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)
Multiplicity of positive solutions for second-order three-point boundary value problems. (English) Zbl 0958.34019
The author studies the second-order three-point boundary value problem $$u''+\lambda h(t)f(u) = 0, \qquad t\in (0, 1), \tag 1$$ $$u(0) = 0, \qquad \lambda u(\eta) = u(1), \tag 2$$ with $\eta : 0 < \eta <1, 0 < \alpha < 1 / \eta .$ The author studies the multiplicity of positive solutions to (1), (2) using the method of upper and lower solutions, the Leray-Schauder degree theory and fixed-point index theorems. Notice, that the main theorem is proved using traditional methods of functional analysis with any monotonicity on $f$.

MSC:
34B18Positive solutions of nonlinear boundary value problems for ODE
34B08Parameter dependent boundary value problems for ODE
34B10Nonlocal and multipoint boundary value problems for ODE
34B27Green functions
WorldCat.org
Full Text: DOI
References:
[1] Il’in, V. A.; Moiseev, E. I.: Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects. Differential equations 23, 803-810 (1987)
[2] Il’in, V. A.; Moiseev, E. I.: Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator. Differential equations 23, No. 8, 979-987 (1987) · Zbl 0668.34024
[3] Gupta, C. P.: Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation. J. math. Anal. appl. 168, 540-551 (1992) · Zbl 0763.34009
[4] Gupta, C. P.: A sharper condition for solvability of a three-point boundary value problem. J. math. Anal. appl. 205, 586-597 (1997) · Zbl 0874.34014
[5] Feng, W.; Webb, J. R. L.: Solvability of a three-point nonlinear boundary value problems at resonance. Nonlinear analysis TMA 30, No. 6, 3227-3238 (1997) · Zbl 0891.34019
[6] Marano, S. A.: A remark on a second order three-point boundary value problem. J. math. Anal. appl. 183, 518-522 (1994) · Zbl 0801.34025
[7] Ma, R.: Existence theorems for a second order three-point boundary value problem. J. math. Anal. appl. 212, 430-442 (1997) · Zbl 0879.34025
[8] Ma, R.: Existence theorems for a second order m-point boundary value problem. J. math. Anal. appl. 211, 545-555 (1997) · Zbl 0884.34024
[9] Ma, R.: Positive solutions of a nonlinear three-point boundary value problem. Electron. J. Diff. eqns. 1999, No. 34, 1-8 (1999) · Zbl 0926.34009
[10] D.R. Dunninger and H. Wang, Multiplicity of positive solutions for a nonlinear differential equation with nonlinear boundary conditions (preprint). · Zbl 0921.34024
[11] Guo, D.; Lakshmikanthan, V.: Nonlinear problems in abstract cone. (1988)