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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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References:
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