Mathematical Sciences: Solutions for Functional DifferentialEquations

小行星9401823 许多自然现象，例如，在生物学中，似乎最好的描述非线性“功能微分方程”或“FDE的”。 粗略地说，FDE是这样的方程，其中出现了一个未知的时间t函数x（t），x（t）随时间的瞬时变化率x '（t）不仅以特定的方式取决于x（t），而且还取决于函数x的过去历史。例如，在某些模型中，一类成熟红细胞在时间t的群体增长率很可能取决于6至10天前这些相同成熟细胞的群体水平。严格的数学理论的非线性FDE的提出了严重的挑战和各种方程的兴趣在应用中被忽视。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。 这个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是给定的函数，a0。方程（*）和方程（*）的推广出现在各种应用中。在与John Mallet-Paret的联合工作中，作者证明了，在关于f和r的自然假设下，对于所有充分小的a，方程（*）有非常数周期解。这些周期解通常具有很强的全局稳定性。 此外，在许多情况下，它已被证明是可能的，以确定这种周期性的解决方案作为一个接近零的形状的限制配置文件。在这个建议中，我们考虑了关于方程（*）的许多问题，并讨论了（*）的结果对更一般的方程类的可能扩展。

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.