Existence of Nonexpansive Retractions for Amenable Semigroups of Nonexpansive Mappings and Nonlinear Ergodic Theorems in Banach Spaces

Authors: Lau A.T-M.¹; Shioji N.²; Takahashi W.³

Source: Journal of Functional Analysis, Volume 161, Number 1, January 1999 , pp. 62-75(14)

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Abstract:

In this paper, we study nonlinear ergodic properties for an amenable semigroup of nonexpansive mappings in a Banach space. We prove that if S is an amenable semigroup and Sscr ={T_t: t isin S} is a nonexpansive semigroup on a closed, convex subset C in a uniformly convex Banach space E such that the set F( Sscr ) of common fixed points of Sscr is nonempty, then there exists a nonexpansive retraction P from C onto F( Sscr ) such that PT_t=T_tP=P for each t isin S and Px isin co{T_tx: t isin S} for each x isin C. In this case, there exists a net {A} of finite averages of Sscr such that for each t isin S and for each bounded subset B of C, lim Verbar AT_tx-Ax Verbar =0 and lim Verbar T_tAx-Ax Verbar =0 uniformly for x isin B. Also, if the norm of E is Fréchet differentiable, then for each x isin C, Px is the unique common fixed point in cap _sS co{ T_tsx: t isin S}. Furthermore, if {} is an asymptotically invariant net of means, then for each x isin C, {Tx} converges weakly to Px. Copyright 1999 Academic Press.

Language: English

Document Type: Research article

Affiliations: 1: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, T6G-2G1, Canada 2: Faculty of Engineering, Tamagawa University, Tamagawa-Gakuen, Machida, Tokyo, 194, Japan 3: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo, 152, Japan

Publication date: 1999-01-01