# 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)
Iterative methods for strict pseudo-contractions in Hilbert spaces. (English) Zbl 1133.47050

This article deals with two iterative algorithms of finding a common fixed points for $N$ strict pseudo-contractions ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ defined on a closed convex subset $C$ of a real Hilbert space $H$ (an operator $T:C\to C$ is a strict pseudo-contraction, if there exists a constant $0\le k<1$ such that ${\parallel Tx-Ty\parallel }^{2}\le {\parallel x-y\parallel }^{2}+k{\parallel \left(I-T\right)x-\left(I-T\right)y\parallel }^{2}$). The first algorithm, called parallel, is defined by the formula

${x}_{n+1}={\alpha }_{n}{x}_{n}+\left(1-{\alpha }_{n}\right)\sum _{i=1}^{N}{\lambda }_{i}^{\left(n\right)}{T}_{i}{x}_{n},\phantom{\rule{4pt}{0ex}}{x}_{0}\in C,\phantom{\rule{4pt}{0ex}}{\lambda }_{i}^{\left(n\right)}>0,\phantom{\rule{4pt}{0ex}}{\lambda }_{1}^{\left(n\right)}+\cdots +{\lambda }_{N}^{\left(n\right)}=1;\phantom{\rule{2.em}{0ex}}\left(1\right)$

the second one, called cyclic, by the formula

${x}_{n+1}={\alpha }_{n}{x}_{n}+\left(1-\alpha \right){T}_{\left[n\right]}{x}_{n},\phantom{\rule{1.em}{0ex}}{x}_{0}\in C,\phantom{\rule{1.em}{0ex}}{T}_{\left[n\right]}={T}_{i},\phantom{\rule{4pt}{0ex}}i=n\left(\phantom{\rule{4.pt}{0ex}}\text{mod}\phantom{\rule{0.166667em}{0ex}}N\right),\phantom{\rule{4pt}{0ex}}1\le i\le N·\phantom{\rule{2.em}{0ex}}\left(2\right)$

The main results describe (provided that $F={\bigcap }_{i=1}^{N}\text{Fix}\left({T}_{i}\right)\ne \varnothing$) conditions on the control sequence $\left\{{\alpha }_{n}\right\}$ so that the approximations ${x}_{n}$ converge weakly to a common fixed point of ${\left\{{T}_{i}\right\}}_{i=1}^{N}$. At the end of the article, some modifications of algorithms (1) and (2) are proposed; it is proved that approximations ${x}_{n}$ for these modified algorithms converge strongly to ${P}_{F}{x}_{0}$, where ${P}_{F}$ is the nearest point projection from $H$ onto $F$.

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties 65J15 Equations with nonlinear operators (numerical methods)