A Degree Theory for Multivalued Perturbations of Densely Defined (S+)-Operators and Application to a Feedback Control Problem
- 1Department of Applied Mathematics, Informatics and Mechanics, Voronezh State University, 394018 Voronezh, Russia
Abstract
We study the topological degree for densely defined operators of class (S+) with compact multivalued perturbations. It is assumed that a perturbation can be represented as a sum of two terms: a multivalued mapping which is itself a composition of a continuous single-valued mapping and an upper semicontinuous multivalued mapping with aspherical images, and a single-valued compact operator. This makes it possible to extend the standard degree theory for convex-valued operators to multivalued mappings whose values may have a complicated geometry, as well as to operators defined not on the whole space but only on a dense subset. Theoretical aspects concerning the definition and main properties of the topological degree for such multivalued mappings are discussed. In particular, it is shown that the introduced degree is homotopy invariant and can serve as a convenient tool for establishing the solvability of the corresponding operator inclusions. As an application, a feedback control problem for a class of nonlinear elliptic equations is considered. Using the proposed topological degree, the solvability of this problem in the weak setting is proved.
Related
- On densely defined linear continuous operators between function spaces
- On wave operators for Hardy operators with and without perturbations
- Smooth spectral statistics of random perturbations of Schrödinger operators near Anosov energy levels
- A long time Gutzwiller trace formula for perturbations of Schrödinger operators near Anosov energy levels
- Perturbations of Truncated Toeplitz operators
BibTeX
@article{baranovskii2026degree,
title = {{A Degree Theory for Multivalued Perturbations of Densely Defined (S+)-Operators and Application to a Feedback Control Problem}},
author = {Evgenii S. Baranovskii and Mikhail A. Artemov},
journal = {Mathematics},
year = {2026},
month = nov,
doi = {10.3390/math14214002},
url = {https://www.mdpi.com/2227-7390/14/21/4002},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - A Degree Theory for Multivalued Perturbations of Densely Defined (S+)-Operators and Application to a Feedback Control Problem
AU - Evgenii S. Baranovskii
AU - Mikhail A. Artemov
PY - 2026
DA - 2026/11/02
JO - Mathematics
DO - 10.3390/math14214002
UR - https://www.mdpi.com/2227-7390/14/21/4002
AB - We study the topological degree for densely defined operators of class (S+) with compact multivalued perturbations. It is assumed that a perturbation can be represented as a sum of two terms: a multivalued mapping which is itself a composition of a continuous single-valued mapping and an upper semicontinuous multivalued mapping with aspherical images, and a single-valued compact operator. This makes it possible to extend the standard degree theory for convex-valued operators to multivalued mappings whose values may have a complicated geometry, as well as to operators defined not on the whole space but only on a dense subset. Theoretical aspects concerning the definition and main properties of the topological degree for such multivalued mappings are discussed. In particular, it is shown that the introduced degree is homotopy invariant and can serve as a convenient tool for establishing the solvability of the corresponding operator inclusions. As an application, a feedback control problem for a class of nonlinear elliptic equations is considered. Using the proposed topological degree, the solvability of this problem in the weak setting is proved.
SN - 2227-7390
ER -