On Ranges of Non-linear Operators

We derive conditions ensuring that the range of a given continuous mapping with a compact convex domain covers a prescribed set. In Fréchet spaces, we consider approximations by one single-valued mapping such that the inverse of it has either convex fibers or admits a continuous single-valued select...

Full description

Saved in:
Bibliographic Details
Published in:Set-valued and variational analysis Vol. 30; no. 2; pp. 789 - 810
Main Authors: Cibulka, Radek, Roubal, Tomáš
Format: Journal Article
Language:English
Published: Dordrecht Springer Netherlands 01.06.2022
Springer Nature B.V
Subjects:
Analysis
Approximation
Blackout
Continuous fibers
Jacobians
Linear operators
Mapping
Mathematics
Mathematics and Statistics
Operators (mathematics)
Optimization
Original Research
Regularity
Security management
Theorems
Pseudo-Jacobian
58C15
Linear openness
49J53
Non-linear image
49J52
Directional regularity
Open mapping theorem
47J07
47J05
Clarke generalized Jacobian
ISSN:1877-0533, 1877-0541
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We derive conditions ensuring that the range of a given continuous mapping with a compact convex domain covers a prescribed set. In Fréchet spaces, we consider approximations by one single-valued mapping such that the inverse of it has either convex fibers or admits a continuous single-valued selection. Subsequently, in Banach and finite-dimensional spaces, we focus on approximations determined by a convex set of bounded linear mappings. We demonstrate that our approach is highly flexible and provides the unified treatment of various, in general non-local, covering properties of possibly non-smooth mappings. In finite-dimensional spaces, we present sufficient conditions for constrained directional semiregularity, metric regularity and strong metric regularity. These conditions cover, unify, and extend such well-known results as Pourciau’s open mapping theorem and Clarke’s inverse and implicit function theorems as well as their generalizations established by V. Jeyakumar and D.T. Luc by using upper semi-continuous unbounded pseudo-Jacobians or by A. Neumaier by considering various interval extensions of the derivative of a smooth mapping. Finally, we provide conditions guaranteeing that the non-linear image of a compact convex set contains a prescribed ordered interval which has direct applications in power network security management such as preventing the electricity blackout.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:1877-0533
1877-0541
DOI:10.1007/s11228-021-00619-8