Brezhneva, Olga A. and Tret'yakov, Alexey A. and Marsden, Jerrold E.
(2008)
*Higher order implicit function theorems and degenerate nonlinear boundary-value problems.*
Communications Pure and Applied Analysis, 7
(2).
pp. 293-315.
ISSN 1534-0392.
http://resolver.caltech.edu/CaltechAUTHORS:20100715-104648317

PDF
- Published Version
Restricted to Repository administrators only See Usage Policy. 354Kb |

Use this Persistent URL to link to this item: http://resolver.caltech.edu/CaltechAUTHORS:20100715-104648317

## Abstract

The first part of this paper considers the problem of solving an equation of the form F(x,y) = 0, for y = φ(x) as a function of x, where F : X x Y → 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 pth-order generalizations of the Implicit Function Theorem for this case. The second part of the paper uses these pth-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).

Item Type: | Article | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Related URLs: |
| |||||||||

Additional Information: | © 2008 AIMS. Received: December 2006. Revised: June 2007. Published: December 2007. The second author is partially supported by the program "Leading Scientic Schools," project no. NSh-2240.2006.1. The third author is partially supported by NSF-ITR Grant ACI-0204932. | |||||||||

Funders: |
| |||||||||

Subject Keywords: | Implicit function theorem, nonlinear boundary-value problem, perturbation method, p-regularity, degeneracy | |||||||||

Classification Code: | MSC 2000:Primary: 34B15, 47J07, 58C15 | |||||||||

Record Number: | CaltechAUTHORS:20100715-104648317 | |||||||||

Persistent URL: | http://resolver.caltech.edu/CaltechAUTHORS:20100715-104648317 | |||||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||

ID Code: | 19073 | |||||||||

Collection: | CaltechAUTHORS | |||||||||

Deposited By: | Ruth Sustaita | |||||||||

Deposited On: | 15 Jul 2010 18:43 | |||||||||

Last Modified: | 26 Dec 2012 12:14 |

Repository Staff Only: item control page