Positive periodic solutions to a nonlinear fourth-order differential equation
The authors prove the existence of positive periodic solutions to a fourth-order differential equation of the type \[ u''''(t)+a(t)u(t)=f(t,u(t)). \] Denoting by \(T\) the period, it is assumed that \(0 \leq a(t) \leq 4(\pi/T)^4\), and a further condition is assumed on the asymptotic behaviour of \(f\) at \(0\) and at \(\infty\). Roughly speaking, the function \(f\) ``jumps'' the first eigenvalue of the associated linear equation. The proof uses the fixed point index theory in a cone.
scientific article; zbMATH DE number 5315153