Stability and convergence of modified Ishikawa iterative sequences

It is a study about equivalence theorems between convergence and stability of modified Ishikawa and modified Mann iterative sequences with mean errors for strictly asymptotically pseudocontractive mappings and asymptotically pseudo-contractive mappings in Banach spaces.It is generally about the study under the conditions of D or {xn} and {Tnxn} boundedness,while D or {xn} and {Tnxn} boundedness conditions,restricts the use of some study results to a certain extent.The purpose of this paper is to establish some sufficient and necessary conditions on some equivalence theorems between convergence and stability of modified Ishikawa and modified Mann iterative sequences with mean errors for strictly asymptotically pseudocontractive mappings and asymptotically pseudocontractive mappings with asymptotically nonexpanisive mappings in the intermediate senseunder the condition of removing the restriction bounder {Tnxn} and {xn},and substituente γn=o(αn) with the weaker condition γn→0(n→∞) in Banach spaces by using a new analytical method.Due to removed {xn} and {Tnxn} boundedness conditions,and no increase in other conditions,so our results essentially extend and improve the corresponding results in some references.