Abstract
The following theorem is proven:if E is a uniformly rotund Banach space with a Fréchet differentiable norm, C is a bounded nonempty closed convex subset of E, and T: C→C is a contraction, then the iterates {T nx} are weakly almost-convergent to a fixed-point of T.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. B. Baillon, untitled preprint.
J. B. Baillon, R. E. Bruck and S. Reich,On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces, to appear in Houston J. Math.
F. E. Browder,Nonlinear operators and nonlinear equationas of evolution in Banach spaces, Proc. Symposia Pure Math. XVIII, pt. 2, Amer. Math. Soc., Providence, R. I., 1976.
R. E. Bruck,Construction of periodic solution of periodic contractive evolution systems, Abstract 747-34-4, Notices Amer. Math. Soc.24 (1977), A-474.
R. E. Bruck,On the almost-convergence of iterates of a nonexpansive mapping in Hilbert space and the structure of the weak ω-limit set, Israel J. Math.29 (1978), 1–16.
R. DeMarr,Common fixed points for commuting contraction mappings, Pacific J. Math.13 (1963), 1139–1141.
G. G. Lorentz,A contribution to the theory of divergent series, Acta Math.80 (1948), 167–190.
Author information
Authors and Affiliations
Additional information
Supported by NSF Grant MCS 76-08217.
Rights and permissions
About this article
Cite this article
Bruck, R.E. A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Israel J. Math. 32, 107–116 (1979). https://doi.org/10.1007/BF02764907
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02764907