Global solution branches of two point boundary value problems
type: Item
instance of: scholarly article
title: Global solution branches of two point boundary value problems
zbMATH Open document ID: 0780.34010
author: Renate Schaaf
published in: Lecture Notes in Mathematics
publication date: 1992-09-17T00:00:00Z
review text: One of the basic problems in this monograph is the study (in Chapter 1) of the global solution branches of two point BVP of the form: (1) \(u''(t)+f(u)=0\), \(u(0)=u(\lambda)=0\), where the nonlinearity \(f\) is such that \(f(0)=0\) and \(f'(0)>0\) up to its first zeros \(>0\) or \(<0\). This is done by using an integral formula for the so called time map \(T\) of \(f\). Conditions on \(f\) are given which result in bifurcating branches that have at most one turning point at the bifurcation point itself (Chafee- Infante, Opial), or at a point away from it. In the latter case, one allows a change of stability along branches, which can happen at most twice. A wide class of the so called \(A-B\)-functions \(f\) has been introduced for which these conditions can be easily verified. Chapter 2 is devoted to the study of the Neumann problem: (2) \(u''(t)+f(u)=0\), \(u'(0)=u'(\lambda)=0\) and of the period map of the Hamiltonian system: (3) \(u'=f_ 2(v)\), \(v'=-f_ 1(u)\). By using the corresponding time map \(T\), it has been shown that some bifurcating branches of (2) have a single turn at the bifurcation point when \(f\) is an \(A-B\)-function. By a similar method, monotonicity results for the period map of (3) have been obtained. After looking at some examples, the author has indicated how these results can be used to prove bifurcation of subharmonic solutions if (3) is perturbed by a small nonautonomous \(t\)-periodic term and the existence and uniqueness of periodic-4-solutions to the time delay equations \(u'(t)+f(u(t-1))=0\) with \(f\) antisymmetric. Positive solution branches of (1) with \(f\geq0\) have been studied in Chapter 3 using again a similar time map formula. Multiple zeros of \(f\) at 0 are allowed and it is indicated that problems with singularities at \(u=0\) could be treated too. For \(A-B\)-functions \(f\), as well as for more general ones, it has been shown that the branch curve is also \(U\)-shaped. Nonbifurcating branches and the case \(f(0)<0\) ha…
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 34-02
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 48412
zbMATH Keywords: global solution branches
zbMATH Keywords: turning point
zbMATH Keywords: bifurcation
zbMATH Keywords: Neumann problem
zbMATH Keywords: period map
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: subharmonic solutions
zbMATH Keywords: time delay equations
reviewed by: P. S. Milojević
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 48412