TY - JOUR

T1 - Asymptotic homogenization for delay-differential equations and a question of analyticity

AU - Mallet-Paret, John

AU - Nussbaum, Roger D.

N1 - Funding Information:
The first author was partially supported by The Center for Nonlinear Analysis at Rutgers University. The second author was partially supported by NSF Grant DMS-1201328 and by The Lefschetz Center for Dynamical Systems at Brown University.
Publisher Copyright:
© 2020 American Institute of Mathematical Sciences. All rights reserved.

PY - 2020

Y1 - 2020

N2 - We consider a class of nonautonomous delay-differential equations in which the time-varying coefficients have an oscillatory character, with zero mean value, and whose frequency approaches +∞ as t → ±∞. Typical simple examples are (Images Presented Here) where q ≥ 2 is an integer. Under various conditions, we show the existence of a unique solution with any prescribed finite limit lim x(t) = x− at −∞. We t→−∞ also show, under appropriate conditions, that any solution of an initial value problem has a finite limit lim x(t) = x+ at +∞, and thus we establish the t→+∞ existence of a class of heteroclinic solutions. We term this limiting phenomenon, and thus the existence of such solutions, “asymptotic homogenization.” Note that in general, proving the existence of a bounded solution of a given delay-differential equation on a semi-infinite interval (−∞, −T] is often highly nontrivial. Our original interest in such solutions stems from questions concerning their smoothness. In particular, any solution x : R → C of one of the equations in (∗) with limits x± at ±∞ is C∞, but it is unknown if such solutions are analytic. Nevertheless, one does know that any such solution of the second equation in (∗) can be extended to the lower half plane {z ∈ C | Im z < 0} as an analytic function.

AB - We consider a class of nonautonomous delay-differential equations in which the time-varying coefficients have an oscillatory character, with zero mean value, and whose frequency approaches +∞ as t → ±∞. Typical simple examples are (Images Presented Here) where q ≥ 2 is an integer. Under various conditions, we show the existence of a unique solution with any prescribed finite limit lim x(t) = x− at −∞. We t→−∞ also show, under appropriate conditions, that any solution of an initial value problem has a finite limit lim x(t) = x+ at +∞, and thus we establish the t→+∞ existence of a class of heteroclinic solutions. We term this limiting phenomenon, and thus the existence of such solutions, “asymptotic homogenization.” Note that in general, proving the existence of a bounded solution of a given delay-differential equation on a semi-infinite interval (−∞, −T] is often highly nontrivial. Our original interest in such solutions stems from questions concerning their smoothness. In particular, any solution x : R → C of one of the equations in (∗) with limits x± at ±∞ is C∞, but it is unknown if such solutions are analytic. Nevertheless, one does know that any such solution of the second equation in (∗) can be extended to the lower half plane {z ∈ C | Im z < 0} as an analytic function.

KW - Analytic solution

KW - Asymptotic behavior

KW - Averaging

KW - Delay-differential equation

KW - Heteroclinic solution

KW - Homogenization

KW - Rapid oscillation

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U2 - 10.3934/dcds.2020044

DO - 10.3934/dcds.2020044

M3 - Article

AN - SCOPUS:85082524426

VL - 40

SP - 3789

EP - 3812

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 6

ER -