Let X be a uniformly smooth Banach space,T:X→X be a continuous strongly accretive operator and Lipschitzian strongly accretive operator,respectively;then we will discuss the Mann sequence iteration and Ishikawa sequence iteration for the fixed point of nonlinear equation Sx=f-Tx+x.

In this paper,it is shown that the convergence of Picard iteration sequence is equivalent to Mann iteration sequence,and the convergence of Mann iteration sequence is equivalent to Ishikawa iteration sequence for Zamfirescu operators in an arbitrary Banach space.

Based on Kikkawa and Takahashi iteration method,the Mann iteration sequence is introduced to revise them by using Hybrid projection method.

In the present paper,by virture of the new inequality and the new approximation methods, we prove some new convergence theorems of the Ishikawa and Mann iteration sequences for strongly accretive and strongly pseudo-contractive operators in Banach spaces.

Several convergence theorems of Mann iterative sequences for generalized Φ-pseudocontractive operators in arbitrary Banach space are proven,which generalizes the recent results.

In this paper, we discuss some problems concerning the fixed points and convergence of Mann iterative sequences for nonlinear mapping in metrizable linear topological spaces.

A Lipschitz strictly pseudocontractive map from a nonempty closed convex subset of a Hilbert space into itself was showed to have fixed points,Ishikawa and Mann iterative sequences with errors of a Lipschitz strictly pseudocontractive map was introduced and the iteration processes strongly converges to a fixed point was proved.