CONVERGENCE THEOREMS OF ITERATIVE ALGORITHMS FOR A FAMILY OF FINITE NONEXPANSIVE MAPPINGS
Jong Soo Jung · Taiwanese Journal of Mathematics · 2007
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 $F := \bigcap_{i = 1}^NFix(T_i)$, 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 $x_t^n = tf(x_t^n) + (1-t)T_{n+N}T_{n+N-1}\cdots T_{n+1}x_t^n$ $(0 < t < 1)$. 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.