Remarks on the approximation methods for nonlinear operator equations of the m-accretive type
Let \(T\) be an m-accretive operator with resolvent \(J_r=(I+rT)^{-1}\) (\(r>0\)). \textit{S. Reich} [J. Math. Anal. Appl. 75, 287-292 (1980; Zbl 0437.47047)] has proved under some assumptions that \(Qx=\lim_{r\to 0}J_rx\) defines a retraction of \(X\) onto \(\overline{D(T)}\). Based on this retraction, approximation methods had been introduced by \textit{L. Zhu} [J. Math. Anal. Appl. 188, No. 2, 410-416 (1994; Zbl 0922.47061)], \textit{C. E. Chidume} and \textit{M. O. Osilike} [J. Math. Anal. Appl. 189, No. 1, 225-239 (1995; Zbl 0824.47050)], and \textit{X. P. Ding} [J. Math. Anal. Appl. 209, No. 1, 191-201 (1997; Zbl 0883.47054)]. The author observes (for bounded \(T\)) that \(Q\) is actually the identity, and so the above approximation methods actually are the classical iteration processes of \textit{W. R. Mann} [Proc. Am. Math. Soc. 4, 506-510 (1953; Zbl 0050.11603)] and \textit{S. Ishikawa} [Proc. Am. Math. Soc. 44, 147-150 (1974; Zbl 0286.47036)]. In the paper also a new iteration process based on the map \(TJ_r\) instead of \(Q\) is introduced.
scientific article; zbMATH DE number 1491638