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An implicit iteration process for nonexpansive mappings. (English) Zbl 0999.47043
Let $C$ be a closed convex subset of a Hilbert space $H$ and $T: C\to C$ be a nonexpansive mapping (i.e., $\|Tx-Ty\|\le \|x-y\|$, $x,y\in C$). It is well known, that for each $t\in (0,1)$, the contraction $T_t: C\to C$ defined by $T_t(x)= tu+(1-t)Tx$, $x\in C$ ($u\in C$ is a fix point) has a unique fixed point $x_t$. {\it F. E. Browder} [Arch. Ration. Mech. Anal. 24, 82-90 (1967; Zbl 0148.13601)] proved: $\{x_t\}$ converges in norm, as $t\to 0$, to a fixed point of $T$. In the present paper the authors study the convergence of an implicit iteration process to a fixed point of a finite family of nonexpansive mappings. The main result of the paper is the following: Theorem 2. Let $C$ be a closed convex subset of a Hilbert space $H$ and $T_1,T_2,\dots, T_N$ be $N$ nonexpansive self-mappings of $C$ such that $\bigcap^N_{i=1} \text{Fix}(T_i)\ne\emptyset$. Let $x_0\in C$ and $\{t_n\}$ be a sequence in $(0,1)$ such that $\lim_{n\to\infty} t_n= 0$. Then the sequence $\{x_n\}$ defined in the following way: $$x_n= t_n x_{n-1}+ (1- t_n) T_n x_n,\quad n\ge 1,$$ where $T_k= T_{k\text{ mod }N}$ (here the $\text{mod }N$ function takes values in $\{1,2,\dots, N\}$) converges weakly to a common fixed point of the mappings $T_1,T_2,\dots, T_N$.

47H09Mappings defined by “shrinking” properties
65J15Equations with nonlinear operators (numerical methods)
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
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