Jui-Chi Huang

Common Fixed Points Iteration Processes for a Finite Family of Asymptotically Nonexpansive Mappings

Abstract:
Let $E$ be a uniformly convex Banach space which satisfies Opial's condition or its dual $E^*$ has the Kadec-Klee property, $C$ a nonempty closed convex subset of $E$, and $T_j:C\rightarrow C$ an asymptotically nonexpansive mapping for each $j=1,2,\ldots,r$. Suppose $\{x_n\}$ is generated iteratively by
$$x_0\in C,\;\; x_{n+1}=(1-\alpha_{n(r)})x_n+\alpha_{n(r)}
{1\over n+1}\sum_{i=0}^n T_r^i U_{n(r-1)}x_n,\;\; n=0,1,2,\ldots,$$
where $U_{n(j)}=(1-\alpha_{n(j)})I+\alpha_{n(j)} {1\over n+1} \sum_{i=0}^n T_j^i U_{n(j-1)}$, $j=1,2,\ldots,r$, $U_{n(0)}=I$, $I$ is the identity map and $\{\alpha_{n(j)}\}$ is a suitable sequence in $[0,1]$. If the set $\cap_{j=1}^r F(T_j)$ of common fixed points of $\{T_j\}_{j=1}^r$ is nonempty, then weak convergence of $\{x_n\}$ to some $p\in \cap_{j=1}^r F(T_j)$ is obtained.

Keywords:
Asymptotically nonexpansive mapping, Fixed point, uniformly convex Banach space, Opial's condition, Kadec-Klee property.

MSC 2000: 47H10.