Abstract:Consider the Mann iteration $x_{n+1} = ( 1 - \alpha_n ) x_n + \alpha_n Tx_n$ for a nonexpansive mapping $T\colon K \to K$ defined on some subset $K$ of the normed space $X$. We present an innovative proof of the Ishikawa almost fixed point principle for nonexpansive mapping that reveals deeper aspects of the behavior of the process. This fact allows us, among other results, to derive convergence of the process under the assumption of existence of an accumulation point of $\{ x_n \}$.
Keywords: Mann iteration; fixed point; nonexpansive mapping
DOI: DOI 10.14712/1213-7243.2020.031
AMS Subject Classification: 47H10