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)
A nonlinear boundary value problem associated with the static equilibrium of an elastic beam supported by sliding clamps. (English) Zbl 0685.34016
Summary: The fourth-order boundary value problem $d\sp 4u/dx\sp 4+f(x)u=e(x)$, $0<x<\pi$; $u'(0)=u'(\pi)=u'''(0)=u'''(\pi)=0$; where $f(x)\le 0$ for $0\le x\le \pi$, describe the unstable static equilibrium of an elastic beam which is supported by sliding clamps at both ends. This paper concerns the nonlinear analogue of this boundary value problem, namely, $-(d\sp 4u/dx\sp 4)+g(x,u)=e(x),$ $0<x<\pi$, $u'(0)=u'(\pi)=u'''(0)=u'''(\pi)=0$, where $g(x,u)u\ge 0$ for a.e. x in $[0,\pi]$ and all $u\in {\bbfR}$ with $\vert u\vert$ sufficiently large. Some resonance and nonresonance conditions on the asymptotic behavior of $u\sp{-1}g(x,u)$, for $\vert u\vert$ sufficiently large, are studied for the existence of solutions of this nonlinear boundary value problem.

MSC:
 34B10 Nonlocal and multipoint boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE 74K10 Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics
Full Text: