Equivalence of the Convergences of some Modified Iterations with Errors for Uniformly Lipschitzian Asymtotically Pseudo-Contractive Maps

Abstract

Certain iterative schemes demonstrate a faster convergence to a fixed point compared to others when used to solve various nonlinear differential equations. We show that the convergence of various iterative schemes, including the modified Mann iteration, modified Mann iteration with errors, modified Ishikawa iteration, modified Ishikawa iteration with errors, modified Noor iteration, modified Noor iteration with errors, modified multistep iteration and modified multi-step iteration with errors are all equivalent when applied to uniformly Lipschitzian asymptotically pseudo-contractive maps in an arbitrary real Banach space. Our results expand and generalize the earlier works of Rhoades and Soltuz [1], Olaleru and Odumosu [2] and Odumosu, Olaleru and Ayodele [3].