Mathematical Sciences: Solutions for Functional DifferentialEquations

[401823]努斯鲍姆许多自然现象，例如，在生物学中，似乎最好用非线性的“泛函微分方程”或“FDE”来描述。大致说来,FDE的方程中未知函数的时间t, x (t),似乎和x (t) x (t)的瞬时变化率随着时间的推移,在指定的方式不仅取决于x (t)也对过去的历史功能x。例如,有些型号的人口增长率的成熟红细胞t时刻很可能取决于人口水平相同的成熟细胞6到10天前。非线性FDE的严格数学理论提出了严峻的挑战，各种应用中感兴趣的方程被忽视了。我们建议研究一些类别的例子，直到最近，被认为是棘手的，但现在似乎有可能获得各种各样的令人惊讶的详细定理。本建议的出发点是研究方程（*），ax'(t) = f(x(t),x(t-r)), r = r(x(t))，其中f和r是给定函数和a0。方程（*）和方程（*）的推广在各种应用中出现。在与John Mallet-Paret的共同工作中，作者证明了在f和r的自然假设下，对于所有足够小的a，方程（*）具有非常周期解。这些周期解通常似乎具有很强的全局稳定性。此外，在许多情况下，当a趋近于零时，可以确定这类周期解的极限形状。在这个提议中，我们考虑了关于方程（*）的许多问题，并讨论了（*）的结果可能推广到更一般的方程类。

9401823 Nussbaum Many natural phenomena, for example, in biology, seem best described by nonlinear "functional differential equations" or "FDE's". Roughly speaking, FDE's are equations in which an unknown function of time t, x(t), appears and x'(t), the instantaneous rate of change of x(t) with time, depends in a specified way not only on x(t) but also on the past history of the function x. For example,in some models the rate of increase of a population of a class of mature red blood cells at time t may well depend on population levels of those same mature cells six to ten days earlier. A rigorous mathematical theory of of nonlinear FDE's poses serious challenges and various equations of interest in applications have been neglected. We propose to study some classes of examples which were, until quite recently, considered intractable, but for which it now seems possible to obtain a wide variety of surprisingly detailed theorems. The starting point of this proposal is the study of the equation (*), ax'(t) = f(x(t),x(t-r)), r = r(x(t)), where f and r are given functions and a0. Equation (*) and generalizations of equation (*) arise in a variety of applications. In joint work with John Mallet-Paret, the author has shown that, under natural assumptions on f and r and for all sufficiently small a, equation (*) has nonconstant periodic solutions. These periodic solutions often seem to have strong global stability properties. Furthermore, in many cases it has proved possible to determine the limiting profile of shape of such periodic solutions as a approaches zero. In this proposal we consider many questions about equation (*), and we discuss possible extensions of results for (*) to much more general classes of equations.