On estimation and control of errors of the Mann iteration process (Q1414236)

Let \(K\) be a nonempty convex subset of a Banach space \(X\). The purpose of this note is to study the Mann iterative process \(x_0 \in K\), \(x_{n+1}=(1-\alpha_n) x_n +\alpha_n {\overline {Tx_n}}\), where \(\alpha_n \in (0,1)\) and the mapping \({\overline T}: X \to X\) is given with errors. Let \(Tx\) be an available approximation of \({\overline {Tx}}\). It is shown that the accumulative error in the process has the order \(O(\sup_{n \geq 0} \| Tx_n- {\overline {Tx_n}} \|)\).