Approximation methods for nonlinear operator equations

Let $E$ be a real normed linear space and $A: E \rightarrow E$ be a uniformly quasi-accretive map. For arbitrary $x_1\in E$ define the sequence $x_n \in E$ by $x_{n+1}:=x_n-\alpha_nAx_n,~n\geq 1,$ where $\{\alpha_n\}$ is a positve real sequence satisfying the following conditions: (i) $\sum\alpha_n=\infty$; (ii) $\lim \alpha_n=0$. For $x^*\in N(A):=\{x\in E:Ax=0\}$, assume that $\sigma :=\inf_{ n\in N_0 } \frac{\psi(\vert\vert x_{n+1}-x^*\vert\vert)}{\vert\vert x_{n+1}-x^*\vert\vert}>0$ and that $\vert\vert Ax_{n+1}-Ax_n\vert\vert\rightarrow 0$, where $N_0:=\{n\in N$ (the set of all positive integers): $x_{n+1}\neq x^*\}$and $\psi:[0,\infty)\rightarrow [0,\infty)$ is a strictly increasing function with $\psi(0)=0$. It is proved that a Mann-type iteration process converges strongly to $x^*$. Furthermore if, in addition, $A$ is a uniformly continuous map, it is proved, without the condition on $\sigma$, that the Mann-type iteration process converges strongly to $x^*$. As a consequence, corresponding convergence theorems for fixed points of hemi-contractive maps are proved.