On Reich's open question (Q1888904)

Let \(E\) be a real Banach space whose norm is uniformly Gâteau differentiable, \(D\) a nonempty closed convex subset of \(E\), \(T\) an asymptotically nonexpansive selfmap of \(E\) with sequence \(\{k_n\} \subset [1, \infty), k_n \to 1, \sum \nolimits (k_n - 1) < \infty\). The modified Wittman-Ishikawa sequence operating on \(T\) is defined by \(x_0, x \in D\), \(x_{n+1} := \alpha_nx + (1 - \alpha_n)T^ny_n\), \(y_n := \beta_nx_n + (1 - \beta_n)T^nx_n\), \(n \geq 0\), where \(\{\alpha_n\}, \{\beta_n\}\) are sequences in \([0, 1]\) satisfying certain conditions. Under some assumptions, including the fact that \(F(T)\), the fixed point set of \(T\), is nonempty, the author proves that \(x_n\) converges strongly to a fixed point of \(T\) iff \(\{x_n\}\) is bounded. There is a similar result for the modified Wittman sequence obtained by setting each \(\beta_n = 1\). The final result obtains sufficient conditions for the iteration scheme \(x_0 \in D\), \(x_{n+1} := \alpha_nx + (1 - \alpha_n)Ty_n\), \(y_n := \beta_nx_n + (1 - \beta_n)Tx_n\), \(n \geq 0\) (or the one with each \(\beta_n = 1\)) to converge strongly to a fixed point for a nonexpansive map with \(F(T) \neq \emptyset\).