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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., $\parallel Tx-Ty\parallel \le \parallel x-y\parallel$, $x,y\in C$). It is well known, that for each $t\in \left(0,1\right)$, the contraction ${T}_{t}:C\to C$ defined by ${T}_{t}\left(x\right)=tu+\left(1-t\right)Tx$, $x\in C$ ($u\in C$ is a fix point) has a unique fixed point ${x}_{t}$. F. E. Browder [Arch. Ration. Mech. Anal. 24, 82-90 (1967; Zbl 0148.13601)] proved: $\left\{{x}_{t}\right\}$ 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 ${T}_{1},{T}_{2},\cdots ,{T}_{N}$ be $N$ nonexpansive self-mappings of $C$ such that ${\bigcap }_{i=1}^{N}\text{Fix}\left({T}_{i}\right)\ne \varnothing$. Let ${x}_{0}\in C$ and $\left\{{t}_{n}\right\}$ be a sequence in $\left(0,1\right)$ such that ${lim}_{n\to \infty }{t}_{n}=0$. Then the sequence $\left\{{x}_{n}\right\}$ defined in the following way:

${x}_{n}={t}_{n}{x}_{n-1}+\left(1-{t}_{n}\right){T}_{n}{x}_{n},\phantom{\rule{1.em}{0ex}}n\ge 1,$

where ${T}_{k}={T}_{k\phantom{\rule{4.pt}{0ex}}\text{mod}\phantom{\rule{4.pt}{0ex}}N}$ (here the $\text{mod}\phantom{\rule{4.pt}{0ex}}N$ function takes values in $\left\{1,2,\cdots ,N\right\}$) converges weakly to a common fixed point of the mappings ${T}_{1},{T}_{2},\cdots ,{T}_{N}$.

##### MSC:
 47H09 Mappings defined by “shrinking” properties 65J15 Equations with nonlinear operators (numerical methods) 47H10 Fixed point theorems for nonlinear operators on topological linear spaces