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Application of p-regularity theory to the Duffing equation

Beata Medak, Alexey A Tret’yakov

Boundary Value Problems > 2017 > 2017 > 1 > 1-9

The paper studies a solution existence problem of the nonlinear Duffing equation of the form F ( x , μ , β ) = x ″ + x + μ x 3 − β sin t = 0 , β > 0 , μ ≠ 0 , $$ F(x,\mu, \beta)=x''+x+\mu x^{3}-\beta\sin t = 0,\quad \beta > 0, \mu\neq0, $$ where F : C 2 [ 0 , 2 π ] × R × R → C [ 0 , 2 π ] $F: \mathcal{C}^{2}[0,2\pi]\times\mathbb{R}\times \mathbb{R}\rightarrow\mathcal{C}[0,2\pi]$...

article

Application of p-regularity theory to nonlinear boundary value problems

Wiesław Grzegorczyk, Beata Medak, Alexey A Tret’yakov

Boundary Value Problems > 2013 > 2013 > 1 > 1-13

The paper studies the question of solution existence to a nonlinear equation in the degenerate case. This question is studied for three particular boundary value problems for ordinary and partial second-order differential equations. The so-called p-regularity theory is applied to these purposes as an effective apparatus to investigate many nonlinear mathematical, physical and numerical problems. All...

article

Generalization of p-regularity notion and tangent cone description in the singular case

Wiesław Grzegorczyk, Beata Medak, Alexey A. Tret’yakov

Annales UMCS, Mathematica > 2012 > 66 > 2 > 63-79

The theory of p-regularity has approximately twenty-five years’ history and many results have been obtained up to now. The main result of this theory is description of tangent cone to zero set in singular case. However there are numerous nonlinear objects for which the p-regularity condition fails, especially for p > 2. In this paper we generalize the p-regularity notion as a starting point for...

article

Generalization of p-regularity notion and tangent cone description in the singular case

Wiesław Grzegorczyk, Beata Medak, Alexey A. Tret’yakov

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica > 2012 > 66 > 2

The theory of p-regularity has approximately twenty-five years’ history and many results have been obtained up to now. The main result of this theory is description of tangent cone to zero set in singular case. However there are numerous nonlinear objects for which the p-regularity condition fails, especially for p > 2. In this paper we generalize the p-regularity notion as a starting point for...

article

The p-Factor Method for Nonlinear Optimization

E.. Szczepanik, A. Prusińska, A. Tret’yakov

Schedae Informaticae > 2012 > Vol. 21 > 141--157

We present the main concept and results of the p-regularity theory (also known as p-factor analysis of nonlinear mappings) applied to nonlinear optimization problems. The approach is based on the construction of p-factor operator. The main result of this theory gives a detailed description of the structure of the zero set of irregular nonlinear mappings. Applications include a new numerical method...

article

High order stability conditions for degenerate optimization problems; elements of p-regularity theory

Ewa Bednarczuk, Agnieszka Prusińska, Alexey A. Tret’yakov

Nonlinear Analysis > 2011 > 74 > 3 > 836-846

This paper is devoted to sensitivity analysis of degenerate equality-constrained optimization problems, when the customary regularity of constraints may be violated. Under the p-regularity assumptions we develop p-order stability theory for this class of problems subject to small perturbations and present the estimations of asymptotic behavior of the solutions to the perturbed problems. We derive...

article

P-order necessary and sufficient conditions for optimality in singular calculus of variations

Agnieszka Prusińska, Alexey Tret'yakov

Discussiones Mathematicae, Differential Inclusions, Control and Optimization > 2010 > 30 > 2 > 269-279

This paper is devoted to singular calculus of variations problems with constraint functional not regular at the solution point in the sense that the first derivative is not surjective. In the first part of the paper we pursue an approach based on the constructions of the p-regularity theory. For p-regular calculus of variations problem we formulate and prove necessary and sufficient conditions for...

article

On existence of solutions to degenerate nonlinear optimization problems

Agnieszka Prusińska, Alexey Tret'yakov

Discussiones Mathematicae, Differential Inclusions, Control and Optimization > 2007 > 27 > 1 > 151-164

We investigate the existence of the solution to the following problemmin φ(x) subject to G(x)=0,where φ: X → ℝ, G: X → Y and X,Y are Banach spaces. The question of existence is considered in a neighborhood of such point x₀ that the Hessian of the Lagrange function is degenerate. There was obtained an approximation for the distance of solution x* to the initial point x₀.

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Search results

Application of p-regularity theory to the Duffing equation

Application of p-regularity theory to nonlinear boundary value problems

Generalization of p-regularity notion and tangent cone description in the singular case

Generalization of p-regularity notion and tangent cone description in the singular case

The p-Factor Method for Nonlinear Optimization

High order stability conditions for degenerate optimization problems; elements of p-regularity theory

P-order necessary and sufficient conditions for optimality in singular calculus of variations

On existence of solutions to degenerate nonlinear optimization problems

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