Mathematical Sciences: Solutions for Functional DifferentialEquations

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 a>0. 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.