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Algebraic-delay differential systems: $ C^0$-extendable submanifolds and linearization


Authors: N. Kosovalić, Y. Chen and J. Wu
Journal: Trans. Amer. Math. Soc. 369 (2017), 3387-3419
MSC (2010): Primary 34K05; Secondary 34A09, 92D25
DOI: https://doi.org/10.1090/tran/6760
Published electronically: January 6, 2017
MathSciNet review: 3605975
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Abstract: Consider the abstract algebraic-delay differential system,

$\displaystyle x'(t)$ $\displaystyle =$ $\displaystyle Ax(t)+F(x(t),a(t)),$  
$\displaystyle a(t)$ $\displaystyle =$ $\displaystyle H(x_t,a_t) .$  

Here $ A$ is a linear operator on $ D(A)\subseteq X$ satisfying the Hille-Yosida conditions, $ x(t)\in \overline {D(A)}\subseteq X$, $ a(t)\in {\mathbf {R}}^n$, and $ X$ is a real Banach space. Let $ C_0\subseteq \overline {D(A)}$ be closed and convex, and $ K\subseteq \mathbf {R}^n$ be a compact set contained in the ball of radius $ h>0$ centered at 0. Under suitable Lipschitz conditions on the nonlinearities $ F$ and $ H$ and a subtangential condition, the system generates a continuous semiflow on a subset of the space of continuous functions $ C([-h,0],C_0\times \mathbf {R}^n)$, which is induced by the algebraic constraint. The object of this paper is to find conditions under which this semiflow is also differentiable with respect to initial data. In the motivating example coming from modelling the dynamics of an age structured population, the nonlinearities $ F$ and $ H$ are not Fréchet differentiable on the sets $ C_0\times K$ and $ C([-h,0],C_0\times K)$, respectively. The main challenge of obtaining the differentiability of the semiflow is to determine the right type of differentiability and the right phase space. We develop a novel approach to address this problem which also shows how the spaces on which the derivatives of solution operators act reflect the model structure.

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Additional Information

N. Kosovalić
Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3
Address at time of publication: Department of Mathematics and Statistics, University of South Alabama, 411 University Boulevard N, Mobile, Alabama 36688-0002
Email: kosovalic@southalabama.edu

Y. Chen
Affiliation: Department of Mathematics, Wilfrid Laurier University, 75 University Avenue West, Waterloo, Ontario, Canada N2L 3C5
Email: ychen@wlu.ca

J. Wu
Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3
Email: wujh@mathstat.yorku.ca

DOI: https://doi.org/10.1090/tran/6760
Keywords: Linearization, nonlinear transport equation, state dependent delay, structured population dynamics, Banach manifold, interpolation space, functional differential equation.
Received by editor(s): December 9, 2013
Received by editor(s) in revised form: May 8, 2015
Published electronically: January 6, 2017
Additional Notes: The research of the second author was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and by the Early Researchers Award Program of Ontario (ERA)
The research of the third author was partially supported by NSERC and by the Canada Research Chairs Program (CRC)
Article copyright: © Copyright 2017 American Mathematical Society