Approximating common fixed points of asymptotically nonexpansive mappings. (English) Zbl 1107.47053

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: \(x_{1} \in C\), \[ \begin{alignedat}{2} z_{n}&=(1-\gamma _{n}-\nu _{n})x_{n}+\gamma _{n}T^{n}_{1}x_{n}+\nu _{n}u_{n}, &&\quad n\geq 1,\\ y_{n}&=(1-\beta _{n}-\mu _{n})x_{n}+\beta _{n}T^{n}_{2}z_{n}+\mu _{n}v_{n}, &&\quad n\geq 1,\\ x_{n+1}&=(1-\alpha _{n}-\lambda _{n})x_{n}+\alpha _{n}T^{n}_{3}y_{n}+\lambda _{n}w_{n}, &&\quad n\geq 1,\end{alignedat} \] 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.


47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
Full Text: DOI