Tensor Products, Positive Linear Operators, and Delay-Differential Equations

Abstract

We develop the theory of compound functional differential equations, which are tensor and exterior products of linear functional differential equations. Of particular interest is the equation $\dot x(t)=-α(t)x(t)-β(t)x(t-1)$ with a single delay, where the delay coefficient is of one sign, say $δβ(t)\ge 0$ with $δ\in{-1,1}$. Positivity properties are studied, with the result that if $(-1)^k=δ$ then the $k$-fold exterior product of the above system generates a linear process which is positive with respect to a certain cone in the phase space. Additionally, if the coefficients $α(t)$ and $β(t)$ are periodic of the same period, and $β(t)$ satisfies a uniform sign condition, then there is an infinite set of Floquet multipliers which are complete with respect to an associated lap number. Finally, the concept of $u_0$-positivity of the exterior product is investigated when $β(t)$ satisfies a uniform sign condition.