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)
Uniqueness and existence results for a third-order nonlinear multi-point boundary value problem. (English) Zbl 1166.34008

This paper studies the existence and uniqueness of solutions for the following nonlinear multi-point third-order boundary value problem (BVP):

x ''' (t)+f(t,x(t),x ' (t),x '' (t))=0,0<t<1,x(0)=0,g(x ' (0),x '' (0),x(ξ 1 ),,x(ξ m-2 ))=A,h(x ' (1),x '' (1),x(η 1 ),,x(η n-2 ))=B,(1·1)

where 0<ξ i ,η j <1,i=1,2,,m-2,j=1,2,,n-2,A,B, and f:[0,1]× 3 ,g: m ,h: n are continuous functions. Assuming some monotonicity conditions on the functions f,g,h, the existence of a solution for the BVP (1.1) is proved by applying the method of lower and upper solutions, and Leray-Schauder degree theory.

In order to establish the uniqueness of the solution for BVP (1.1), the authors make use of the following auxiliary boundary value problem:

x ''' (t)+a(t)x '' (t)+b(t)x ' (t)+c(t)x(t)=0,0<t<1,x(0)=0,p 1 x ' (0)+q 1 x '' (0)+ i=1 m-2 r i x(ξ i )=0,p 2 x ' (1)+q 2 x '' (1)+ j=1 n-2 R j x(η j )=0,

where a(t),b(t),c(t)C[0,1],c(t)0,t[0,1],p 1 ,p 2 ,q 1 ,q 2 ,r i ,R j with q 1 0,q 2 0,r i 0,R j 0· Some illustrative examples are also presented.

MSC:
34B10Nonlocal and multipoint boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations