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

• DOI: 10.1007/s11784-018-0530-9
• Publication Date: 2018-02-19T02:50:24Z
• Authors: Ruyun Ma; Jinxiang Wang; Yan Long
• Keywords: 0101 mathematics; 01 natural sciences

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.