Taiwanese Journal of Mathematics

CONVERGENCE THEOREMS OF ITERATIVE ALGORITHMS FOR A FAMILY OF FINITE NONEXPANSIVE MAPPINGS

Jong Soo Jung

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Abstract

Let $E$ be a Banach space, $C$ a nonempty closed convex subset of $E$, $f : C \to C$ a contraction, and $T_i : C \to C$ a nonexpansive mapping with nonempty $F := \bigcap_{i = 1}^N Fix(T_i)$, where $N \ge 1$ is an integer and $Fix(T_i)$ is the set of fixed points of $T_i$. Let $\{x_t^n\}$ be the sequence defined by $x_t^n = tf(x_t^n) + (1-t) T_{n+N} T_{n+N-1} \cdots T_{n+1} x_t^n$ ($0 \lt t \lt 1$). First, it is shown that as $t \to 0$, the sequence $\{x_t^n\}$ converges strongly to a solution in $F$ of certain variational inequality provided $E$ is reflexive and has a weakly sequentially continuous duality mapping. Then it is proved that the iterative algorithm $x_{n+1} = \lambda_{n+1} f(x_n) + (1-\lambda_{n+1}) T_{n+1} x_n$ ($n \ge 0$) converges strongly to a solution in $F$ of certain variational inequality in the same Banach space provided the sequence $\{\lambda_n\}$ satisfies certain conditions and the sequence $\{x_n\}$ is weakly asymptotically regular. Applications to the convex feasibility problem are included.

Article information

Source
Taiwanese J. Math., Volume 11, Number 3 (2007), 883-902.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500404763

Digital Object Identifier
doi:10.11650/twjm/1500404763

Mathematical Reviews number (MathSciNet)
MR2340169

Zentralblatt MATH identifier
1219.47117

Subjects
Primary: 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc. 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30] 47J20: Variational and other types of inequalities involving nonlinear operators (general) [See also 49J40] 47J25: Iterative procedures [See also 65J15] 49M05: Methods based on necessary conditions

Keywords
iterative algorithms nonexpansive mapping contraction viscosity approximation method common fixed points weakly sequentially continuous duality mapping variational inequality sunny and nonexpansive retraction weakly asymptotically regular

Citation

Jung, Jong Soo. CONVERGENCE THEOREMS OF ITERATIVE ALGORITHMS FOR A FAMILY OF FINITE NONEXPANSIVE MAPPINGS. Taiwanese J. Math. 11 (2007), no. 3, 883--902. doi:10.11650/twjm/1500404763. https://projecteuclid.org/euclid.twjm/1500404763


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