A geometric criterion for equation x'=∑a_i(t)x^i having at most m isolated periodic solutions
A geometric criterion for equation x'=∑a_i(t)x^i having at most m isolated periodic solutions
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方程 x'=�a_i(t)x^i 的几何准则最多具有 m 个孤立周期解
DOI：
10.1016/j.jde.2019.11.032
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发表时间：
2020
影响因子：
2.4
通讯作者：
Haihua Liang
中科院分区：
文献类型：
--
作者：
Jianfeng Huang;Haihua Liang
本文研究广义Abel方程x·=S(x，t)=∑i=0 mi(T)xi，其中a i∈C∞([0，1]).解x(T)称为周期解，如果x(0)=x(1)。为了估计方程孤立周期解的个数，我们提出了一个仅与S(x，t)在m条直线上有关的假设(H)：存在m个实数λ1<；⋯<；λm使得当(−1)i⋅S(λi，t)≥0当i=1，⋯，m时，或(−1)i⋅S(λi，t)≤0当i=1，⋯，m时.借助于拉格朗日插值公式，我们证明了当假设(H)成立时，方程至多有m个孤立周期解(重计数)，且上界是尖锐的.此外，这一结论在一些较弱的几何假设下也是成立的。应用我们的主要结果，对于一次系数的三角广义Abel方程，给出了孤立周期解个数的上界的一个判据。这一标准与假设(H)“几乎等同”，可以更有效地加以检验。
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.