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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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##### References:
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