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Zbl 1107.47053
Cho, Yeol Je; Kang, Jung Im; Zhou, Haiyun
Approximating common fixed points of asymptotically nonexpansive mappings.
(English)
[J] Bull. Korean Math. Soc. 42, No. 4, 661-670 (2005). ISSN 1015-8634

The authors study approximations of common fixed points of iterative sequences with errors for three asymptotically nonexpansive mappings in uniformly convex Banach space generated by the following scheme: $x_{1} \in C$, \alignat2 z_{n}&=(1-\gamma _{n}-\nu _{n})x_{n}+\gamma _{n}T^{n}_{1}x_{n}+\nu _{n}u_{n}, &&\quad n\geq 1,\\ y_{n}&=(1-\beta _{n}-\mu _{n})x_{n}+\beta _{n}T^{n}_{2}z_{n}+\mu _{n}v_{n}, &&\quad n\geq 1,\\ x_{n+1}&=(1-\alpha _{n}-\lambda _{n})x_{n}+\alpha _{n}T^{n}_{3}y_{n}+\lambda _{n}w_{n}, &&\quad n\geq 1,\endalignat where $0<a\leq \alpha _{n},\beta _{n},\gamma _{n}<1$, $\sum ^{\infty }_{n=1}\lambda _{n}<+\infty$, $\sum^{\infty }_{n=1}\mu _{n}<+\infty$, and $\sum ^{\infty }_{n=1}\nu _{n}<+\infty$, $\{u_{n}\}$ and $\{v_{n}\}$ are two bounded sequences in a subset $C$ of a uniformly convex Banach space. By Theorem 6 (with the correction: $F=\bigcap ^{3}_{i=1}T_{i} \neq \emptyset$), the authors prove the weak convergence of $\{x_{n}\}$ as generated by the above scheme to a common fixed point of $T_{i}$ $(i=1,2,3)$ in a subset of a uniformly convex Banach space satisfying Opial's condition. In Theorem 7, in a compact convex subset of uniformly convex Banach space, the authors prove the strong convergence of the sequence $\{x_{n}\}$ to a common fixed point of $T_i$ $(i=1,2,3)$ by dropping Opial's condition.
[Edward Prempeh (Kumasi)]
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H09 Mappings defined by "shrinking" properties

Keywords: uniformly convex Banach space; asymptotically nonexpansive mapping; approximate fixed point; Opial's condition

Cited in: Zbl 1163.47308

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