Iterative methods for nonlinear Lipschitz pseudocontractive operators
Considerable research efforts in the last 15 years have been devoted to iterative methods for approximating the solution of \(Au= 0\) when \(A\) is accretive (or, correspondingly, fixed points of pseudocontractions \(T= I-A\), since \(A\) is accretive if and only if \(T\) is pseudocontractive). The problem is relevant due to its relation to the initial-value evolution system \[ du/dt+ Au=0,\quad u(0)= w \] [\textit{F. E. Browder}, Proc. Natl. Acad. Sci., USA 61, 388-393 (1968; Zbl 0167.15205)]. In the special case the operator \(T\) is Lipschitz and strongly pseudocontractive, the iterative Mann process [\textit{W. R. Mann}, Proc. Am. Math. Soc. 4, 506-510 (1953; Zbl 0050.11603)] has been successfully employed in approximating fixed points of such mappings. It is a long standing question whether or not the Mann process converges when \(T\) is a Lipschitz pseudocontractions even in optimal cases (i.e. \(T\) defined on a compact convex subset of a Hilbert space) [\textit{T. L. Hicks} and \textit{J. D. Kubicek}, J. Math. Anal. Appl. 59, 498-504 (1977; Zbl 0361.65057)]. The author exhibits a simple example, constructed by the author himself and Mutangadura, answering negatively to the question. Other results about different iterative methods are summarized. The concept of acceptably paired sequences, due in the real case to R. E. Bruck, is extended to the context of real normed spaces. A couple of questions of interest are raised at the conclusion of the paper.
scientific article; zbMATH DE number 1545036