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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Bibliographic Details
Published in:Entropy (Basel, Switzerland) Vol. 27; no. 5; p. 518
Main Authors: Bednarczuk, Ewa, Brezhneva, Olga, Leśniewski, Krzysztof, Prusińska, Agnieszka, Tret’yakov, Alexey A.
Format: Journal Article
Language:English
Published: Switzerland MDPI AG 12.05.2025
MDPI
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:1099-4300
1099-4300
DOI:10.3390/e27050518