A topological method for vector-valued and n th-order nonlinear boundary value problems
type: Item
instance of: scholarly article
title: A topological method for vector-valued and n th-order nonlinear boundary value problems
zbMATH Open document ID: 0558.34015
DOI: 10.1016/0022-247X(84)90188-4
published in: Journal of Mathematical Analysis and Applications
publication date: 1984-01-01T00:00:00Z
review text: Topological methods are used to obtain the existence of solution to second order boundary value problems in \(R^ n\) of the form \(x''=f(t,x,x')\), \(g(x(c),x'(c))=0\), \(h(x(c),x'(c),x(d),x'(d))=0\) where f,g,h are continuous. The main techniques used include the Wazewski method, the maximum principle, and the Kneser-Hukuhara continuum theorem. A common ingredient in these approaches is the use of upper and lower solutions to obtain bounds on the solutions. The results generalize the scalar case as well as extend some results of others in \(R^ n\). Analytical conditions are imposed on f which ensure apriori bounds on the solution as well as on its derivative. Geometric conditions are given on the two sets \(\{\) (x,y): \(g(x,y)=0\}\) and \(\{\) (x,y,z,w): \(h(x,y,z,w)=0\}\) which allow one to apply the Wazewski method.
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 3889926
zbMATH Keywords: upper solutions
zbMATH Keywords: Wazewski method
zbMATH Keywords: maximum principle
zbMATH Keywords: Kneser-Hukuhara continuum theorem
zbMATH Keywords: lower solutions
author: S. R. Bernfeld
author: Panos K. Palamides
MaRDI profile type: Publication
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full work available at URL: 0022-247x(84)90188-4
OpenAlex ID: W1980585156
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description: scientific article; zbMATH DE number 3889926