Lower and upper solution method for the problem of elastic beam with hinged ends (Q1742568)

The authors develop the method of lower and upper solutions for the fourth-order ordinary differential equation \[ y^{(4)}(x)+(k_1+k_2)\, y''(x)+ k_1\, k_2 \, y(x)=f(x,y(x)), \, x\in (0,1); \] with the boundary conditions \[ y(0)=y(1)=y''(0)=y''(1)=0, \] \(0<k_1<k_2\) are given constants. To this end, they obtain the exact expression of the Green's function of the related linear operator, and prove that it is positive on \((0,1) \times (0,1)\) for all \(k_1, \; k_2 \in (0,\pi)\). Moreover, they obtain an upper bound for \(k_1\) and \(k_2\) for which the linear equation is disconjugate on \((0,1)\). For such choosen parameters, they prove that the solutions of a linear equation with suitable inhomogeneous boundary conditions are also positive on \((0,1)\). By combining all the previous properties, they prove the existence of a solution lying between a pair of well ordered lower and upper solutions.