Mathematical Sciences: Solutions for Functional DifferentialEquations

9401823 Nussbaum 许多自然现象，例如生物学中的现象，似乎可以通过非线性“函数微分方程”或“FDE”来最好地描述。 粗略地说，FDE 是其中出现时间 t 的未知函数 x(t) 的方程，并且 x'(t)（x(t) 随时间的瞬时变化率）以特定方式不仅取决于 x(t)，还取决于函数 x 的过去历史。例如，在一些模型中，一类成熟红细胞的种群在时间 t 的增长率很可能取决于六到十天前这些相同成熟细胞的种群水平。非线性 FDE 的严格数学理论提出了严峻的挑战，并且忽略了应用中感兴趣的各种方程。我们建议研究某些类别的例子，直到最近，这些例子还被认为是棘手的，但现在似乎有可能获得各种各样的令人惊讶的详细定理。 该提案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 (*) 的推广出现在各种应用中。在与 John Mallet-Paret 的合作中，作者证明，在 f 和 r 的自然假设下以及对于所有足够小的 a，方程 (*) 具有非常数周期解。这些周期解通常看起来具有很强的全局稳定性。 此外，在许多情况下，已经证明可以确定当 a 接近零时这种周期解的形状的极限轮廓。在本提案中，我们考虑了有关方程 (*) 的许多问题，并讨论了 (*) 结果可能扩展到更一般的方程类别。

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.