A Characterization of the Invariance of Positivity for Functional Differential Equations

Consider a system of functional differential equations of the form x = f(t,x,), t > to, */,, = , (1) in which x is a real n-vector valued function of t, x denotes dx/dt, $ is an initial condition function, and x, denotes the function defined by xi(s) = x(t + s) for s to whenever is positive in, for example, the sense that