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March  2008, 7(2): 293-315. doi: 10.3934/cpaa.2008.7.293

Higher--order implicit function theorems and degenerate nonlinear boundary-value problems

Olga A. Brezhneva 1, , Alexey A. Tret’yakov 2, and  Jerrold E. Marsden 3,

1. 

Department of Mathematics and Statistics, 123 Bachelor Hall, Miami University, Oxford, OH 45056, United States
2. 

System Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland, University of Podlasie in Siedlce, 3 Maja 54, 08-110 Siedlce, Poland
3. 

Control and Dynamical Systems 107-81, California Institute of Technology, Pasadena, CA 91125, United States

Received  December 2006 Revised  June 2007 Published  December 2007

The first part of this paper considers the problem of solving an equation of the form $F(x, y)=0$, for $y = \varphi (x)$ as a function of $x$, where $F: X \times Y \rightarrow Z$ is a smooth nonlinear mapping between Banach spaces. The focus is on the case in which the mapping $F$ is degenerate at some point $(x^*, y^*)$ with respect to $y$, i.e., when $F'_y (x^*, y^*)$, the derivative of $F$ with respect to $y$, is not invertible and, hence, the classical Implicit Function Theorem is not applicable. We present $p$th-order generalizations of the Implicit Function Theorem for this case. The second part of the paper uses these $p$th-order implicit function theorems to derive sufficient conditions for the existence of a solution of degenerate nonlinear boundary-value problems for second-order ordinary differential equations in cases close to resonance. The last part of the paper presents a modified perturbation method for solving degenerate second-order boundary value problems with a small parameter.The results of this paper are based on the constructions of $p$-regularity theory, whose basic concepts and main results are given in the paper Factor--analysis of nonlinear mappings: $p$--regularity theory by Tret'yakov and Marsden (Communications on Pure and Applied Analysis, 2 (2003), 425--445).
Keywords: Implicit function theorem, nonlinear boundary-value problem, degeneracy., $p$-regularity, perturbation method.
Mathematics Subject Classification: Primary: 34B15, 47J07, 58C1.
Citation: Olga A. Brezhneva, Alexey A. Tret’yakov, Jerrold E. Marsden. Higher--order implicit function theorems and degenerate nonlinear boundary-value problems. Communications on Pure & Applied Analysis, 2008, 7 (2) : 293-315. doi: 10.3934/cpaa.2008.7.293
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