A geometric criterion for equation $\dot{x}=\sum^m_{i=0}a_i(t)x^i$ having at most $m$ isolated periodic solutions

TL;DR

This paper introduces a geometric criterion under hypothesis (H) that bounds the number of isolated periodic solutions of generalized Abel equations to at most m, with the bound proven to be sharp and applicable under weaker conditions.

Contribution

The paper establishes a new geometric criterion (hypothesis H) that provides a sharp upper bound of m for the number of isolated periodic solutions of generalized Abel equations, extending previous results.

Findings

Maximum of m isolated periodic solutions under hypothesis (H)

Criterion for bounding solutions in trigonometrical Abel equations

Weaker geometric conditions still ensure the bound

Abstract

This paper is devoted to the investigation of generalized Abel equation $\dot{x}=S(x,t)=\sum^m_{i=0}a_i(t)x^i$, where $a_i\in\mathrm C^{\infty}([0,1])$. A solution $x(t)$ is called a {\em periodic solution} if $x(0)=x(1)$. In order to estimate the number of isolated periodic solutions of the equation, we propose a hypothesis (H) which is only concerned with $S(x,t)$ on $m$ straight lines: There exist $m$ real numbers $\lambda_1<\cdots<\lambda_m$ such that either $(-1)^i\cdot S(\lambda_i,t)\geq0$ for $i=1,\cdots,m$, or $(-1)^i\cdot S(\lambda_i,t)\leq0$ for $i=1,\cdots,m$. By means of Lagrange interpolation formula, we proves that the equation has at most $m$ isolated periodic solutions (counted with multiplicities) if hypothesis (H) holds, and the upper bound is sharp. Furthermore, this conclusion is also obtained under some weaker geometric hypotheses. Applying our main result for the trigonometrical generalized Abel equation with coefficients of degree one, we give a criterion to obtain the upper bound for the number of isolated periodic solutions. This criterion is "almost equivalent" to hypothesis (H) and can be much more effectively checked.