A cantilever beam problem with small deflections and perturbed boundary data
In this paper, the authors study the cantilever beam problem modeled by a fourth order nonlinear differential equation subject to integral boundary conditions of the form \[ \begin{cases} u^{(4)}(x)=\lambda f\Big(x, u(x), u'(x), u''(x), u'''(x)\Big)=0, \quad 0\leq x\leq 1,\\ u(0)=u'(0)=\int_0^1p(x)u(x)dx,~ u''(1)=u'''(1)=\int_0^1q(x)u''(x)dx, \end{cases} \] where \(f\) is continuous on \([0,1]\times [0,\infty)\times [0,\infty)\times (-\infty,0]\times (-\infty,0]\) and \(p,q\in L^1[0,1].\) An existence theorem is established via Schauder's fixed point theorem, which implies that small deflections continue to exist on the beam whether it is perfectly cantilevered or not.
fully fourth order nonlinear boundary value problem
scientific article; zbMATH DE number 7446746
scientific article