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

Shimizu, Tomoo; Takahashi, Wataru

J. Math. Anal. Appl. 211, No. 1, 71-83 (1997).

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.

Reviewer: A.A.Melentsov (Ekaterinburg)

MSC:

47H10	Fixed-point theorems
47H20	Semigroups of nonlinear operators

Keywords:

set of fixed points; nonexpansive semigroup; common fixed point; metric projection

References:

[1]	Baillon, J. B., C.R. Acad. Sci. Paris Sér. A-B, 280, 1511-1514 (1975)
[2]	Baillon, J. B.; Brézis, H., Une remarque sur le comportement asymptotique des semigroupes non linéaires, Houston J. Math., 2, 5-7 (1976) · Zbl 0318.47039
[3]	Brézis, H.; Browder, F. E., Nonlinear ergodic theorems, Bull. Amer. Math. Soc., 82, 959-961 (1976) · Zbl 0339.47029
[4]	Halpern, B., Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73, 957-961 (1967) · Zbl 0177.19101
[5]	Krengel, U., Ergodic Theorems (1985), de Gruyter: de Gruyter Berlin/New York · Zbl 0471.28011
[6]	Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73, 591-597 (1967) · Zbl 0179.19902
[7]	Reich, S., Some problems and results in fixed point theory, Contemp. Math., 21, 179-187 (1983) · Zbl 0531.47048
[8]	Takahashi, W., Nonlinear Functional Analysis (1988), Kindaikagakusha: Kindaikagakusha Tokyo
[9]	Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. Math., 58, 486-491 (1992) · Zbl 0797.47036