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Boundary layer phenomena for differential-delay equations with state-dependent time lags, I.

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Abstract

In this paper we begin a study of the differential-delay equation

$$\varepsilon x'(t) = - x(t) + f(x(t - r)), r = r(x(t))$$

.

We prove the existence of periodic solutions for 0<ɛ<ɛ 0, where ɛ 0 is an optimal positive number. We investigate regularity and monotonicity properties of solutions x(t) which are defined for all t and of associated functions like η(t)=t−r(x(t)). We begin the development of a Poincaré-Bendixson theory and phase-plane analysis for such equations. In a companion paper these results will be used to investigate the limiting profile and corresponding boundary layer phenomena for periodic solutions as ɛ approaches zero.

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Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, 02912, Providence, Rhode Island

    John Mallet-Paret & Roger D. Nussbaum

  2. Mathematics Department, Rutgers University, 08903, New Brunswick, New Jersey

    John Mallet-Paret & Roger D. Nussbaum

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  1. John Mallet-Paret
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  2. Roger D. Nussbaum
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Communicated by B. D. Coleman

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Mallet-Paret, J., Nussbaum, R.D. Boundary layer phenomena for differential-delay equations with state-dependent time lags, I.. Arch. Rational Mech. Anal. 120, 99–146 (1992). https://doi.org/10.1007/BF00418497

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