Journal of Dynamics and Differential Equations

, Volume 25, Issue 4, pp 843–905 | Cite as

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

Article
First Online:
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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
$$\begin{aligned} \dot{x}(t)=-\alpha (t)x(t)-\beta (t)x(t-1) \end{aligned}$$
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.

Keywords

Delay-differential equation Tensor product Floquet theory Positive operator Compound differential equation Lap number 

Mathematics Subject Classification (2010)

Primary 34K08 46B28 47B65 Secondary 34K06 47G10 
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Notes

Acknowledgments

John Mallet-Paret was partially supported by NSF DMS-0500674 and by The Center for Nonlinear Analysis at Rutgers University and Roger D. Nussbaum was partially supported by NSF DMS-0701171 and by The Lefschetz Center for Dynamical Systems at Brown University.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA
  2. 2.Department of MathematicsRutgers UniversityPiscatawayUSA

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