A reconsideration on convergence of three-step iterations for asymptotically nonexpansive mappings (Q2383887)

Let \(X\) be a uniformly convex Banach space and \(C\) be some nonempty convex part of it. Further, let \(T:C \to C\) be an asymptotically nonexpansive map. Sufficient conditions are given so as the iterative scheme \[ z_n=a_nT^nx_n+(1-a_n)x_n,\quad y_n=b_nT^nz_n+c_nT^nx_n+(1-b_n-c_n)x_n, \] \[ x_{n+1}=\alpha_nT^ny_n+\beta_nT^nz_n+\gamma_nT^nx_n+ (1-\alpha_n-\beta_n-\gamma_n)x_n,\quad n\geq 1, \] should converge strongly to a fixed point of \(T\). These are complementary to the ones recently established by the authors [Appl. Math. Comput. 181, 1026--1034 (2006; Zbl 1104.65053)].