On the generalized pantograph functional-differential equation
On the generalized pantograph functional-differential equation
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DOI：
10.1017/s0956792500000966
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发表时间：
1993-03
影响因子：
1.9
通讯作者：
A. Iserles
中科院分区：
文献类型：
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作者：
A. Iserles
广义比例方程y′（t）= Ay（t）+ By（qt）+ Cy′（qt），y（0）= y0，其中q ∈（0，1），有着广泛的应用，同时也是求解更一般的单调时滞泛函微分方程的一个有用的范例.虽然许多特殊的情况下，已经被广泛的研究，这个方程的一般理论是缺乏它的发展和阐述是本文件的目的。在推导出A，B，C ∈ d×d上适定性的等价条件后，研究了Dirichlet级数中y的展开式.这为渐近行为的研究提供了一种非常富有成效的形式，我们适时地导出了limt →∞y（t）= 0的条件。稳定边界上的行为没有全面的解释，但我们能够证明，沿着边界的重要部分，y几乎是周期性的，如果q是合理的，它几乎是旋转对称的。本文还详细分析了标量方程y′（t）= by（qt），y（0）= 1，高阶比例方程，A的特征值的特殊构形所出现的类似共振的现象，以及方程Y′（t）= AY（t）+ Y（qt）B，Y（0）= Y0。
The generalized pantograph equation y′(t) = Ay(t) + By(qt) + Cy′(qt), y(0) = y0, where q ∈ (0, 1), has numerous applications, as well as being a useful paradigm for more general functional-differential equations with monotone delay. Although many special cases have been already investigated extensively, a general theory for this equation is lacking–its development and exposition is the purpose of the present paper. After deducing conditions on A, B, C ∈ ℂd×d that are equivalent to well-posedness, we investigate the expansion of y in Dirichlet series. This provides a very fruitful form for the investigation of asymptotic behaviour, and we duly derive conditions for limt⋅→∞y(t) = 0. The behaviour on the stability boundary possesses no comprehensive explanation, but we are able to prove that, along an important portion of that boundary, y is almost periodic and, provided that q is rational, it is almost rotationally symmetric. The paper also addresses itself to a detailed analysis of the scalar equation y′(t) = by(qt), y(0) = 1, to high-order pantograph equations, to a phenomenon, similar to resonance, that occurs for specific configurations of eigenvalues of A, and to the equation Y′(t) = AY(t) + Y(qt) B, Y(0) = Y0.