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)
Global bifurcation and multiple results for Sturm-Liouville problems. (English) Zbl 1223.34030

The following nonlinear Sturm-Liouville boundary value problem

-(p(t)u ' (t)) ' +q(t)u(t)=λa(t)f(u(t)),0<t<1,α 1 u(0)+β 1 u ' (0)=0,α 2 u(1)+β 2 u ' (1)=0

is considered, where f: is a continuous function and there exists f 0 , f (0,) such that

f 0 =lim |x|0 f(x) x,f =lim |x| f(x) x·

A global bifurcation result is obtained, and then the existence of solutions having exactly k-1 zeros in (0,1) is derived, where k.

MSC:
34B24Sturm-Liouville theory
34C23Bifurcation (ODE)
References:
[1]Rabinowitz, P. H.: Some global results for nonlinear eigenvalue problems, J. funct. Anal 7, 487-513 (1971) · Zbl 0212.16504 · doi:10.1016/0022-1236(71)90030-9
[2]Rabinowitz, P. H.: On bifurcation from infinity, J. differential equations 14, No. 3, 462-475 (1973) · Zbl 0272.35017 · doi:10.1016/0022-0396(73)90061-2
[3]Ma, Ruyun; Thompson, Bevan: Nodal solutions for nonlinear eigenvalue problems, Nonlinear anal. 59, 707-718 (2004) · Zbl 1059.34013 · doi:10.1016/j.na.2004.07.030
[4]Ma, Ruyun; Thompson, Bevan: Multiplicity results for second-order two-point boundary value problems with nonlinearities across several eigenvalues, Appl. math. Lett. 18, 587-595 (2005) · Zbl 1074.34016 · doi:10.1016/j.aml.2004.09.011
[5]Gulgowski, Jacek: Global bifurcation and multiplicity results for Sturm–Liouville problems, Nonlinear differential equations appl. 14, 559-568 (2007) · Zbl 1137.34324 · doi:10.1007/s00030-007-5006-3
[6]Gulgowski, Jacek: Bifurcation of solutions of nonlinear Sturm–Liouville problems, J. inequal. Appl. 6, 483-506 (2001) · Zbl 1088.34518 · doi:10.1155/S1025583401000303
[7]Naito, Y.; Tanaka, S.: On the existence of multiple solutions of the boundary value problem for nonlinear second-order differential equations, Nonlinear anal. 56, 919-935 (2004) · Zbl 1046.34038 · doi:10.1016/j.na.2003.10.020
[8]Rynne, B. P.: Global bifurcation for 2mth-order boundary value problems and infinitely many solutions of superlinear problems, J. differential equations 188, 461-472 (2003) · Zbl 1029.34015 · doi:10.1016/S0022-0396(02)00146-8
[9]Rynne, B. P.: Global bifurcation in generic systems of nonlinear Sturm–Liouville problems, Proc. amer. Math. soc. 127, No. 1, 155-165 (1999) · Zbl 0905.34021 · doi:10.1090/S0002-9939-99-04763-2
[10]Sun, Jingxian: Existence of positive solutions for nonlinear Hammerstein integral equations and applications, Chinese ann. Math. ser. A 9, No. 1, 90-96 (1988) · Zbl 0699.45005
[11]Walter, W.: Ordinary differential equations, (1998)
[12]Guo, Dajun; Sun, Jingxian: Nonlinear integral equations, (1987)
[13]Liu, Z.; Li, F.: Multiple positive solutions of nonlinear two-point boundary value problems, J. math. Anal. appl. 203, 610-625 (1996) · Zbl 0878.34016 · doi:10.1006/jmaa.1996.0400
[14]Amann, H.: Fixed point equation and nonlinear eigenvalue problems in ordered Banach spaces, SIAM rev. 18, 620-709 (1976) · Zbl 0345.47044 · doi:10.1137/1018114
[15]Erbe, L. H.: Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems, Math. comput. Modelling 32, 529-539 (2000) · Zbl 0970.34019 · doi:10.1016/S0895-7177(00)00150-3
[16]Erbe, L. H.; Mathsen, R. M.: Positive solutions for singular nonlinear boundary value problems, Nonlinear anal. 46, 979-986 (2001) · Zbl 1007.34020 · doi:10.1016/S0362-546X(00)00147-4
[17]Zhao, Z.: Positive solutions of boundary value problems for nonlinear singular differential equations, Acta math. Sinica 43, 179-188 (2000) · Zbl 1018.34018