Approximation methods for nonlinear operator equations

Abstract:

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 $\sigma :=\inf _{ n\in N_0 } \frac {\psi (||x_{n+1}-x^*||)}{||x_{n+1}-x^*||}>0$ and that $||Ax_{n+1}-Ax_n||\rightarrow 0$, 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.