Taiwanese Journal of Mathematics

STRONG CONVERGENCE OF A HYBRID VISCOSITY APPROXIMATION METHOD WITH PERTURBED MAPPINGS FOR NONEXPANSIVE AND ACCRETIVE OPERATORS

Lu-Chuan Ceng and Hong-Kun Xu

Full-text: Open access

Abstract

Recently, H. K. Xu [J. Math. Anal. Appl. 314 (2006) 631-643] considered the iterative method for approximation to zeros of an $m$-accretive operator $A$ in a Banach space $X$. In this paper, we propose a hybrid viscosity approximation method with perturbed mapping that generates the sequence $\{x_n\}$ by the algorithm $x_{n+1} = \alpha_n(u+f(x_n)) + (1-\alpha_n) [J_{r_n} x_n - \lambda_n F(J_{r_n} x_n)]$, where $\{\alpha_n\}$, $\{r_n\}$ and $\{\lambda_n\}$ are three sequences satisfying certain conditions, $f$ is a contraction on $X$, $J_r$ denotes the resolvent $(I+rA)^{-1}$ for $r \gt 0$, and $F$ is a perturbed mapping which is both $\delta$-strongly accretive and $\lambda$-strictly pseudocontractive with $\delta + \lambda \geq 1$. Under the assumption that $X$ either has a weakly continuous duality map or is uniformly smooth, we establish some strong convergence theorems for this hybrid viscosity approximation method with perturbed mapping.

Article information

Source
Taiwanese J. Math., Volume 11, Number 3 (2007), 661-682.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500404751

Digital Object Identifier
doi:10.11650/twjm/1500404751

Mathematical Reviews number (MathSciNet)
MR2340157

Zentralblatt MATH identifier
1219.47102

Subjects
Primary: 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc. 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30] 47H17

Keywords
hybrid viscosity approximation method with perturbed mapping $m$-accretive operator nonexpansive mapping weakly continuous duality map uniformly smooth Banach space

Citation

Ceng, Lu-Chuan; Xu, Hong-Kun. STRONG CONVERGENCE OF A HYBRID VISCOSITY APPROXIMATION METHOD WITH PERTURBED MAPPINGS FOR NONEXPANSIVE AND ACCRETIVE OPERATORS. Taiwanese J. Math. 11 (2007), no. 3, 661--682. doi:10.11650/twjm/1500404751. https://projecteuclid.org/euclid.twjm/1500404751


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