On the convergence of some iteration processes in uniformly convex Banach spaces

Abstract

For the approximation of fixed points of a nonexpansive operator T in a uniformly convex Banach space E the convergence of the Mann-Toeplitz iteration x n + 1 = α n T ( x n ) + ( 1 − α n ) x n {x_{n + 1}} = {\alpha _n}T({x_n}) + (1 - {\alpha _n}){x_n} is studied. Strong convergence is established for a special class of operators T. Via regularization this result can be used for general nonexpansive operators, if E possesses a weakly sequentially continuous duality mapping. Furthermore strongly convergent combined regularization-iteration methods are presented.