×

Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces. (English) Zbl 0935.47039

The authors consider a closed convex subset \(C\) of a Hilbert space \(H\), a given semigroup \(S\) and an asymptotically nonexpansive semigroup \(\mathcal S = \{T_t\); \(t \in S \}\) on \(C\) (i.e. for any \(t \in S\), \(T_t\) is a mapping from \(C\) into itself, \(\|T_tx-T_ty\|\leq k_t \|x-y\|\), \(\sup k_t < \infty\), \(T_{ts}x = T_tT_sx\)). It is supposed that \(F(\mathcal S) \neq \emptyset\) where \(F(\mathcal S)= \cap_{t\in S} \{x\in C\); \(T_tx=x\}\) – the set of common fixed points of \(\mathcal S\). For any \(x \in C\) and \(x, y_0 \in C\), iteration processes defined by \[ x_n = a_n x + (1-a_n)T_{\mu_n}x_n, \quad \text{and}\quad y_{n+1} = b_n x + (1-b_n)T_{\mu_n}y_n \] are studied. Here \(0<a_n\leq 1\), \(\lim a_n =0\), \(0\leq b_n\leq 1\), \(\lim b_n =0\), \(\sum b_n = \infty\), \(\mu_n\) is a sequence of means on a certain subspace \(X\) of \(B(S)\) containing 1 (i.e. \(\mu_n \in X^*\), \(\|\mu_n \|= \mu_n(1)=1\)), \(T_{\mu}x\) is the element of \(C\) for which \(\mu (f_{x,y}) = (T_{\mu}x,y)\) for all \(y \in H\), \(f_{x,y} \in X\), \(f_{x,y}(t)=(T_tx,y)\). It is proved that under some assumptions, for any \(x \in C\) or \(x\), \(y_0 \in C\), the iteration processes mentioned converge strongly to the element of \(F(\mathcal S)\) which is the nearest to \(x\) in \(F(\mathcal S)\). By a special choice of the means, the authors prove extensions of the results of Shimizu and Takahashi concerning the convergence of some iteration processes to the fixed points of asymptotically nonexpansive mappings in a closed convex subset of a Hilbert space.
Reviewer: M.Kučera (Praha)

MSC:

47H20 Semigroups of nonlinear operators
47H10 Fixed-point theorems
65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baillon, J. B., Un théorème de type ergodic pour les contractions non linéares dans un espace de Hilbert, C. r. hebd. Séanc, Acad. Sci. Paris, 280, 1511-1514 (1975) · Zbl 0307.47006
[2] Brézis, H.; Browder, F. E., Nonlinear ergodic theorems, Bull. Am. Math. Soc., 82, 959-961 (1976) · Zbl 0339.47029
[3] Browder, F. E., Convergence of approximants to fixed points of non-expansive non-linear mappings in Banach spaces, Arch. Rational Mech. Anal., 24, 82-90 (1967) · Zbl 0148.13601
[4] Goebel, K.; Kirk, W. A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. Am. Math. Soc., 35, 171-174 (1972) · Zbl 0256.47045
[5] Hirano, N.; Kido, K.; Takahashi, W., Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces, Nonlinear Anal., 12, 1269-1281 (1988) · Zbl 0679.47031
[6] Rodé, G., An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space, J. Math. Anal. Appl., 85, 172-178 (1982) · Zbl 0485.47041
[7] Shimizu, T.; Takahashi, W., Strong convergence theorem for asymptotically nonexpansive mappings, Nonlinear Anal., 26, 265-272 (1996) · Zbl 0861.47030
[8] Shimizu, T.; Takahashi, W., Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl., 211, 71-83 (1997) · Zbl 0883.47075
[9] Takahashi, W., A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space, Proc. Am. Math. Soc., 81, 253-256 (1981) · Zbl 0456.47054
[10] Takahashi, W., Fixed point theorem and nonlinear ergodic theorem for nonexpansive semigroups without convexity, Canadian J. Math., 44, 880-887 (1992) · Zbl 0786.47047
[11] Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. Math., 58, 486-491 (1992) · Zbl 0797.47036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.