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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:
34B10Nonlocal and multipoint boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
74K10Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics
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