Periodic solutions with nonconstant sign in Abel equations of the second kind

TL;DR

This paper investigates the existence, uniqueness, and stability of periodic solutions with nonconstant sign in Abel equations of the second kind, extending the understanding beyond constant sign solutions and their transformations.

Contribution

It provides new sufficient conditions for the existence of nonconstant sign periodic solutions in Abel equations of the second kind, including their zeros, and analyzes their uniqueness and stability.

Findings

Established conditions for existence of nonconstant sign periodic solutions.

Proved uniqueness and stability properties of these solutions.

Extended the analysis beyond solutions obtainable via first kind Abel equations.

Abstract

The study of periodic solutions with constant sign in the Abel equation of the second kind can be made through the equation of the first kind. This is because the situation is equivalent under the transformation $x\mapsto x^{-1}$, and there are many results available in the literature for the first kind equation. However, the equivalence breaks down when one seeks for solutions with nonconstant sign. This note is devoted to periodic solutions with nonconstant sign in Abel equations of the second kind. Specifically, we obtain sufficient conditions to ensure the existence of a periodic solution that shares the zeros of the leading coefficient of the Abel equation. Uniqueness and stability features of such solutions are also studied.