Generic Properties of Nonexpansive Mappings
(FWF P 32523)

Project Abstract for a General Audience

The project “Generic properties of nonexpansive mappings“ is concerned with the study of properties of functions which do no increase distances. These functions appear naturally in various areas of mathematics, for example in optimization, where they play an important role. In this project, we do not consider properties which are shared by all of these functions but with properties which are typical for them. In this context being typical means that most of the functions do have this property. Since there are infinitely many nonexpansive mappings, we cannot decide what is typical by counting the functions with a certain property and compare their number with the number of all nonexpansive mappings but we have to use a different approach. One possibility of describing this approach is to look at the points of a disk in the plane: there are infinitely many such points but it seems clear that most of them do not lie on the boundary of the disc. A formal way of checking this statement is to see that arbitrarily close to every point on the boundary of the disk, there is a smaller disk which lies entirely outside the circle. The characterization we use for typical nonexpansive mappings is very similar to the corresponding characterization of the points in the interior of the disk.More precisely, we want to consider the question of finding geometric conditions such that the typical nonexpansive mapping has a fixed point, i.e. a point which is not changed by this mapping. Another property we are interested in the behavior of the typical “slope” of such a function.In some situations, e.g. when taking into account uncertainties, it makes sense to consider functions whose values do not consists of single points but of certain small sets. For such functions, if their values are compact, we want to consider methods to find fixed points in an iterative fashion. Also in this setting, it turns out that these methods do not work for all nonexpansive mapping and we consider the question of whether they work for typical such functions.

Project Members

Christian Bargetz

Emir Medjic

Katriin Pirk

Davide Ravasini

Daylen Thimm

Publications

C. Bargetz, M. Dymond, E. Medjic and S. Reich, On the existence of fixed points for typical nonexpansive mappings on spaces with positive curvature. Topol. Methods Nonlinear Anal. 57(2): 621–634, 2021, DOI: 10.12775/TMNA.2020.040, arXiv:2004.02567
D. Ravasini, Compactivorous Sets in Separable Banach Spaces. Proc. Amer. Math. Soc. 150(5): 2121–2129. 2022, DOI: 10.1090/proc/15851, arXiv:2104.02695
D. Ravasini, Haar null closed and convex sets in separable Banach spaces. Bull. London Math. Soc., 55: 137-148. DOI: 10.1112/blms.12716, arXiv:2110.05250
E. Medjic: On Successive Approximations for Compact-Valued Nonexpansive Mappings. Set-Valued and Variational Analysis 31, article number 24, 2023 DOI: 10.1007/s11228-023-00684-1, arXiv:2203.03470
C. Bargetz, S. Reich, D. Thimm: Generic properties of nonexpansive mappings on unbounded domains. J. Math. Anal. Appl. 526(1): Article 127179, 2023. DOI: 10.1016/j.jmaa.2023.127179, arXiv:2204.10279
D. Ravasini, A Topological Characterisation of Haar Null Convex Sets. Proc. Amer. Math. Soc. 151(12): 5325–5333. 2023, DOI: 10.1090/proc/16535, arXiv:2210.15545
C. Bargetz, E. Medjic, and K. Pirk, On generic convergence of successive approximations of mappings with convex and compact point images. Monatsh. Math. 202: 659–683, 2023. DOI: 10.1007/s00605-022-01813-y, arXiv:2211.02298
D. Ravasini, Generic uniformly continuous mappings on unbounded hyperbolic spaces, Preprint, 2023, arXiv:2308.15277

Presentations

D. Ravasini, Compactivorous Sets in Separable Banach Spaces. Banach Afternoon. 16/04/2021. Milano, Italy (Online)
E. Medjic, On the existence of fixed points for typical nonexpansive mappings on spaces with positive curvature. DMV-ÖMG Jahrestagung 2021, Passau (online), 30/09/2021
E. Medjic, On successive approximations of compact-valued mappings. Invited Talk, Universität Leipzig, 05/05/2022
D. Ravasini, Haar null sets in Banach spaces: some recent developments. Functional Analysis in Lille. June 27th - July 1st 2022, Lille, France.
D. Ravasini, Compactivorous sets and Haar negligibility. Progress in Functional Analysis: Methods and Application. September 19th - 21st 2022, Lecce, Italy.
K. Pirk, About fixed points and tricks how to prove generic properties. Seminar of Functional Analysis in the University of Tartu (Online). 16/12/2022
C. Bargetz, Generic properties of nonexpansive mappings on unbounded domains. 50th Winter School in Abstract Analysis. January 7th - 14th 2023, Sněžné, Czech Republic.
K. Pirk, On generic convergence of successive approximations of mappings with convex and compact point imagess. 50th Winter School in Abstract Analysis. January 7th - 14th 2023, Sněžné, Czech Republic.
D. Ravasini, A topological characterisation of Haar null convex sets. 50th Winter School in Abstract Analysis. January 7th - 14th 2023, Sněžné, Czech Republic.
K. Pirk, Extremal mappings among nonexpansive mappings, 45th Summer Symposium in Real Analysis, Caserta, June 19th-23rd, 2023
C. Bargetz, Typical properties of nonexpansive mappings, Bremen-Hamburg²-Kiel Seminar, July 14th 2023, Hamburg
D. Ravasini, Generic properties of uniformly continuous and nonexpansive mappings. Infinite dimensional convexity and geometry of Banach spaces. Università Cattolica del Sacro Cuore, MIlano, Italy. November 9–10, 2023.

Posters