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 $$|Tx-Ty|\leq|x-y|$$ 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) $$|S(t)x-S(t)y|\leq|x-y|$$ 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 $x_{n+1}=\alpha_nx+(1-\alpha_n) {2\over(n+1)(n+2)} \sum^n_{k=0} \sum_{i+j=k} S^iT^ix_n,\quad n\geq 0,$ 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 $z_{n+1}= \beta_nz+(1- \beta_n) {1\over t_n} \int^{t_n}_0 S(u)z_ndu,\quad n\geq 0,$ 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 47H20 Semigroups of nonlinear operators
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References:

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