泛函微分方程的次指数增长率
Subexponential Growth Rates in Functional Differential Equations
摘要：
研究了一类具有有限时滞的纯量自治非线性泛函微分方程的增长率，其中右端是$f(x)$的正连续线性泛函.我们假设$f$是次线性增长的，并且是这样的解决方案应该表现出比多项式增长快，但比指数增长慢。在$f$的某些技术条件下，证明了只要$f$比$l(x)=x/\log x$增长得慢，则该泛函微分方程的解渐近于辅助自治常微分方程的解，该辅助自治常微分方程的右端与$f$成比例（比例常数等于与线性泛函相关的有限测度的质量）.这种线性-对数增长率也被证明是至关重要的：如果$f$增长更迅速，$l$，ODE占主导地位的FDE;如果$f$是渐近的$l$的一个常数倍，FDE和ODE以相同的速度增长，模一个常数非单位因子。
英文摘要：
This paper determines the rate of growth to infinity of a scalar autonomous nonlinear functional differential equation with finite delay, where the right hand side is a positive continuous linear functional of $f(x)$. We assume $f$ grows sublinearly, and is such that solutions should exhibit growth faster than polynomial, but slower than exponential. Under some technical conditions on $f$, it is shown that the solution of the functional differential equation is asymptotic to that of an auxiliary autonomous ordinary differential equation with righthand side proportional to $f$ (with the constant of proportionality equal to the mass of the finite measure associated with the linear functional), provided $f$ grows more slowly than $l(x)=x/\log x$. This linear--logarithmic growth rate is also shown to be critical: if $f$ grows more rapidly than $l$, the ODE dominates the FDE; if $f$ is asymptotic to a constant multiple of $l$, the FDE and ODE grow at the same rate, modulo a constant non--unit factor.
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