zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Upper and lower solutions in the theory of ODE boundary value problems: Classical and recent results. (English) Zbl 0889.34018
Zanolin, F. (ed.), Non linear analysis and boundary value problems for ordinary differential equations. Wien: Springer. CISM Courses Lect. 371, 1-78 (1996).

This article surveys results on the existence of solutions to boundary value problems for scalar ordinary differential equations of the type

u ¨+f(t,u)=0

by means of the method of upper and lower solutions. The notion of W 2,1 -upper and lower solutions is introduced which allows angles in the graph of these functions. The authors study periodic boundary value problems (u(0)=u(2π),u ˙(0)=u ˙(2π)) and Dirichlet boundary value problems (u(0)=u(2π)=0). The Dirichlet problem is studied also for equations with singularities. Boundary value problems depending on parameters (Ambrosetti-Prodi problem) are included in this survey. Applications to mechanical problems with singular forces and to Landesman-Laser conditions are given. The relation with degree theory and the variational approach is used to derive multiplicity results. Together with ordered upper and lower solutions the authors consider also upper and lower solutions in the reversed order and without ordering. In this connection, monotone methods play a crucial role. Historical and bibliographical notes are given in the last section. The references contain 121 items.


MSC:
34B15Nonlinear boundary value problems for ODE
34-02Research monographs (ordinary differential equations)