A general result to the existence of a periodic solution to an indefinite equation with a weak singularity
The paper investigates the existence of \(T\)-periodic solutions for the singular equation \[ u'' = \frac{h(t)}{u^\lambda}, \qquad u \in \mathbb{R}, \] where \(h(t)\) is a \(T\)-periodic integrable function with \(\int_0^T h(t)\,dt < 0\) and \(\lambda \in (0,1)\) (the so-called weak singularity case). A quite general result is first proved, by the use of Leray-Schauder degree theory, assuming that the weight function \(h(t)\) satisfies a rather technical condition, related to its oscillatory and symmetry properties. Some corollary are later provided, in the cases when \(h(t)\) is piecewise constant and piecewise affine.
scientific article; zbMATH DE number 7047674