On the equivalence of the convergence criteria between modified Mann-Ishikawa and multi-step iterations with errors for successively strongly pseudo-contractive operators

In this paper, the equivalence of the convergence between the modified Mann-Ishikawa and multi-step Noor iterations with errors is proven for the successively strongly pseudo-contractive operators without Lipschitzian assumption. Our results generalize the recent results of the paper [B.E. Rhoades, S.M. Soltuz, The equivalence between Mann-Ishikawa iterations and multi-step iteration, Nonlinear Anal. 58 (2004) 219-228; B.E. Rhoades, S.M. Soltuz, The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudo-contractive maps, J. Math. Anal. Appl. 289 (2004) 266-278] by extending to the more generalized multi-step iterations with errors and hence improve the corresponding results of all the references in bibliography by providing the equivalences of convergence between all of these up-to-date iteration schemes.