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Zbl 1203.47053
Jung, Jong Soo
Convergence on composite iterative schemes for nonexpansive mappings in Banach spaces.
(English)
[J] Fixed Point Theory Appl. 2008, Article ID 167535, 14 p. (2008). ISSN 1687-1812/e

Let $E$ be a reflexive Banach space with a uniformly Gateaux differentiable norm. Suppose that every weakly compact convex subset of $E$ has the fixed point property for nonexpansive mappings. Let $C$ be a nonempty closed convex subset of $E$, $f: C \to C$ be a contractive mapping (or a weakly contractive mapping), and $T: C \to C$ be a nonexpansive mapping with nonempty fixed point set $F(T)$. Let the sequence $\{x_n\}$ be generated by the following composite iterative scheme: \left\{\aligned &y_n=\lambda_n f(x_n)+(1-\lambda_n)Tx_n,\\ &x_{n+1}=(1-\beta_n)y_n+\beta_n Ty_n,\endaligned\right.\quad n\geq0. It is proved that $\{x_n\}$ converges strongly to a point in $F(T)$, which is a solution of a certain variational inequality, provided that the sequence $\{\lambda_n\}\subset(0,1)$ satisfies $\lim_{n\rightarrow\infty}\lambda_n=0$ and $\sum_{n=1}^\infty\lambda_n=\infty$, $\{\beta_n\}\subset[0,a)$ for some $0<a<1$, and the sequence $\{x_n\}$ is asymptotically regular.
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H09 Mappings defined by "shrinking" properties

Cited in: Zbl 1226.47080

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