On linear and nonlinear fourth-order eigenvalue problems with nonlocal boundary condition (Q1951093)

The author uses the the Krein-Rutman theorem to investigate the principal eigenvalue of the fourth-order linear eigenvalue problem \[ \begin{aligned} u^4(t)+\beta u''(t)&=\mu[u(t)-u''(t)],\\u(0)=u(1)&=\int^1_0 p(s)u(s)\,ds,\\u''(0)=u''(1)&=\int^1_0 q(s)u(s)\,ds,\end{aligned} \] where \(0<\beta<\pi^2\) and \(p,q\in L^1[0,1]\). He also proves the existence of positive solutions for the corresponding nonlinear problem via fixed point index theory.