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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 \(|Tx-Ty|\leq|x-y|\) 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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