zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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 for a singular second-order three-point boundary value problem. (English) Zbl 1138.34015
Summary: This paper is concerned with the existence of positive solutions for the singular three-point boundary value problem $$u''(t)+a(t)u'(t)+b(t)u(t)+h(t)f(t,u)=0,\quad 0<t<1\quad u(0)=0,\quad u(1)=\alpha u(\eta),$$ where $h(t)$ is allowed to be singular at $t=0,1$ and $f$ may be singular at $u = 0$. Existence criteria for positive solutions are established by applying the fixed point index theorem under some weaker conditions concerning the first eigenvalue corresponding to the relevant linear operator.

34B18Positive solutions of nonlinear boundary value problems for ODE
34B10Nonlocal and multipoint boundary value problems for ODE
34B16Singular nonlinear boundary value problems for ODE
47H11Degree theory (nonlinear operators)
Full Text: DOI
[1] Amann, A.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach space. SIAM rev. 18, 620-709 (1976) · Zbl 0345.47044
[2] Anderson, D.: Multiple positive solutions for a three-point boundary value problems. Math. comput. Model. 27, 49-57 (1998) · Zbl 0906.34014
[3] Gupta, C. P.: Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equations. J. math. Anal. appl. 168, 540-551 (1988) · Zbl 0763.34009
[4] Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cone. (1988) · Zbl 0661.47045
[5] Liu, B.: Positive solutions of a nonlinear three-point boundary value problem. Comput. math. Appl. 44, 201-211 (2002) · Zbl 1008.34014
[6] Li, J.; Shen, J.: Multiple positive solutions for a second-order three-point boundary value problem. Appl. math. Comput. 182, 258-268 (2006) · Zbl 1109.34014
[7] Ma, R.; Wang, H.: Positive solutions of nonlinear three-point boundary-value problems. J. math. Anal. appl. 279, 216-227 (2003) · Zbl 1028.34014
[8] Sun, Y.; Liu, L.: Solvability for nonlinear second-order three-point boundary value problem. J. math. Anal. appl. 296, 265-275 (2004) · Zbl 1069.34018
[9] Webb, J. R. L.: Positive solutions of some three-point boundary value problems via fixed point index theory. Nonlinear anal. 47, 4319-4332 (2001) · Zbl 1042.34527
[10] Yao, Q.: Positive solutions of a class of singular sublinear two-point boundary value problem. Acta math. Appl. sinica 24, 522-526 (2001) · Zbl 0998.34017
[11] Zhang, G.; Sun, J.: Positive solutions of m-point boundary value problems. J. math. Anal. appl. 291, 406-418 (2004) · Zbl 1069.34037