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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 $\{T_\lambda \}_{\lambda \in \Lambda}$ and $\{S_\lambda \}_{\lambda \in \Lambda}$ be two infinite families of asymptotically nonexpansive mappings from $K$ to itself with $F := \{x \in K : T_\lambda x = x = S_\lambda x$, $\lambda \in \Lambda \} \neq \emptyset$. For an arbitrary initial point $x_0 \in K$, $\{x_n\}$ is defined as follows: $x_n = \alpha_nx_{n-1} + \beta_n(T^\ast_{n-1})^{m_{n-1}}x_{n-1} + \gamma_n(T^\ast_n)^{m_n}y_n$, $y_n = \alpha'_nx_n + \beta'_n(S^\ast_{n-1})^{m_{n-1}}x_{n-1} + \gamma'_n(S^\ast_n)^{m_{n}}x_n$, $n = 1, 2, 3, \dots$, where $T^\ast_n = T_{\lambda_{i_n}}$ and $S^\ast_n = S_{\lambda_{i_n}}$ with $i_n$ and $m_n$ satisfying the positive integer equation: $n = i + (m - 1)m/2$, $m \geq i$; $\{T_{\lambda_i}\}^\infty_{i=1}$ and $\{S_{\lambda_i}\}^\infty_{i=1}$ are two countable subsets of $\{T_\lambda\}_{\lambda \in \Lambda}$ and $\{S_\lambda\}_{\lambda \in \Lambda}$, respectively; $\{\alpha_n\}$, $\{\beta_n\}$, $\{\gamma_n\}$, $\{\alpha'_n\}$, $\{\beta'_n\}$, and $\{\gamma'_n\}$ are sequences in $[\delta, 1 - \delta]$ for some $\delta \in (0, 1)$, satisfying $\alpha_n + \beta_n + \gamma_n = 1$ and $\alpha'_n + \beta'_n + \gamma'_n = 1$. Under some suitable conditions, a strong convergence theorem for common fixed points of the mappings $\{T_\lambda\}_{\lambda \in \Lambda}$ and $\{S_\lambda\}_{\lambda \in \Lambda}$ 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
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##### References:
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