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)

  • Volume: 66, Issue: 2, page 63-79
  • ISSN: 2083-7402

Abstract

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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 more detailed consideration based on different p-factor operators constructions.

How to cite

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Wiesław Grzegorczyk, Beata Medak, and Alexey A. Tret’yakov. "Generalization of p-regularity notion and tangent cone description in the singular case." Annales UMCS, Mathematica 66.2 (2012): 63-79. <http://eudml.org/doc/267945>.

@article{WiesławGrzegorczyk2012,
abstract = {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 more detailed consideration based on different p-factor operators constructions.},
author = {Wiesław Grzegorczyk, Beata Medak, Alexey A. Tret’yakov},
journal = {Annales UMCS, Mathematica},
keywords = {p-regularity; singularity; nonlinear mapping; p-factor operator; curves; surfaces.; -regularity; -factor operator; surfaces},
language = {eng},
number = {2},
pages = {63-79},
title = {Generalization of p-regularity notion and tangent cone description in the singular case},
url = {http://eudml.org/doc/267945},
volume = {66},
year = {2012},
}

TY - JOUR
AU - Wiesław Grzegorczyk
AU - Beata Medak
AU - Alexey A. Tret’yakov
TI - Generalization of p-regularity notion and tangent cone description in the singular case
JO - Annales UMCS, Mathematica
PY - 2012
VL - 66
IS - 2
SP - 63
EP - 79
AB - 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 more detailed consideration based on different p-factor operators constructions.
LA - eng
KW - p-regularity; singularity; nonlinear mapping; p-factor operator; curves; surfaces.; -regularity; -factor operator; surfaces
UR - http://eudml.org/doc/267945
ER -

References

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  1. [1] Brezhneva, O. A., Tret’yakov, A. A., New Methods for Solving Singular Nonlinear Problems, Moscow, 2000 (Russian). 
  2. [2] Brezhneva, O. A., Tret’yakov, A. A., Optimality conditions for degenerate extremum problems with equality contractions, SIAM J. Control Optim. 42, no. 2 (2003), 725-745. 
  3. [3] Ioffe, A. D., Tihomirov, V. M., Theory of Extremal Problems, North-Holland, Studies in Mathematics and its Applications, Amsterdam, 1979. Zbl0407.90051
  4. [4] Izmailov, A. F., Tret‘yakov, A. A., Factor-analysis of Nonlinear Mappings, Nauka, Moscow, 1994 (Russian). 
  5. [5] Izmailov, A. F., Tret’yakov, A. A., 2-regular solutions of nonlinear problems, Theory and numerical methods, Fizmatlit., Nauka, Moscow, 1999 (Russian). 
  6. [6] Musielak, A., Linear Operators, PWN, Warszawa, 1987 (Polish). 
  7. [7] Niczyporowicz, E., The Flat Curves. Selected Problems in Analytic and Differential Geometry, PWN, Warszawa, 1991 (Polish). 
  8. [8] Prusińska, A., Trety’akov, A., A remark on existence of solutions to nonlinear equations with degenerate mappings, Set-Valued Var. Anal. 16 (2008), 93-104. 
  9. [9] Tret’yakov, A. A., Marsden, J. E., Factor-analysis of nonlinear mappings: p-regularity theory, Commun. Pure Appl. Anal. 2, no. 4 (2003), 425-445. 
  10. [10] Tret’yakov, A. A., Necessary and Sufficient Conditions for Optimality of p-th Order, Control and Optimization, Moscow, MSU, 1983, 28-35 (Russian). 
  11. [11] Tret’yakov, A. A., Necessary and sufficient conditions for optimality of p-th order, Zh. Vychisl. Mat. i Mat. Fiz. 24 (1984), 123-127 (Russian). 

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