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Zbl 1123.47048
Lau, Anthony To-Ming; Miyake, Hiromichi; Takahashi, Wataru
Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 67, No. 4, A, 1211-1225 (2007). ISSN 0362-546X

The authors investigate the iterative scheme of Halpern's type $$x_{n+1}=\alpha_nx+(1-\alpha_n)T(\mu_n)x_n,\quad n=1,2,\dots,\ x_1=x,\tag1$$ and the iterative scheme of Browder's type $$x_n=\alpha_n x+(1-\alpha_n)T(\mu_n)x_n,\quad n=1,2,\dots,\tag2$$ for a semigroup ${\cal S}=\{T(s):s\in S\}$ of nonexpansive mappings on a compact convex subset $C$ of a smooth Banach space with respect to a sequence $(\mu_n)_n$ of strongly asymptotically invariant means defined on an appropriate invariant subspace of the space of bounded real-valued functions on a left reversible semigroup $S$. For a given sequence $(\alpha_n)_n\subset [0,1]$ which satisfies some assumptions and $x\in C$, the solutions $(x_n)_n$ of the problems (1) and (2) converge strongly to $Px$, where $P$ is the unique sunny nonexpansive retraction of $C$ onto the fixed point set $F(\cal S)$. Some applications to the additive semigroup of nonnegative real numbers and to the commuting pairs of nonexpansive mappings are also presented.
[Rodica Luca Tudorache (Iaşi)]
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
43A60 Almost periodic functions on groups, etc.
47H09 Mappings defined by "shrinking" properties
47H20 Semigroups of nonlinear operators

Keywords: iterative scheme; left reversible semigroup; amenable subspace; strongly left regular sequence of means; left asymptotically invariant sequence of means; sunny retraction; strong convergence

Cited in: Zbl 1178.47036 Zbl 1222.47086

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