Equivalence of Modified Mann and Multi-step Noor Iteration Methods with Errors

Abstract

The equivalence of the strong convergence between the modified Mann and multi-step Noor iterations with errors is proved for uniformly Lipchitzian generalized strongly successively asymptotically Q-pseudocontractive operators in arbitrary real Banach space. These results generalize those recent ones due to Rhoades and Soltuz in 2003 and 2004 by extending to the more generalized modified multi-step Noor iterations with errors and the most general class of operators, and hence improve the corresponding results of the references given in this paper by providing the equivalences of convergence between all of these iteration schemes for any initial points x1, u1 in arbitrary real Banach space. Our proof methods are quite different and simple.