# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
General iterative methods for a one-parameter nonexpansive semigroup in Hilbert space. (English) Zbl 1177.47075
Let $H$ be a Hilbert space, $f$ a fixed contractive mapping with coefficient $0<\alpha<1$, $A$ a strongly positive linear bounded operator with coefficient $\overline\gamma>0$. Consider two iterative methods that generate the sequences $\{x_n\}$, $\{y_n\}$ by $$x_n=(1-\alpha_nA)\frac{1} {t_n}\int^{t_n}_0T(s)x_n\, ds+\alpha_n\gamma f(x_n),\tag I$$ $$y_{n+1}=(I-\alpha_n A)\frac{1} {t_n}\int^{t_n}_0T(s)y_n\,ds+\alpha_n\gamma f(y_n),\tag II$$ where $\{\alpha_n\}$ and $\{t_n\}$ are two sequences satisfying certain conditions, and ${\germ I}=\{T(s):s\ge 0\}$ is a one-parameter nonexpansive semigroup on $H$. It is proved that the sequences $\{x_n\}$, $\{y_n\}$ generated by the iterative methods (I) and (II), respectively, converge strongly to a common fixed point $x^*\in F({\germ I})$ which solves the variational inequality $$\langle(A-\gamma f)x^*,x^*-z\rangle\le 0\ z\in F({\germ I}).$$

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H20 Semigroups of nonlinear operators 47J20 Inequalities involving nonlinear operators 65J15 Equations with nonlinear operators (numerical methods)
Full Text:
##### References:
 [1] Deutsch, F.; Yamada, I.: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numer. funct. Anal. optim. 19, 33-56 (1998) · Zbl 0913.47048 [2] Xu, H. K.: Iterative algorithms for nonlinear operators. J. London math. Soc. 66, 240-256 (2002) · Zbl 1013.47032 [3] Xu, H. K.: An iterative approach to quadratic optimization. J. optim. Theory appl. 116, 659-678 (2003) · Zbl 1043.90063 [4] Yamada, I.: The hybrid steepest descent method for the variational inequality problem of the intersection of fixed point sets for nonexpansive mappings. Inherently parallel algorithm for feasibility and optimization, 473-504 (2001) · Zbl 1013.49005 [5] Yamada, I.; Ogura, N.; Yamashita, Y.; Sakaniwa, K.: Quadratic approximation of fixed points of nonexpansive mappings in Hilbert spaces. Numer. funct. Anal. optim. 19, 165-190 (1998) · Zbl 0911.47051 [6] Geobel, K.; Kirk, W. A.: Topics in metric fixed point theory. Cambridge stud. Adv. math. 28 (1990) [7] Moudafi, A.: Viscosity approximation methods for fixed points problems. J. math. Anal. appl. 241, 46-55 (2000) · Zbl 0957.47039 [8] Xu, H. K.: Viscosity approximation methods for nonexpansive mappings. J. math. Anal. appl. 298, 279-291 (2004) · Zbl 1061.47060 [9] Marino, Giuseppe; Xu, Hong Kun: A general iterative method for nonexpansive mappings in Hilbert spaces. J. math. Anal. appl. 318, 43-52 (2006) · Zbl 1095.47038 [10] 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 [11] Tan, K. K.; Xu, H. K.: The nonlinear ergodic theorem for asymptotically nonexapansive mappings in Banach spaces. Proc. amer. Math. soc. 114, 399-404 (1992) · Zbl 0781.47045 [12] Baillon, J. B.: On the’ore’me de type ergodique pour LES contractions online’aires dans un espace de Hilbert. C. R. Acad. sci. Paris 280, 1511-1514 (1975) · Zbl 0307.47006 [13] Reich, S.: Almost convergence and nonlinear ergodic theorems. J. approx. Theory 24, 269-272 (1978) · Zbl 0404.47032 [14] Reich, S.: A note on the mean ergodic theorem for nonlinear semigroups. J. math. Anal. appl. 91, 547-551 (1983) · Zbl 0521.47034 [15] Kaczor, W.; Kuczumow, T.; Reich, S.: A mean ergodic theorem for nonlinear semigroups which are asymptotically nonexpansive in the intermediate sense. J. math. Anal. appl. 246, 1-27 (2000) · Zbl 0981.47037