zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Halpern’s iteration in CAT(0) spaces. (English) Zbl 1197.54074
Let $C$ be a closed convex subset of a complete CAT(0) space $(X,d)$ and $\{T_1,\dots,T_N\}$ be nonexpansive maps with $F:=\cap\{F(T_i); i=1,\dots,N\}=F(T_N\circ\dots\circ T_1)$. In addition, let $(a_n)\subset (0,1)$ be a sequence satisfying (i) $a_n\to 0$, (ii) $\sum_na_n=\infty$, (iii) $\sum_n|a_n-a_{n+N}|< \infty$ or $\lim_n(a_n/a_{n+N})=1$. Then, for each $u,x_1\in C$, the iterative method given by $$x_{n+1}=a_nu\oplus(1-a_n)T_{n(\text{modulo} N)}x_n,\quad n\ge 1,$$ converges to some $z\in F$ which is nearest to $u$. An extension of this result to countable families $(T_n)$ of such maps is also given, but under stronger conditions such as (a) $\sum_n\sup\{d(T_nz,T_{n+1}z); z\in B\}< \infty$, for each bounded subset $B$ of $C$, (b) $F(T)=\cap_n\{F(T_n)\}$, where $Tx=\lim_nT_nx$, $x\in C$.

54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
47J25Iterative procedures (nonlinear operator equations)
Full Text: DOI EuDML