Generic aspects of metric fixed point theory (Q2761641)

Baire's well-known theorem states that the intersection of every countable collection of open dense subsets of a complete metric space \(X\) is dense in \(X\). This motivates to consider the following: ``Given a property which elements of \(X\) may have, it is of interest to determine whether this property is generic, that is, whether the set of elements which do enjoy this property contains a countable intersection of open sets. Such an approach, when a certain property is investigated for the whole space \(X\) and not just for a single point in \(X\), has already been successfully applied in many areas of analysis.'' NEWLINENEWLINENEWLINEThe present exposition embodies several results in metric fixed point theory which exhibit generic phenomena. The authors' recent results [Math. Comput. Modelling 32, 1423-1431 (2000; Zbl 0977.47046)] concerning the successive approximation of super-regular mappings are presented in Section 3. The subsequent section deals with contractive mappings, and some of the fundamental results from the authors [C. R. Math. Acad. Sci., Soc. R. Can. 22, 118-124 (2000; Zbl 0971.47039)] are nicely presented. Section 5 deals with asymptotic behaviour of (random) infinite products of generic sequences of nonexpansive as well as uniformly continuous operators on closed convex subsets of a complete hyperbolic space. Besides other results, the formulation leads to the presentation of a weak ergodic theorem from a recent work by the authors [Nonlinear Anal., Theory Methods Appl. 36A, 1049-1065 (1999; Zbl 0932.47043)]. The intention of sections 6-8 is to present some results concerning (F)-attracting mappings, contractive set-valued mappings and nonexpansive set-valued mappings. The last section discusses porous sets which are near in nature to and yet different from nowhere dense sets in metric spaces. Using the notion of porosity, a kind of approximation theorem for nonexpansive mappings from \textit{F. S. De Blasi} and \textit{J. Myjak} [C. R. Acad. Sci., Paris, Sér. I 308, 51-54 (1989; Zbl 0647.47053)] is presented.NEWLINENEWLINEFor the entire collection see [Zbl 0970.54001].