Weak stability of Mann and Ishikawa iterations with errors for image-hemicontractive operators Original Research Article

Let EE be an arbitrary Banach space and T:E→ET:E→E be a Lipschitzian and ϕϕ-hemicontractive mapping. This paper proves that, without the property liminfn→∞ϕ(t)/t>0liminfn→∞ϕ(t)/t>0, the Mann and Ishikawa iterative sequences with errors are weakly TT-stable. The related result deals with the weak TT-stability of these sequences with errors to the unique solution of the equation f=Txf=Tx when T:E→ET:E→E is Lipschitzian and ϕϕ-strongly accretive operator. Our results improve and generalize the recent results of Zhou et al. [H.Y. Zhou, S.S. Chang, Y.J. Cho, Weak stability of Ishikawa iteration procedures for ϕϕ-hemicontractive and accretive operators, Appl. Math. Lett. 14 (2001) 949–954] without the strict requirement liminfn→∞ϕ(t)/t>0liminfn→∞ϕ(t)/t>0 and extend the results to the Mann and Ishikawa iterations with errors.