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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 $F(T)$ 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 $\{S(t)\}_{t\in R^+}$ of mappings of $C$ into itself is called a nonexpansive semigroup of $C$, if it satisfies the following conditions: (1) $S(t_1+t_2)x=S(t_1)S(t_2)x$ for each $t_1,t_2\in R^+$ and $x\in C$; (2) $S(0)x= x$ for each $x\in C$; (3) for each $x\in C$, $t\to S(t)x$ is continuous; (4) $|S(t)x-S(t)y|\le|x-y|$ 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 $F(S)\cap F(T)$ is nonempty. Suppose that $\{\alpha_n\}^\infty_{n= 0}\subseteq[0, 1]$ satisfies $$\lim_{n\to\infty} \alpha_n= 0,\qquad\text{and}\qquad\sum^\infty_{n=0} \alpha_n=\infty.$$ Then, for an arbitrary $x\in C$, the sequence $\{x_n\}^\infty_{n=0}$ generated by $x_0=x$ and $$x_{n+1}=\alpha_nx+(1-\alpha_n) {2\over(n+1)(n+2)} \sum^n_{k=0} \sum_{i+j=k} S^iT^ix_n,\quad n\ge 0,$$ converges strongly to a common fixed point $Px$ of $S$ and $T$, where $P$ is the metric projection of $H$ onto $F(S)\cap F(T)$. 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 $\{S(t)\}_{t\in R^+}$ be a nonexpansive semigroup on $C$ such that $\bigcap_{t\in R^+} F(S(T))$ is nonempty. Suppose that $\{\beta_n\}^\infty_{n= 0}$ satisfies $$\lim_{n\to\infty} \beta_n=0,\qquad\text{and} \qquad\sum^\infty_{n= 0}\beta_n= \infty.$$ Then, for an arbitrary $z\in C$, the sequence $\{z_n\}^\infty_{n=0}$ generated by $z_0=z$ and $$z_{n+1}= \beta_nz+(1- \beta_n) {1\over t_n} \int^{t_n}_0 S(u)z_ndu,\quad n\ge 0,$$ converges strongly to a common fixed point $Pz$ of $S(t)$, $t\in R^+$, where $P$ is the metric projection of $H$ onto $\bigcap_{t\in R^+} F(S(T))$ and $\{t_n\}^\infty_{n= 0}$ is a positive real divergent sequence.

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