Global solution curves for boundary value problems, with linear part at resonance
type: Item
instance of: scholarly article
title: Global solution curves for boundary value problems, with linear part at resonance
zbMATH Open document ID: 1185.34021
author: Philip Korman
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-07-24T00:00:00Z
review text: The exact number of solutions is obtained for both the Dirichlet problem \[ \begin{cases} u''(x)+u(x)+kg(u(x))=\mu \sin x +e(x) \;\;\text{ for } x\in (0, \pi),\\ u(0)=u(\pi)=0\end{cases} \tag{1} \] and the Neumann problem \[ \begin{cases} u''(x)+kg(u(x))=\mu +e(x) \;\;\text{ for } x\in (0, \pi),\\ u'(0)=u'(\pi)=0\end{cases}\tag{2} \] by a dynamical version of the classical Liapunov-Schmidt procedure. Moreover, the paper performs continuation in the subspace orthogonal to the kernel after decomposing the space. The main results include uniqueness and infinite multiplicity. Theorem 4.1, one of the main results is as follows: Assume that \(g \in C^{1}(\mathbb R)\) in (1) satisfies \(|g(u)|\leq M_{0}\) for all \(u \in\mathbb R\) , \(|g'(u)|\leq M \) for all \(u \in\mathbb R\) \ with some positive constants \(M_{0}\)\ and \(M\). Assume that \(k\leq \frac{3}{M},\;e\in L^{2}_{1} \) is a given function with \( \int^{\pi}_{0}e(x)\sin xdx=0\). In addition, assume that \(\mu\) satisfies the necessary condition \(\frac{4}{\pi}kg(-\infty)<\mu<\frac{4}{\pi}kg(\infty)\), \ where \(g(\infty)=\lim_{u\to \infty}g(u)\)\ and \(g(-\infty)=\lim_{u\to -\infty}g(u)\). Then problem (1) has a solution. Moreover, all solutions of (1) lie on a unique solution curve, which is given by an equation \(\mu = \phi (\xi)\), where \(\phi(\xi)\) is a bounded continuous function, having limits at infinity: \(\phi (-\infty)=\frac{4}{\pi}kg(-\infty)\) and \(\phi (\infty)=\frac{4}{\pi}kg(\infty)\).
reviewed by: Chun-Lei Tang
Mathematics Subject Classification ID: 34B08
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 5586782
zbMATH Keywords: boundary value problem
zbMATH Keywords: global solution curves
zbMATH Keywords: resonance
zbMATH Keywords: Lyapunov-Schmidt procedure
zbMATH Keywords: continuation
MaRDI profile type: Publication
full work available at URL: j.na.2009.01.128
OpenAlex ID: W2063785437
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DOI: 10.1016/J.NA.2009.01.128
description: scientific article; zbMATH DE number 5586782