Existence and uniqueness results for the bending of an elastic beam equation at resonance (Q1109597)

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.