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An implicit iteration process for common fixed points of two infinite families of asymptotically nonexpansive mappings in Banach spaces. (English) Zbl 1266.47090
Summary: Let $K$ be a nonempty, closed, and convex subset of a real uniformly convex Banach space $E$. Let ${\left\{{T}_{\lambda }\right\}}_{\lambda \in {\Lambda }}$ and ${\left\{{S}_{\lambda }\right\}}_{\lambda \in {\Lambda }}$ be two infinite families of asymptotically nonexpansive mappings from $K$ to itself with $F:=\left\{x\in K:{T}_{\lambda }x=x={S}_{\lambda }x$, $\lambda \in {\Lambda }\right\}\ne \varnothing$. For an arbitrary initial point ${x}_{0}\in K$, $\left\{{x}_{n}\right\}$ is defined as follows: ${x}_{n}={\alpha }_{n}{x}_{n-1}+{\beta }_{n}{\left({T}_{n-1}^{*}\right)}^{{m}_{n-1}}{x}_{n-1}+{\gamma }_{n}{\left({T}_{n}^{*}\right)}^{{m}_{n}}{y}_{n}$, ${y}_{n}={\alpha }_{n}^{\text{'}}{x}_{n}+{\beta }_{n}^{\text{'}}{\left({S}_{n-1}^{*}\right)}^{{m}_{n-1}}{x}_{n-1}+{\gamma }_{n}^{\text{'}}{\left({S}_{n}^{*}\right)}^{{m}_{n}}{x}_{n}$, $n=1,2,3,\cdots$, where ${T}_{n}^{*}={T}_{{\lambda }_{{i}_{n}}}$ and ${S}_{n}^{*}={S}_{{\lambda }_{{i}_{n}}}$ with ${i}_{n}$ and ${m}_{n}$ satisfying the positive integer equation: $n=i+\left(m-1\right)m/2$, $m\ge i$; ${\left\{{T}_{{\lambda }_{i}}\right\}}_{i=1}^{\infty }$ and ${\left\{{S}_{{\lambda }_{i}}\right\}}_{i=1}^{\infty }$ are two countable subsets of ${\left\{{T}_{\lambda }\right\}}_{\lambda \in {\Lambda }}$ and ${\left\{{S}_{\lambda }\right\}}_{\lambda \in {\Lambda }}$, respectively; $\left\{{\alpha }_{n}\right\}$, $\left\{{\beta }_{n}\right\}$, $\left\{{\gamma }_{n}\right\}$, $\left\{{\alpha }_{n}^{\text{'}}\right\}$, $\left\{{\beta }_{n}^{\text{'}}\right\}$, and $\left\{{\gamma }_{n}^{\text{'}}\right\}$ are sequences in $\left[\delta ,1-\delta \right]$ for some $\delta \in \left(0,1\right)$, satisfying ${\alpha }_{n}+{\beta }_{n}+{\gamma }_{n}=1$ and ${\alpha }_{n}^{\text{'}}+{\beta }_{n}^{\text{'}}+{\gamma }_{n}^{\text{'}}=1$. Under some suitable conditions, a strong convergence theorem for common fixed points of the mappings ${\left\{{T}_{\lambda }\right\}}_{\lambda \in {\Lambda }}$ and ${\left\{{S}_{\lambda }\right\}}_{\lambda \in {\Lambda }}$ is obtained. The results extend those of the authors whose related works are restricted to the situation of finite families of asymptotically nonexpansive mappings.
##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties