Monotone positive solutions for a fourth order equation with nonlinear boundary conditions
Consider the boundary value problems \[ u^{(4)}(t)=f(t,u,u'),\;0<t<1,\tag{1} \] \[ u(0)=u'(0)=0,\;u'''(1)=g(u(1)),\tag{2} \] \[ u''(1)=0, \tag{3} \] \[ u'(1)=0\tag{4} \] describing bending equilibra of elastic beams. Using the method of lower and upper solutions coupled with monotone iteration, the authors derive conditions on the functions \(f\) and \(g\) such that the boundary value problems \((1),(2),(3)\) and \((1),(2),(4)\) have monotone positive solutions. They also present numerical simulations of the problem \((1),(2),(3)\).
scientific article; zbMATH DE number 5577450