Periodic solutions for indefinite singular equations with singularities in the spatial variable and non-monotone nonlinearity
The author's aim is to prove the existence of a periodic solution for a scalar differential equation of the type \[ (\phi(u'))'=h(t)g(u), \] where \(\phi:(-1,1)\to{\mathbb R}\) is a strictly increasing odd function, \(g:(\alpha,\beta)\to{\mathbb R}\) is continuous, while \(h:{\mathbb R}\to{\mathbb R}\) is locally integrable and \(T\)-periodic. Here, \(\alpha\) and \(\beta\) are two singularities satisfying \[ \lim_{x\to\alpha^+}g(x)=\lim_{x\to\beta^-}g(x)=+\infty. \] The function \(h\) needs to change sign, and a rather technical condition on its behaviour at the points where it changes sign is proposed, related to the order of the singularity \(\alpha\). The proof of the existence of a periodic solution is based on the Leray-Schauder continuation principle.
scientific article; zbMATH DE number 6923442
scientific article