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

Autor: John Mallet-Paret, Roger D. Nussbaum
Rok vydání: 2013
Předmět:
Zdroj: Journal of Dynamics and Differential Equations. 25:843-905
ISSN: 1572-9222
1040-7294
DOI: 10.1007/s10884-013-9318-1
Popis: 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.
Comment: 84 pages
Databáze: OpenAIRE
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