Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Weak convergence theorems for nonexpansive mappings and semigroups in Banach spaces with Opial's property


Author: T. Kuczumow
Journal: Proc. Amer. Math. Soc. 93 (1985), 430-432
MSC: Primary 47H20; Secondary 47H09
DOI: https://doi.org/10.1090/S0002-9939-1985-0773996-8
MathSciNet review: 773996
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X$ be a Banach space with Opial's property, $ C$ a weakly compact subset of $ X$, $ x \in C$ and $ S$ a nonexpansive semigroup on $ C$. Then $ {\left\{ {S(t)x} \right\}_{t \geqslant 0}}$ converges weakly to a common fixed point of $ S$ iff $ S(t + h)x - S(t)x \rightharpoonup 0$ as $ t \to \infty $ for all $ h > 0$.


References [Enhancements On Off] (What's this?)

  • [1] J. B. Baillon, R. E. Bruck and S. Reich, On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces, Houston J. Math. 4 (1978), 1-9. MR 0473932 (57:13590)
  • [2] N. Hirano, Nonlinear ergodic theorems and weak convergence theorems, J. Math. Soc. Japan 34 (1982), 35-46. MR 639803 (83d:47059)
  • [3] L. A. Karlovitz, On nonexpansive mappings, Proc. Amer. Math. Soc. 55 (1976), 321-325. MR 0405182 (53:8976)
  • [4] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597. MR 0211301 (35:2183)
  • [5] A. Pazy, On the asymptotic behavior of semigroups of nonlinear contractions in Hilbert space, J. Funct. Anal. 27 (1978), 292-307. MR 0477932 (57:17432)
  • [6] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach space, J. Math. Anal. Appl. 67 (1979), 274-276. MR 528688 (80d:47090)
  • [7] -, A note on the mean ergodic theorem for nonlinear semigroups, J. Math. Anal. Appl. 91 (1983), 547-551. MR 690887 (84g:47057)
  • [8] S. Rolewicz, Metric linear spaces, PWN, Warsaw, 1972. MR 808176 (88i:46004b)
  • [9] R. Schoeneberg, Weak convergence of trajectories of nonexpansive semigroups in Hilbert space, Israel J. Math. 30 (1978), 130-132. MR 508258 (80m:47066)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47H20, 47H09

Retrieve articles in all journals with MSC: 47H20, 47H09


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0773996-8
Keywords: Nonexpansive mappings, nonexpansive semigroups, fixed points, Opial's property
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society