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)
Positive solutions of semilinear differential equations with singularities. (English) Zbl 0909.34013

The authors deal with the existence of positive solutions to a second-order differential equation of the form (1): z '' +g(t)f(z)=0 with suitable boundary conditions and gL 1 (0,1). The function f is supposed to satisfy either 0limsup x0 f(x)/x<a and b<liminf x f(x)/x or 0limsup x f(x)/x<a and b<liminf x0 f(x)/x for suitable a and b.

The main idea is to change (1) into a Hammerstein integral equation of the form z(t)= 0 1 k(t,s)g(s)f(z(s))ds. The compactness of A is proved. This allows one to apply fixed point index theory for compact maps.

Two applications are given: the existence of eigenvalues to the equation z '' +λg(t)f(z)=0 and the existence of positive radial solutions to the equation Δu+h(|x|)f(u)=0.


MSC:
34B15Nonlinear boundary value problems for ODE
45G10Nonsingular nonlinear integral equations