Global Attractivity of the Zero Solution for Wright's Equation
Global Attractivity of the Zero Solution for Wright's Equation
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DOI：
10.1137/120904226
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发表时间：
2014-03
期刊：
影响因子：
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通讯作者：
B. Bánhelyi;T. Csendes;T. Krisztin;A. Neumaier
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文献类型：
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作者：
B. Bánhelyi;T. Csendes;T. Krisztin;A. Neumaier
1955年，E.M. Wright证明了延迟微分方程$\dot x（t）= -\alpha（e^{x（t-1）}-1）$的所有解收敛于0，并且证明了对于$\alpha\in（0，\pi/2）$这也是正确的。本文证明了关于$\alpha\在[1.5，1.5706]$中的猜想（与$\pi/2=1.570796\ldots$相比）.证明的第一部分证明了保证慢振荡周期解不存在是足够的，并且证明了具有小振幅的慢振荡周期解不存在。在第二部分中，给出了一个计算机辅助的证明，以排除大振幅的慢振荡周期解。
In 1955 E.M. Wright proved that all solutions of the delay differential equation $\dot x(t) = -\alpha (e^{x(t-1)}-1)$ converge to zero as $t\to\infty$ for $\alpha\in(0,3/2]$ and conjectured that this is even true for $\alpha\in(0,\pi/2)$. The present paper proves the conjecture for $\alpha\in[1.5,1.5706]$ (compare with $\pi/2=1.570796\ldots$). The first part of the proof verifies that it is sufficient to guarantee the nonexistence of slowly oscillating periodic solutions, and it shows that slowly oscillating periodic solutions with small amplitudes cannot exist. In the second part a computer-assisted proof is given to exclude slowly oscillating periodic solutions with large amplitudes.