Abstract This paper is devoted to the investigation of generalized Abel equation x ˙ = S ( x , t ) = ∑ i = 0 m a i ( t ) x i , where a i ∈ C ∞ ( [ 0 , 1 ] ) . A solution x ( t ) is called a 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 λ 1 ⋯ λ m such that either ( − 1 ) i ⋅ S ( λ i , t ) ≥ 0 for i = 1 , ⋯ , m , or ( − 1 ) i ⋅ S ( λ i , t ) ≤ 0 for i = 1 , ⋯ , m . By means of Lagrange interpolation formula, we prove 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 valid 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.

中文翻译：

---

方程 x˙=∑i=0mai(t)xi 的几何准则至多具有 m 个孤立周期解

摘要 本文致力于研究广义阿贝尔方程x˙=S(x,t)=∑i=0mai(t)xi，其中ai∈C∞([0,1])。如果 x ( 0 ) = x ( 1 ) ，则解 x ( t ) 称为周期解。为了估计方程的孤立周期解的数量，我们提出了一个假设 (H)，它只与 m 条直线上的 S ( x , t ) 有关： 存在 m 个实数 λ 1 ⋯ λ m 使得要么( − 1 ) i ⋅ S ( λ i , t ) ≥ 0 for i = 1 , ⋯ , m ，或 ( − 1 ) i ⋅ S ( λ i , t ) ≤ 0 for i = 1 ,⋯, m 。借助拉格朗日插值公式，我们证明，如果假设（H）成立，则方程至多有 m 个孤立周期解（以重数计），且上界尖锐。此外，这个结论在一些较弱的几何假设下也是成立的。应用我们对系数为 1 的三角广义 Abel 方程的主要结果，我们给出了获得孤立周期解数量上限的准则。该标准“几乎等同于”假设 (H)，可以更有效地进行检查。