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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 involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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