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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 \[ x_{n+1}=a_nu\oplus(1-a_n)T_{n(\text{modulo} N)}x_n,\quad n\geq 1, \] 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 (topological aspects)
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J25 Iterative procedures involving nonlinear operators
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References:

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