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 y(4)(x)+(k1+k2)y′′(x)+k1k2y(x)=f(x,y(x)),x∈(0,1),y(0)=y(1)=y′′(0)=y′′(1)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&y^{(4)}(x)+(k_1+k_2) y''(x)+k_1k_2 y(x)=f(x,y(x)), \ \ \ \ x\in (0,1),\\&y(0) = y(1) = y''(0) = y''(1) = 0\\ \end{aligned}$$\end{document}under the condition 0<k1<k2<x12≈4.11585\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<k_1<k_2<x_1^2\approx 4.11585$$\end{document}, where x1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_1$$\end{document} is the first positive solution of the equation xcos(x)+sin(x)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\cos (x)+\sin (x)=0$$\end{document}. The main tools are Schauder fixed point theorem and the Elias inequality.