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Strong convergence to common fixed points of families of nonexpansive mappings. (English) Zbl 0883.47075

Let $H$ be a real Hilbert space, and let $C$ be a nonempty closed convex subset of $H$. A mapping $T$ of $C$ into itself is said to be nonexpansive, if $|Tx-Ty|\le |x-y|$ for each $x,y\in C$.

For a mapping $T$ of $C$ into itself, we denote by $F\left(T\right)$ the set of fixed points of $T$. We also denote by $N$ and ${R}^{+}$ the set of positive integers and nonnegative real numbers, respectively. A family ${\left\{S\left(t\right)\right\}}_{t\in {R}^{+}}$ of mappings of $C$ into itself is called a nonexpansive semigroup of $C$, if it satisfies the following conditions:

(1) $S\left({t}_{1}+{t}_{2}\right)x=S\left({t}_{1}\right)S\left({t}_{2}\right)x$ for each ${t}_{1},{t}_{2}\in {R}^{+}$ and $x\in C$;

(2) $S\left(0\right)x=x$ for each $x\in C$;

(3) for each $x\in C$, $t\to S\left(t\right)x$ is continuous;

(4) $|S\left(t\right)x-S\left(t\right)y|\le |x-y|$ for each $t\in {R}^{+}$ and $x,y\in C$.

Convergence theorem for a finite mapping.

Convergence theorem for two commutative mappings in a Hilbert space.

Theorem 1. Let $H$ be a Hilbert space, and let $C$ be a nonempty closed convex subset of $H$. Let $S$ and $T$ be nonexpansive mappings of $C$ into itself such that $ST=TS$ and $F\left(S\right)\cap F\left(T\right)$ is nonempty. Suppose that ${\left\{{\alpha }_{n}\right\}}_{n=0}^{\infty }\subseteq \left[0,1\right]$ satisfies

$\underset{n\to \infty }{lim}{\alpha }_{n}=0,\phantom{\rule{2.em}{0ex}}\text{and}\phantom{\rule{2.em}{0ex}}\sum _{n=0}^{\infty }{\alpha }_{n}=\infty ·$

Then, for an arbitrary $x\in C$, the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\infty }$ generated by ${x}_{0}=x$ and

${x}_{n+1}={\alpha }_{n}x+\left(1-{\alpha }_{n}\right)\frac{2}{\left(n+1\right)\left(n+2\right)}\sum _{k=0}^{n}\sum _{i+j=k}{S}^{i}{T}^{i}{x}_{n},\phantom{\rule{1.em}{0ex}}n\ge 0,$

converges strongly to a common fixed point $Px$ of $S$ and $T$, where $P$ is the metric projection of $H$ onto $F\left(S\right)\cap F\left(T\right)$.

Convergence theorem for nonexpansive semigroups.

Convergence theorem for a nonexpansive semigroup in a Hilbert space.

Theorem 2. Let $H$ be a Hilbert space and let $C$ be a nonempty closed convex subset of $H$. Let ${\left\{S\left(t\right)\right\}}_{t\in {R}^{+}}$ be a nonexpansive semigroup on $C$ such that ${\bigcap }_{t\in {R}^{+}}F\left(S\left(T\right)\right)$ is nonempty. Suppose that ${\left\{{\beta }_{n}\right\}}_{n=0}^{\infty }$ satisfies

$\underset{n\to \infty }{lim}{\beta }_{n}=0,\phantom{\rule{2.em}{0ex}}\text{and}\phantom{\rule{2.em}{0ex}}\sum _{n=0}^{\infty }{\beta }_{n}=\infty ·$

Then, for an arbitrary $z\in C$, the sequence ${\left\{{z}_{n}\right\}}_{n=0}^{\infty }$ generated by ${z}_{0}=z$ and

${z}_{n+1}={\beta }_{n}z+\left(1-{\beta }_{n}\right)\frac{1}{{t}_{n}}{\int }_{0}^{{t}_{n}}S\left(u\right){z}_{n}du,\phantom{\rule{1.em}{0ex}}n\ge 0,$

converges strongly to a common fixed point $Pz$ of $S\left(t\right)$, $t\in {R}^{+}$, where $P$ is the metric projection of $H$ onto ${\bigcap }_{t\in {R}^{+}}F\left(S\left(T\right)\right)$ and ${\left\{{t}_{n}\right\}}_{n=0}^{\infty }$ is a positive real divergent sequence.

##### MSC:
 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H20 Semigroups of nonlinear operators