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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 $Tx=\lim_nT_nx$, $x\in C$.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties 47J25 Iterative procedures (nonlinear operator equations)
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##### References:
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