Factor analysis of nonlinear mappings: p-regularity theory

Jerrold E. Marsden; Alexey Tret'yakov

doi:10.3934/cpaa.2003.2.425

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2003, Volume 2, Issue 4: 425-445. Doi: 10.3934/cpaa.2003.2.425

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Factor analysis of nonlinear mappings: p-regularity theory

Jerrold E. Marsden^1, and
Alexey Tret'yakov^2,

1.
Control and Dynamical Systems 107-81, California Institute of Technology, Pasadena, CA 91125
2.
Computing Center of the Russian Academy of Sciences, Vavilova 40, Moscow, GSP-1

Received: November 30, 2002

Revised: May 31, 2003

Published: November 2003

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Abstract

The paper presents recent advances in $p$-regularity theory, which has been developing successfully for the last twenty years. The main result of this theory gives a detailed description of the structure of the zero set of an irregular nonlinear mapping. We illustrate the theory with an application to degenerate problems in different fields of mathematics, which substantiates the general applicability of the class of $p$-regular problems. Moreover, the connection between singular problems and nonlinear mappings is shown. Amongst the applications, the structure of $p$-factor-operators is used to construct numerical methods for solving degenerate nonlinear equations and optimization problems.

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Mathematics Subject Classification: 58C15, 58K05, 35B32, 37G10, 49K27,46N10. Secondary: 49N60, 90C30, 65J15.

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