Wright方程慢振荡周期解的稳定性和唯一性
Stability and Uniqueness of Slowly Oscillating Periodic Solutions to Wright's Equation
摘要：
本文证明了Wright方程$y'(t) = - αy(t-1) \{1 + y(t)\}$对所有参数值$α\in [ 1.9,6.0]$都有唯一的慢振荡周期解（SOPS），直到时间平移.我们的证明基于Xie [27]早期采用的相同策略;证明每个SOPS都是渐近稳定的。我们首先介绍了一个分支定界算法来控制所有的SOP使用边界函数在所有的参数值$α\in [ 1.9,6.0]$。一旦构建的边界函数，我们然后控制Floquet乘数的所有可能的SOP严格解决本征值问题，再次使用配方介绍谢。利用这两个主要步骤，我们证明了Wright方程的所有SOPS对于$α\in [ 1.9,6.0]$是渐近稳定的，证明如下。这个结果是向1962年提出的琼斯猜想的证明迈进了一步。
英文摘要：
In this paper, we prove that Wright's equation $y'(t) = - αy(t-1) \{1 + y(t)\}$ has a unique slowly oscillating periodic solution (SOPS) for all parameter values $α\in [ 1.9,6.0]$, up to time translation. Our proof is based on a same strategy employed earlier by Xie [27]; show that every SOPS is asymptotically stable. We first introduce a branch and bound algorithm to control all SOPS using bounding functions at all parameter values $α\in [ 1.9,6.0]$. Once the bounding functions are constructed, we then control the Floquet multipliers of all possible SOPS by solving rigorously an eigenvalue problem, again using a formulation introduced by Xie. Using these two main steps, we prove that all SOPS of Wright's equation are asymptotically stable for $α\in [ 1.9,6.0]$, and the proof follows. This result is a step toward the proof of the Jones' Conjecture formulated in 1962.
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