Iterative approximation of common fixed points of two nonself asymptotically nonexpansive mappings (Q541332)

Summary: Suppose that \(K\) is nonempty closed convex subset of a uniformly convex and smooth Banach space \(E\) with \(P\) as a sunny nonexpansive retraction and \(F := F(T_1) \cap F(T_2) = \{x \in K : T_1x = T_2x = x\} \neq \emptyset\). Let \(T_1, T_2 : K \to E\) be two weakly inward nonself asymptotically nonexpansive mappings with respect to \(P\) with two sequences \(\{k^{(i)}_{n}\} \subset [1, \infty)\) satisfying \(\sum^{\infty}_{n=1} (k^{(i)}_{n} - 1) < \infty\) \((i = 1, 2)\), respectively. For any given \(x_1 \in K\), suppose that \(\{x_n\}\) is a sequence generated iteratively by \(x_{x+1} = (1 - \alpha_{n}) (PT_1)^{n} y_{n} + \alpha_{n} (PT_{2})^{n} y_{n}\), \(y_{n} = (1 - \beta_{n}) x_{n} + \beta_{n} (PT_{1})^{n} x_{n}\), \(n \in \mathbb N\), where \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are sequences in \([a, 1 - a]\) for some \(a \in (0, 1)\). Under some suitable conditions, strong and weak convergence theorems of \(\{x_n\}\) to a common fixed point of \(T_1\) and \(T_2\) are obtained.