On the number of unbounded solution branches in a neighborhood of an asymptotic bifurcation point
type: Item
instance of: scholarly article
title: On the number of unbounded solution branches in a neighborhood of an asymptotic bifurcation point
zbMATH Open document ID: 1128.47055
publication date: 2007-06-08T00:00:00Z
review text: This paper concerns abstract equations of the type \[ x= \lambda Ax+F_0(\lambda)+ F_1(Ax,\lambda), \quad \lambda\in\mathbb R,\;x\in B. \tag{1} \] Here, \(H\) is a Hilbert space with scalar product \(\langle\cdot,\cdot\rangle\), \(B\hookrightarrow H\) is a Banach space, \(A:H\to B\) is linear, completely continuous in \(B\) and selfadjoint in \(H\), and \(H_0:\mathbb R\to H\) and \(F_1:H\times\mathbb R\to\mathbb R\) are continuous. Further, it is supposed that \(F_1\) is asymptotically homogeneous, that \(\lambda_0\in\mathbb R\) is a characteristic value of \(A\) of multiplicity two, that \(F_0(\lambda_0)\perp\ker (I-\lambda_0A)\), and that \(F_1\) is ``coordinated with \(A\)'' (see the paper for this notion). Then each proper solution of a certain scalar equation on \(\{x\in H: x= \lambda_0Ax\), \(\|x\|=1\}\) generates a directed unbounded branch of solutions to (1). Applications to second order ODEs (forced oscillations, two-point boundary value problems) are also given.
reviewed by: Lutz Recke
Mathematics Subject Classification ID: 47J15
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 58E07
zbMATH DE Number: 5163295
zbMATH Keywords: asymptotic bifurcation point
zbMATH Keywords: solution branch
zbMATH Keywords: asymptotically homogeneous operator
zbMATH Keywords: periodic oscillations
zbMATH Keywords: subharmonic
author: Alexander Krasnosel'skij
author: Dmitrii Rachinskii
MaRDI profile type: Publication
full work available at URL: s10688-005-0038-0
OpenAlex ID: W2049021809
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DOI: 10.1007/S10688-005-0038-0
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 5163295