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Semiflows for Neutral Equations with State-Dependent Delays

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Infinite Dimensional Dynamical Systems

Part of the book series: Fields Institute Communications ((FIC,volume 64))

Abstract

We show that under mild hypotheses neutral functional differential equations where delays may be state-dependent generate continuous semiflows, a larger one on a thin set in a Banach space of C1-functions and a smaller one, with better smoothness properties, on a closed subset in a Banach manifold of C2-functions. The hypotheses are satisfied for a prototype equation of the form

$$x^{\prime}(t) = a\,x^{\prime}(t + d(x(t))) + f(x(t))$$

with−h<d(x(t))<0, which for certain d and f models the interaction between following a trend and negative feedback with respect to some equilibrium state.

Mathematics Subject Classification (2010): Primary 34K40, 37L05; Secondary 34K05, 58B99

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Authors and Affiliations

  1. Mathematisches Institut, Universitaet Giessen, Arndtstr. 2, D 35392, Giessen, Germany

    Hans-Otto Walther

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  1. Hans-Otto Walther
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Correspondence to Hans-Otto Walther .

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Editors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, RI, USA

    John Mallet-Paret

  2. Department of Mathematics and Statistics, York University, Toronto, ON, Canada

    Jianhong Wu  & Huaiping Zhu  & 

  3. Georgia Institute of Technology, School of Mathematics, Atlanta, GA, USA

    Yingfei Yi

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Walther, HO. (2013). Semiflows for Neutral Equations with State-Dependent Delays. In: Mallet-Paret, J., Wu, J., Yi, Y., Zhu, H. (eds) Infinite Dimensional Dynamical Systems. Fields Institute Communications, vol 64. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4523-4_9

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