Strong and weak convergence theorems for asymptotically nonexpansive mappings. (Q1873668)

This article deals with iterations \(x_n\) defined by the formula \[ x_{n+1} = P(((1 - \alpha_n)x_n + \alpha_nT(PT)^{n-1}x_n), \quad n = 1,2,\ldots, \;x_1 \in K \] where \(K\) is a nonempty closed convex nonexpansive retract in a real uniformly convex Banach space \(E\), \(P\) a nonexpansive retraction, \(T: \;K \to E\) an asymptotically nonexpansive nonself-map such that \[ \| T^nx - T^ny\| \leq k_n\| x - y\| , \quad n = 1,2, \dots, \;x, y \in K, \] (\(k_n \to 1\) and \(\sum_{n=1}^\infty (k_n^2 - 1) < \infty\)) and such that \(F(T) = \{x \in K: \;Tx = x\} \neq \emptyset\), and \((\alpha_n)\) a sequence such that \(\varepsilon < 1 - \alpha_n < 1 - \varepsilon\) for some \(\varepsilon > 0\). It is proved the strong convergence of \((x_n)\) to some \(x_* \in F(T)\) provided that \(T\) is completely continuous and the weak convergence of \((x_n)\) to some \(x_* \in F(T)\) provided that the norm in \(E\) is Fréchet differentiable.