Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2009; 46(6): 1151-1164

Printed November 1, 2009

https://doi.org/10.4134/JKMS.2009.46.6.1151

Copyright © The Korean Mathematical Society.

Strong convergence of composite iterative methods for nonexpansive mappings

Jong Soo Jung

Dong-A University

Abstract

Let $E$ be a reflexive Banach space with a weakly sequentially continuous duality mapping, $C$ be a nonempty closed convex subset of $E$, $f : C \to C$ a contractive mapping (or a weakly contractive mapping), and $T : C \to C$ a nonexpansive mapping with the fixed point set $F(T) \neq \emptyset$. Let $\{x_n\}$ be generated by a new composite iterative scheme: $y_n = \lambda_nf(x_n) + (1 - \lambda_n)Tx_n$, $x_{n+1} = (1 - \beta_n)y_n + \beta_nTy_n$, $(n \ge 0)$. It is proved that $\{x_n\}$ converges strongly to a point in $F(T)$, which is a solution of certain variational inequality provided the sequence $\{\lambda_n\} \subset (0,1)$ satisfies $\lim_{n\to \infty}\lambda_n = 0$ and $\sum_{n =0}^{\infty}\lambda_n = \infty$, $\{\beta_n\} \subset [0,a)$ for some $0 < a < 1$ and the sequence $\{x_n\}$ is asymptotically regular.

Keywords: viscosity approximation method, nonexpansive mapping, composite iterative scheme, contractive mapping, weakly contractive mapping, weakly sequentially continuous duality mapping, variational inequality

MSC numbers: 47H09, 47H10, 47J20, 47J25, 49M05