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Zbl 1102.47056
Fixed point solutions of variational inequalities for asymptotically nonexpansive mappings in Banach spaces.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 64, No. 3, A, 558-567 (2006); erratum ibid. 66, No. 12, A, 2980-2981 (2007). ISSN 0362-546X

Let $E$ be a real Banach space with uniformly Gâteaux differentiable norm, $K$ a nonempty bounded closed convex subset of $E$, and $T: K \to K$ an asymptotically nonexpansive mapping, i.e., $\Vert T^n x- T^n y \Vert \leq k_n \Vert x-y \Vert$ for all $x, y \in K$, where $k_n \in [1, \infty)$, $\lim_{n \to \infty} k_n=1$. Let $f: K \to K$ be a contraction. The authors study approximating properties of the sequence $\{ x_n\}$ defined by $x_n=(1-k_n^{-1} t_n) f(x_n)+k_n^{-1} t_n T^n x_n$ ($t_n>0$, $\lim_{n \to \infty} t_n=1$) with respect to the set of fixed points of the mapping $T$.
[Mikhail Yu. Kokurin (Yoshkar-Ola)]
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H09 Mappings defined by "shrinking" properties
47H06 Accretive operators, etc. (nonlinear)
47J20 Inequalities involving nonlinear operators

Keywords: viscosity approximation; asymptotically nonexpansive mapping; variational inequality; normal structure

Cited in: Zbl 1237.47074

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