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

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.

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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).

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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.
Funding AgencyGrant Number
Leading Scientific SchoolsNSh-2240.2006.1
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
Record Number:CaltechAUTHORS:20100715-104648317
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19073
Deposited By: Ruth Sustaita
Deposited On:15 Jul 2010 18:43
Last Modified:08 Nov 2021 23:49

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