Generic Properties of Nonexpansive Mappings
Generic Properties of Nonexpansive Mappings
Disciplines
Mathematics (100%)
Keywords
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Lipschitz constant,
Set-Valued Mappings,
Nonexpansive Mappings,
Successive Approximations,
Baire category,
Fixed Points
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. A focus of this project is to study the connection between the typical behavior of nonexpansive mappings and the geometry of the underlying space. For the problems described above this is a completely new approach.
- Universität Innsbruck - 100%
Research Output
- 10 Citations
- 14 Publications
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2022
Title Haar null closed and convex sets in separable Banach spaces DOI 10.1112/blms.12716 Type Journal Article Author Ravasini D Journal Bulletin of the London Mathematical Society Pages 137-148 Link Publication -
2023
Title Generic properties of nonexpansive mappings on unbounded domains DOI 10.1016/j.jmaa.2023.127179 Type Journal Article Author Bargetz C Journal Journal of Mathematical Analysis and Applications Pages 127179 Link Publication -
2022
Title On generic convergence of successive approximations of mappings with convex and compact point images DOI 10.48550/arxiv.2211.02298 Type Preprint Author Bargetz C -
2022
Title A Topological Characterisation of Haar Null Convex Sets DOI 10.48550/arxiv.2210.15545 Type Preprint Author Ravasini D -
2022
Title On generic convergence of successive approximations of mappings with convex and compact point images DOI 10.1007/s00605-022-01813-y Type Journal Article Author Bargetz C Journal Monatshefte für Mathematik Pages 659-683 Link Publication -
2022
Title On Successive Approximations for Compact-Valued Nonexpansive Mappings DOI 10.48550/arxiv.2203.03470 Type Preprint Author Medjic E -
2022
Title Generic properties of nonexpansive mappings on unbounded domains DOI 10.48550/arxiv.2204.10279 Type Preprint Author Bargetz C -
2022
Title Compactivorous sets in Banach spaces DOI 10.1090/proc/15851 Type Journal Article Author Ravasini D Journal Proceedings of the American Mathematical Society Pages 2121-2129 Link Publication -
2021
Title Compactivorous Sets in Banach Spaces DOI 10.48550/arxiv.2104.02695 Type Preprint Author Ravasini D -
2023
Title On Successive Approximations for Compact-Valued Nonexpansive Mappings DOI 10.1007/s11228-023-00684-1 Type Journal Article Author Medjic E Journal Set-Valued and Variational Analysis Pages 24 Link Publication -
2023
Title Generic uniformly continuous mappings on unbounded hyperbolic spaces DOI 10.48550/arxiv.2308.15277 Type Preprint Author Ravasini D -
2023
Title A topological characterisation of Haar null convex sets DOI 10.1090/proc/16535 Type Journal Article Author Ravasini D Journal Proceedings of the American Mathematical Society Pages 5325-5333 Link Publication -
2020
Title On the existence of fixed points for typical nonexpansive mappings on spaces with positive curvature DOI 10.12775/tmna.2020.040 Type Journal Article Author Bargetz C Journal Topological Methods in Nonlinear Analysis Pages 1 Link Publication -
2021
Title Haar Null Closed and Convex Sets in Separable Banach Spaces DOI 10.48550/arxiv.2110.05250 Type Preprint Author Ravasini D