Towards Nonlinearity: The p-Regularity Theory

We present recent advances in the analysis of nonlinear problems involving singular (degenerate) operators. The results are obtained within the framework of p-regularity theory, which has been successfully developed over the past four decades. We illustrate the theory with applications to degenerate...

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Vydané v:	Entropy (Basel, Switzerland) Ročník 27; číslo 5; s. 518
Hlavní autori:	Bednarczuk, Ewa, Brezhneva, Olga, Leśniewski, Krzysztof, Prusińska, Agnieszka, Tret’yakov, Alexey A.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Switzerland MDPI AG 12.05.2025 MDPI
Predmet:	Analysis Banach spaces Boundary value problems Computational mathematics degenerate problems Differential equations Dynamical systems implicit function theorem Mathematical analysis nonlinear differential equations Nonlinear equations nonlinear optimization Nonlinearity Numerical analysis Numerical methods operator equations Operators (mathematics) Optimization Parameters Partial differential equations Regularity New York United States Germany Russia numerical methods p-regularity singularities degenerate problems operator equations nonlinear differential equations implicit function theorem nonlinear optimization
ISSN:	1099-4300, 1099-4300
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Abstract	We present recent advances in the analysis of nonlinear problems involving singular (degenerate) operators. The results are obtained within the framework of p-regularity theory, which has been successfully developed over the past four decades. We illustrate the theory with applications to degenerate problems in various areas of mathematics, including optimization and differential equations. In particular, we address the problem of describing the tangent cone to the solution set of nonlinear equations in singular cases. The structure of p-factor operators is used to propose optimality conditions and to construct novel numerical methods for solving degenerate nonlinear equations and optimization problems. The numerical methods presented in this paper represent the first approaches targeting solutions to degenerate problems such as the Van der Pol differential equation, boundary-value problems with small parameters, and partial differential equations where Poincaré’s method of small parameters fails. Additionally, these methods may be extended to nonlinear degenerate dynamical systems and other related problems.
AbstractList	We present recent advances in the analysis of nonlinear problems involving singular (degenerate) operators. The results are obtained within the framework of p-regularity theory, which has been successfully developed over the past four decades. We illustrate the theory with applications to degenerate problems in various areas of mathematics, including optimization and differential equations. In particular, we address the problem of describing the tangent cone to the solution set of nonlinear equations in singular cases. The structure of p-factor operators is used to propose optimality conditions and to construct novel numerical methods for solving degenerate nonlinear equations and optimization problems. The numerical methods presented in this paper represent the first approaches targeting solutions to degenerate problems such as the Van der Pol differential equation, boundary-value problems with small parameters, and partial differential equations where Poincaré’s method of small parameters fails. Additionally, these methods may be extended to nonlinear degenerate dynamical systems and other related problems. We present recent advances in the analysis of nonlinear problems involving singular (degenerate) operators. The results are obtained within the framework of -regularity theory, which has been successfully developed over the past four decades. We illustrate the theory with applications to degenerate problems in various areas of mathematics, including optimization and differential equations. In particular, we address the problem of describing the tangent cone to the solution set of nonlinear equations in singular cases. The structure of -factor operators is used to propose optimality conditions and to construct novel numerical methods for solving degenerate nonlinear equations and optimization problems. The numerical methods presented in this paper represent the first approaches targeting solutions to degenerate problems such as the Van der Pol differential equation, boundary-value problems with small parameters, and partial differential equations where Poincaré's method of small parameters fails. Additionally, these methods may be extended to nonlinear degenerate dynamical systems and other related problems. We present recent advances in the analysis of nonlinear problems involving singular (degenerate) operators. The results are obtained within the framework of p-regularity theory, which has been successfully developed over the past four decades. We illustrate the theory with applications to degenerate problems in various areas of mathematics, including optimization and differential equations. In particular, we address the problem of describing the tangent cone to the solution set of nonlinear equations in singular cases. The structure of p-factor operators is used to propose optimality conditions and to construct novel numerical methods for solving degenerate nonlinear equations and optimization problems. The numerical methods presented in this paper represent the first approaches targeting solutions to degenerate problems such as the Van der Pol differential equation, boundary-value problems with small parameters, and partial differential equations where Poincaré's method of small parameters fails. Additionally, these methods may be extended to nonlinear degenerate dynamical systems and other related problems.We present recent advances in the analysis of nonlinear problems involving singular (degenerate) operators. The results are obtained within the framework of p-regularity theory, which has been successfully developed over the past four decades. We illustrate the theory with applications to degenerate problems in various areas of mathematics, including optimization and differential equations. In particular, we address the problem of describing the tangent cone to the solution set of nonlinear equations in singular cases. The structure of p-factor operators is used to propose optimality conditions and to construct novel numerical methods for solving degenerate nonlinear equations and optimization problems. The numerical methods presented in this paper represent the first approaches targeting solutions to degenerate problems such as the Van der Pol differential equation, boundary-value problems with small parameters, and partial differential equations where Poincaré's method of small parameters fails. Additionally, these methods may be extended to nonlinear degenerate dynamical systems and other related problems.
Audience	Academic
Author	Prusińska, Agnieszka Tret’yakov, Alexey A. Bednarczuk, Ewa Brezhneva, Olga Leśniewski, Krzysztof
AuthorAffiliation	1 Department of CAD/CAM Systems Design and Computer-Aided Medicine, Faculty of Mathematics and Information Sciences, Warsaw University of Technology, 00-661 Warszawa, Poland; ewa.bednarczuk@pw.edu.pl 4 Faculty of Science, University of Siedlce, 08-110 Siedlce, Poland; aprus@uws.edu.pl (A.P.); alexey.tretiyakov@uws.edu.pl (A.A.T.) 3 System Research Institute, Polish Academy of Sciences, 02-106 Warsaw, Poland 5 Dorodnicyn Computing Center, Federal Research Center “Computer Science and Control”, Russian Academy of Sciences, Moscow 119333, Russia 2 Department of Mathematics, Miami University, Oxford, OH 45056, USA
AuthorAffiliation_xml	– name: 1 Department of CAD/CAM Systems Design and Computer-Aided Medicine, Faculty of Mathematics and Information Sciences, Warsaw University of Technology, 00-661 Warszawa, Poland; ewa.bednarczuk@pw.edu.pl – name: 4 Faculty of Science, University of Siedlce, 08-110 Siedlce, Poland; aprus@uws.edu.pl (A.P.); alexey.tretiyakov@uws.edu.pl (A.A.T.) – name: 2 Department of Mathematics, Miami University, Oxford, OH 45056, USA – name: 5 Dorodnicyn Computing Center, Federal Research Center “Computer Science and Control”, Russian Academy of Sciences, Moscow 119333, Russia – name: 3 System Research Institute, Polish Academy of Sciences, 02-106 Warsaw, Poland
Author_xml	– sequence: 1 givenname: Ewa surname: Bednarczuk fullname: Bednarczuk, Ewa – sequence: 2 givenname: Olga orcidid: 0000-0001-6247-9780 surname: Brezhneva fullname: Brezhneva, Olga – sequence: 3 givenname: Krzysztof surname: Leśniewski fullname: Leśniewski, Krzysztof – sequence: 4 givenname: Agnieszka orcidid: 0000-0002-6091-6884 surname: Prusińska fullname: Prusińska, Agnieszka – sequence: 5 givenname: Alexey A. orcidid: 0000-0003-3474-8458 surname: Tret’yakov fullname: Tret’yakov, Alexey A.
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Snippet	We present recent advances in the analysis of nonlinear problems involving singular (degenerate) operators. The results are obtained within the framework of...
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SubjectTerms	Analysis Banach spaces Boundary value problems Computational mathematics degenerate problems Differential equations Dynamical systems implicit function theorem Mathematical analysis nonlinear differential equations Nonlinear equations nonlinear optimization Nonlinearity Numerical analysis Numerical methods operator equations Operators (mathematics) Optimization Parameters Partial differential equations Regularity
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Title	Towards Nonlinearity: The p-Regularity Theory
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