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Best approximation to common fixed points of a semigroup of nonexpansive operators. (English) Zbl 1071.41031
Let $C$ be a nonempty closed convex subset of a Hilbert space $H,$ $\Gamma=\{T_{t}:t\in G\},$ $G\subset\Bbb{R}_{+},$ a semigroup of nonexpansive operators $T_{t}:C\rightarrow C,$ $t\in G,$ and $F=\cap_{t\in G}Fix\left( T_{t}\right) $ the set of common fixed points of the operators from $\Gamma.$ For a fixed $u\in C$ one looks for a best approximation element of $u$ in $F,$ denoted by $P_{F}u.$ To solve this problem the author proposes the following algorithm: step $0:$ $x_{0}\in C$ is arbitrary; step $n+1:$ $x_{n+1}=\alpha_{n}u+\left( 1-\alpha_{n}\right) T_{r_{n}}x_{n},$ for all $n\geq0,$ where $0\leq\alpha_{n}\leq1$ and $\{r_{n}\}_{n\geq0}\subset G$ is a given sequence. The semigroup $\Gamma$ is called a uniformly asymptotically regular semigroup of nonexpansive operators on $C$ if $\lim_{n\rightarrow \infty}\left( \sup_{x\in C}\left\Vert T_{s}T_{r}x-T_{r}x\right\Vert \right) =0.$ The sequence $\{\alpha_{n}\}$ $_{n\geq0}$ is called a steering sequence if it has the following properties: $\left( 1\right) $ $\alpha_{n}\in \lbrack0;1],\ n\geq0$ and $\alpha_{n}\rightarrow0;$ $\left( 2\right) $ $\sum_{n=0}^{+\infty}\alpha_{n}=+\infty$ (or, equivalently, $\prod _{n=0}^{\infty}\left( 1-\alpha_{n}\right) =0);$ $\left( 3\right) $ $\sum_{n=0}^{\infty}\left\vert \alpha_{n+1}-\alpha_{n}\right\vert <\infty.$ The authors prove that for given $u\in C$, if the semigroup $\Gamma$ is uniformly asymptotically regular, $\{\alpha_{n}\}$ is a steering sequence and $\{r_{n}\}_{n\geq0}\subset G$ satisfies the conditions: $\left( a\right) $ $0\leq r_{0}\leq r_{1}\leq$...$\leq r_{n}\leq...$ and $\lim_{n\rightarrow \infty}r_{n}=+\infty$; $\left( b\right) $ $\sum_{n=0}^{\infty}\sup_{x\in C}\left\Vert T_{s}T_{r_{n}}x-T_{r_{n}}x\right\Vert <\infty$ uniformly for all $s\in G,$ then the proposed algorithm is strongly convergent to $P_{F}u.$

MSC:
41A65Abstract approximation theory
47H20Semigroups of nonlinear operators