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)
A note on multi-point boundary value problems. (English) Zbl 1127.34006
This work is concerned with solvability of the multipoint boundary value problem $$x''(t) = f(t,x(t),x'(t)), \quad t \in (0,1),$$ $$x(0) = \sum_{j=1}^n \alpha_j x(\xi_j), \quad x(1) = \sum_{j=1}^n \beta_j x(\eta_j),$$ where $0 < \eta_j, \xi_j < 1$, $\eta_j, \xi_j \in \Bbb{R}$, $j = 1,2,\dots,n$, $n \geq 2$, $e \in L^1[0,1]$, and $f:[0,1] \times \Bbb{R}^2 \to \Bbb{R}$ is a Carathéodory function. The additional assumptions $$\sum_{j=1}^n \alpha_j = 1 = \sum_{j=1}^n \beta_j, \quad \sum_{j=1}^n \alpha_j \xi_j = 0 = \sum_{j=1}^n \beta_j \eta_j$$ are “critical”, that is, responsible for resonance. This note complements the result by {\it N. Kosmatov} [Nonlinear Anal., Theory Methods Appl. 65, 622--633 (2006; Zbl 1121.34023)] and extends the result by the first author [Appl. Math. Comput. 143, 275--299 (2003; Zbl 1071.34014)]. The existence result follows from the celebrated coincidence degree theorem due to {\it J. Mawhin} [Topological degree methods in nonlinear boundary value problems (Regional Conference Series in Mathematics, No. 40, Providence, R.I., The American Mathematical Society) (1979; Zbl 0414.34025)].

34B10Nonlocal and multipoint boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
47H11Degree theory (nonlinear operators)
Full Text: DOI
[1] Liu, B.: Solvability of multi-point boundary value problem at resonance (IV). Appl. math. Comput. 143, 275-299 (2003) · Zbl 1071.34014
[2] Mawhin, J.: Topological degree and boundary value problems for nonlinear differential equations. Lecture notes in math. 1537, 74-142 (1993) · Zbl 0798.34025
[3] Mawhin, J.: Topological degree methods in nonlinear boundary value problems. NSFCBMS regional conference series in mathematics (1979)
[4] Kosmatov, N.: A multi-point boundary value problem with two critical conditions. Nonlinear anal. 65, 622-633 (2006) · Zbl 1121.34023
[5] Bitsadze, A. V.: On the theory of nonlocal boundary value problems. Soviet. math. Dock. 30, 8-10 (1964) · Zbl 0586.30036
[6] Bitsadze, A. V.; Samarskii, A. A.: Some elementary generalizations of linear elliptic boundary value problems. Dokil. akad. Nauk SSSR 185, 739-740 (1969) · Zbl 0187.35501
[7] Il’in, V. A.; Moiseev, E. I.: Nonlocal boundary value problems of the second kind for a Sturm-Liouville operator. Differ. equ. 23, 979-987 (1987) · Zbl 0668.34024
[8] Feng, W.; Webb, J. R. L.: Solvability of m-point boundary value problems with nonlinear growth. J. math. Anal. appl. 212, 467-480 (1997) · Zbl 0883.34020
[9] Feng, W.; Webb, J. R. L.: Solvability of three point boundary value problems at resonance. Nonlinear anal. 30, 3227-3238 (1997) · Zbl 0891.34019
[10] Liu, B.: Solvability of multi-point boundary value problem at resonance (III). Appl. math. Comput. 129, 119-143 (2002) · Zbl 1054.34033