Existence and uniqueness results for the bending of an elastic beam equation at resonance. (English) Zbl 0655.73001

The purpose of this paper is to study the following nonlinear analogue of the boundary-value problem for bending of an elastic beam which is simply supported at both ends and is at resonance: \[ (1)\quad d^ 4u/dx^ 4- \pi^ 4u+g(x,u)=e(x),\quad 0<x<1,\quad u(0)=u(1)=u''(0)=u''(1)=0, \] where g: [0,1]\(\times R\to R\) satisfies Caratheodory’s conditions and e(x)\(\in L^ 1[0,1]\) with \(\int^{1}_{0}e(x)\) sin \(\pi\) x dx\(=0\). It is shown that (1) has at least one solution if g(x,u)u\(\geq 0\) for all x in [0,1] and u in R. It is also proven that (1) has a unique solution if g(x,u) is strictly increasing in u for every x in [0,1] and \(\int^{1}_{0}g(x,0)\sin \pi x dx=0.\)
Another boundary-value problem studied in the same paper is defined by \[ (2)\quad -d^ 4u/dx^ 4+\pi^ 4u+g(x,u)=e(x),\quad 0<x<1,\quad u(0)=u(1)=u''(0)=u''(1)=0. \] It is proven that a solution of (2) exists when \(e(x)\in L^ 1[0,1]\) with \(\int^{1}_{0}e(x)\quad \sin \pi x dx=0\) and g(x,u) satisfies two supplementary conditions. A proof is also given for the uniqueness of the solutions of (2) when \(e(x)\in L^ 1(0,1)\) with \(\int^{1}_{0}e(x)\sin \pi x dx=0\) and g satisfies two specified conditions.
Reviewer: W.A.Bassali


74G30 Uniqueness of solutions of equilibrium problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
34B15 Nonlinear boundary value problems for ordinary differential equations
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