**
**

**Abstract**

We show how to solve a large class of Functional Differential Equations (FDEs) of the form

*x'(t) = F(t,x(t),x(f(t)))*
If the argument

*f(t) *has certain properties then these equations are suitable for determining the initial conditions for problems from various fields such as control theory, traffic flow, spread of epidemics, and age-structured population growth. The required property on

*f* is that it be idempotent, that is,

*f(f(t))=t*. We extend the methods given here to apply to more general cases for

*f*, solving equations that describe the behavior of a quantity whose reate of change depends upon several type of deviations of the time variable such as reverse time flows or periodic time flows, for example.

**
See Volume 6
Number ****3 October**** 2003****
issue of The Journal of Interdisciplinary Mathematics, pages 279-289**

I will be happy to send interested parties a copy of a reprint of this article. E-mail me at clement.falbo@sonoma.edu.

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