Periodic solutions for indefinite singular equations with applications to the weak case

TL;DR

This paper establishes conditions for the existence of periodic solutions to a second order differential equation with a weak singularity, broadening understanding of such equations with sign-changing coefficients.

Contribution

It provides new efficient conditions for the existence of T-periodic solutions to singular differential equations with sign-changing functions, focusing on the weak singularity case.

Findings

Established existence of T-periodic solutions under new conditions

Extended results to equations with sign-changing and weak singularities

Provided analytical framework for singular differential equations

Abstract

As a consequence of the main result of this paper efficient conditions guaranteeing the existence of a $T-$periodic solution to the second order differential equation \begin{equation*} u"=\frac{h(t)}{u^{\lambda}} \end{equation*} are established. Here, $h\in L(\mathbb{R}/T\mathbb{Z})$ is a piecewise-constant sign-changing function where the non-linear term presents a weak singularity at 0 (i.e. $\lambda\in (0,1)$).