, 280 (1975), 1511–1514.Google Scholar
2. Baillon, J.B.,Quelques propriétés de convergence asymptotique pour les semigroups de contractions impaires, C.R. Acad. Sci. Paris Sér. A-B, 283 (1976), 75–78.Google Scholar
3. Brézis, H. and Browder, F.E., Nonlinear ergodic theorems, Bull Amer. Math. Soc, 82 (1976), 959–961.Google Scholar
4. Brézis, H., Remarks on nonlinear ergodic theory, Adv. in Math., 25 (1977), 165–177.Google Scholar
5. Hirano, N. and Takahashi, W., Nonlinear ergodic theorems for nonexpansive mappings in Hilbert space, KodaiMath. J., 2 (1979), 11–25.Google Scholar
6. Ishihara, H. and Takahashi, W., Fixed point theorems for uniformly lipschitzian semigroups in Hilbert spaces, J. Math. Anal. Appl., 127 (1987), 206–210.Google Scholar
7. Ishihara, H., Fixed point theorems for lipschitzian semigroups, Canad. Math. Bull., 32 (1989), 90–97.Google Scholar
8. Lau, A.T., Semigroup of nonexpansive mappings on a Hilbert space, J. Math. Anal. Appl., 105 (1985), 514–522.Google Scholar
9. Mitchell, T., Topological semigroups and fixed points, Illinois J. Math., 14 (1970), 630–641.Google Scholar
10. Mizoguchi, N. and Takahashi, W., On the existence of fixed points and ergodic retractions for Lipschitzian semigroups in Hilbert spaces, Nonlinear Analysis, 14 (1990), 69–80.Google Scholar
11. Pazy, A., On the asymptotic behavior of iterates of nonexpansive mappings in Hilbert space, Israel J. Math., 26 (1977), 197–204.Google Scholar
12. Pazy, A. , On the asymptotic behavior of semigroups of nonlinear contractions in Hilbert space, J. Funct. Anal., 27 (1978), 292–307.Google Scholar
13. Phelps, R.P., Convex sets and nearest points, Proc. Amer. Math. Soc, 8 (1957), 790–797.Google Scholar
14. Rodé, G., An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space, J. Math. Anal. Appl., 85 (1982), 172–178.Google Scholar
15. Takahashi, W., A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space, Proc. Amer. Math. Soc, 81 (1981), 253–256.Google Scholar
16. Rodé, G., Fixed point theorems for families of nonexpansive mappings on unbounded sets, J. Math. Soc. Japan, 36 (1984), 543–553.Google Scholar
17. Rodé, G., A nonlinear ergodic theorem for a reversible semigroup of nonexpansive mappings in a Hilbert space, Proc. Amer. Math. Soc, 96 (1986), 55–58.Google Scholar