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Strong convergence to common fixed points of a finite family of asymptotically nonexpansive mappings. (Strong convergence to common fixed points of a finite family of asymptotically nonexpansive map.) (English) Zbl 1219.47135
Summary: Suppose that $E$ is a real Banach space with uniform normal structure and suppose that $E$ has a uniformly Gâteaux differentiable norm. Let $C$ be a nonempty closed convex and bounded subset of $E$. Let $T_1,T_2,\dots,T_r:C\to C$ be a finite family of asymptotically nonexpansive mappings. In this paper, we suggest and analyze an iterative algorithm for $\{T_i\}_{i=1}^r$. We show the convergence of the proposed algorithm to a common fixed point $p\in\bigcap_{i=1}^r F(T_i)$ which is the unique solution of some variational inequality. Our results can be considered as an refinement and improvement of many known results.

47J25Iterative procedures (nonlinear operator equations)
47H05Monotone operators (with respect to duality) and generalizations
47H10Fixed-point theorems for nonlinear operators on topological linear spaces