On the existence of positive solutions for a fourth-order boundary value problem (Q1674083)

From the text and introduction: In this paper, we consider the following fourth-order ordinary differential equation boundary value problem: \[ \begin{gathered} u^{(4)}(t)= f(t,u(t), u'(t), u''(t), u'''(t)),\;t\in [0,1],\\ u(0)= u'(0)= u''(1)= u'''(1)= 0,\end{gathered} \] where \(f:[0,1]\times \mathbb{R}^3_+\times \mathbb{R}_-\to \mathbb{R}\), is continuous. By using the method of order reduction and the fixed point index, the existence of positive solutions is studied. We provide conditions under which the existence results hold. Such conditions are related to the frst eigenvalue corresponding to the relevant linear differential equation with dependence on the derivatives of the unknown function.