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

We develop the method of lower and upper solutions for the fourth-order differential equation which models the stationary states of the deflection of an elastic beam, whose both ends simply supported

$$\begin{aligned}&y^{(4)}(x)+(k_1+k_2) y''(x)+k_1k_2 y(x)=f(x,y(x)), \ \ \ \ x\in (0,1),\\&y(0) = y(1) = y''(0) = y''(1) = 0\\ \end{aligned}$$

under the condition \(0<k_1<k_2<x_1^2\approx 4.11585\), where \(x_1\) is the first positive solution of the equation \(x\cos (x)+\sin (x)=0\). The main tools are Schauder fixed point theorem and the Elias inequality.