Communications on Pure and Applied Analysis (CPAA)

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

Pages: 293 - 315, Volume 7, Issue 2, March 2008      doi:10.3934/cpaa.2008.7.293

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Olga A. Brezhneva - Department of Mathematics and Statistics, 123 Bachelor Hall, Miami University, Oxford, OH 45056, United States (email)
Alexey A. Tret’yakov - 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 (email)
Jerrold E. Marsden - Control and Dynamical Systems 107-81, California Institute of Technology, Pasadena, CA 91125, United States (email)

Abstract: 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, perturbation method, $p$-regularity, degeneracy.
Mathematics Subject Classification:  Primary: 34B15, 47J07, 58C15.

Received: December 2006;      Revised: June 2007;      Available Online: December 2007.