THE EQUIVALENCE BETWEEN THE CONVERGENCES OF MULTI-STEP ITERATIONS WITH ERRORS FOR UNIFORMLY GENERALIZED LIPSCHITZ CONTINUOUS OPERATORS
具一致广义Lipschitz连续算子的带误差的多步迭代间的收敛等价性

In this paper, the equivalence of the convergence of modified Mann iteration and multi-step Noor iteration with errors is invesgated for uniformly generalized Lipschitz continuous and successively asymptotically $\Phi$-strongly pseudo-contractive type operators in uniformly smooth Banach spaces. In 2007, Zhenyu Huang showed the equivalence of the convergence criteria between modified Mann and Ishikawa iterations with errors for successively $\Phi$-strongly pseudo-contractive operators with bounded range in uniformly smooth Banach spaces. The results obtained in this paper generalize the results of Huang in 2007. The results give an affirmative answer to the conjection raised by Rhoades and Soltuz in 2003.