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\(\Delta\)-convergence problems for asymptotically nonexpansive mappings in CAT(0) spaces. (English) Zbl 1273.47114
Summary: New \(\Delta\)-convergence theorems of iterative sequences for asymptotically nonexpansive mappings in CAT(0) spaces are obtained. Consider an asymptotically nonexpansive self-mapping \(T\) of a closed convex subset \(C\) of a CAT(0) space \(X\). Consider the iteration process \(\{x_n\}\), where \(x_0 \in C\) is arbitrary and \(x_{n + 1} = \alpha_n x_n \oplus (1 - \alpha_n)T^n y_n\) or \(x_{n + 1} = \alpha_n T^n x_n \oplus (1 - \alpha_n)y_n\), \(y_n = \beta_n x_n \oplus (1 - \beta_n)T^n x_n\) for \(n \geq 1\), where \(\{\alpha_n\}, \{\beta_n\} \subset (0, 1)\). It is shown that, under certain appropriate conditions on \(\alpha_n, \beta_n,\, \{x_n\}\, \Delta\)-converges to a fixed point of \(T\).

MSC:
47J25 Iterative procedures involving nonlinear operators
54H25 Fixed-point and coincidence theorems (topological aspects)
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