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Common fixed points for asymptotic pointwise nonexpansive mappings in metric and Banach spaces. (English) Zbl 1295.47060
Summary: Let $$C$$ be a nonempty bounded closed convex subset of a complete CAT(0) space $$X$$. We prove that the common fixed point set of any commuting family of asymptotic pointwise nonexpansive mappings on $$C$$ is nonempty closed and convex. We also show that, under some suitable conditions, the sequence $$\{x_k\}^\infty_{k=1}$$ defined by
\begin{aligned} x_{k+1} &= (1 - t_{mk})x_k \oplus t_{mk} T^{n_k}_m y_{(m-1)k},\\ y_{(m-1)k} &= (1 - t_{(m-1)k}) x_k \oplus t_{(m-1)k} T^{n_k}_{m-1} y_{(m-2)k},\\ y_{(m-2)k} &= (1 - t_{(m-2)k}) x_k \oplus t_{(m-2)k} T^{n_k}_{m-2} y_{(m-3)k},\\ &\vdots\\ y_{2k} &= (1 - t_{2k}) x_k \oplus t_{2k} T^{n_k}_2 y_{1k},\\ y_{1k} &= (1 - t_{1k}) x_k \oplus t_{1k} T^{n_k}_1 y_{0k},\\ y_{0k} &= x_k,\end{aligned}
$$k \in \mathbb N$$, converges to a common fixed point of $$T_1, T_2, \dots, T_m$$ which are asymptotic pointwise nonexpansive mappings on $$C$$, while $$\{t_{ik}\}^\infty_{k=1}$$ are sequences in [0,1] for all $$i = 1, 2, \dots, m$$, and $$\{n_k\}$$ is an increasing sequence of natural numbers. The related results for uniformly convex Banach spaces are also included.

##### MSC:
 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 54H25 Fixed-point and coincidence theorems (topological aspects) 54E40 Special maps on metric spaces
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