## Taiwanese Journal of Mathematics

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

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

#### 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