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

TL;DR

This paper develops the theory of tensor and exterior products of linear functional differential equations, focusing on positivity properties, Floquet multipliers, and $u_0$-positivity in delay-differential systems.

Contribution

It introduces a framework for compound functional differential equations, analyzing positivity, Floquet theory, and $u_0$-positivity for systems with delays and sign conditions.

Findings

Exterior products generate positive linear processes under certain sign conditions.

Periodic coefficients lead to a complete set of Floquet multipliers.

$u_0$-positivity is established for systems with uniform sign conditions on coefficients.

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 $\dot x(t)=-\alpha(t)x(t)-\beta(t)x(t-1)$ with a single delay, where the delay coefficient is of one sign, say $\delta\beta(t)\ge 0$ with $\delta\in{-1,1}$. Positivity properties are studied, with the result that if $(-1)^k=\delta$ 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 $\alpha(t)$ and $\beta(t)$ are periodic of the same period, and $\beta(t)$ 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 $\beta(t)$ satisfies a uniform sign condition.