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. doi:10.3934/cpaa.2008.7.293. https://resolver.caltech.edu/CaltechAUTHORS:20100715-104648317
![]() |
PDF
- Published Version
Restricted to Repository administrators only See Usage Policy. 362kB |
Use this Persistent URL to link to this item: https://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 | |||||||||
Issue or Number: | 2 | |||||||||
Classification Code: | MSC 2000:Primary: 34B15, 47J07, 58C15 | |||||||||
DOI: | 10.3934/cpaa.2008.7.293 | |||||||||
Record Number: | CaltechAUTHORS:20100715-104648317 | |||||||||
Persistent URL: | https://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: | 08 Nov 2021 23:49 |
Repository Staff Only: item control page