On the approximation problem of fixed points for asymptotically nonexpansive mappings
This article deals with the Ishikawa and Mann approximations with errors \[ x_{n+1}= (1-\alpha_n-\gamma_n) x_n+\alpha_nT^ny_n +\gamma_nw_n,\quad y_n=(1-\beta_n-\delta_n)x_n+ \beta_nT^nx_n+ \delta_n z_n, \tag{1} \] \[ x_{n+1}= (1-\alpha_n -\gamma_n)x_n+ \alpha_n T^nx_n+ \gamma_n w_n,\tag{2} \] for asymptotically nonexpansive mappings \(T\) in a real uniformly convex Banach space \(E\). The author formulates sufficient conditions on \(\alpha_n,\beta_n, \gamma_n, \delta_n\) and Lipschitzian constants \(k_n\) of \(T^n\) for the convergence of approximations (1) and (2) and states that the corresponding theorems improve and generalize results by K. Goebel and W. A. Kirk, B. E. Rhoades, J. Schu and correct results by Z. Y. Huang.
Ishikawa and Mann approximations with errors
scientific article; zbMATH DE number 1785895