Positive solutions to a boundary value problem for the beam equation (Q2454953)

This paper is concerned with the boundary value problem \[ u''''(t) = g(t) f(u(t)), \quad 0 \leq t \leq 1, \] \[ u(0) = u''(0) = u''(1) = u(1), \] which describes the deformations of a simply supported beam. The author establishes the existence and nonexistence of positive solutions by applying the Guo-Krasnosel'skii fixed point theorem. The main assumptions are \(f \in C([0,\infty),[0,\infty))\), \(g \in C([0,1],[0,\infty))\), and \(g(t) \not \equiv 0\). Some quite interesting a priori estimates are obtained for solutions. These estimates enable the author to significantly improve solvability criteria known in the literature.