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.
Jong Soo Jung
Dong-A University
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
2008; 45(2): 377-392
2021; 58(3): 525-552
2017; 54(3): 1031-1047
2015; 52(6): 1287-1303
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd