Algebraic-delay differential systems: $C^0$-extendable submanifolds and linearization
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- by N. Kosovalić, Y. Chen and J. Wu PDF
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Abstract:
Consider the abstract algebraic-delay differential system, \begin{eqnarray*} x’(t) &=& Ax(t)+F(x(t),a(t)), \\ a(t) &=& H(x_t,a_t) . \end{eqnarray*} 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.References
- Odo Diekmann, Stephan A. van Gils, Sjoerd M. Verduyn Lunel, and Hans-Otto Walther, Delay equations, Applied Mathematical Sciences, vol. 110, Springer-Verlag, New York, 1995. Functional, complex, and nonlinear analysis. MR 1345150, DOI 10.1007/978-1-4612-4206-2
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
- Jack Hale, Theory of functional differential equations, 2nd ed., Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977. MR 0508721
- Ferenc Hartung, Tibor Krisztin, Hans-Otto Walther, and Jianhong Wu, Functional differential equations with state-dependent delays: theory and applications, Handbook of differential equations: ordinary differential equations. Vol. III, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2006, pp. 435–545. MR 2457636, DOI 10.1016/S1874-5725(06)80009-X
- M. L. Hbid, M. Louihi, and E. Sánchez, A threshold state-dependent delayed functional equation arising from marine population dynamics: modelling and analysis, J. Evol. Equ. 10 (2010), no. 4, 905–928. MR 2737164, DOI 10.1007/s00028-010-0075-x
- K. Korvasova, Linearized stability in case of state-dependent delay: a simple test example, Master’s thesis, Universiteit Utrecht, 2011.
- N. Kosovalić, F. M. G. Magpantay, Y. Chen, and J. Wu, Abstract algebraic-delay differential systems and age structured population dynamics, J. Differential Equations 255 (2013), no. 3, 593–609. MR 3053479, DOI 10.1016/j.jde.2013.04.025
- Tibor Krisztin, A local unstable manifold for differential equations with state-dependent delay, Discrete Contin. Dyn. Syst. 9 (2003), no. 4, 993–1028. MR 1975366, DOI 10.3934/dcds.2003.9.993
- Serge Lang, Differential manifolds, 2nd ed., Springer-Verlag, New York, 1985. MR 772023, DOI 10.1007/978-1-4684-0265-0
- Pierre Magal and Shigui Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc. 202 (2009), no. 951, vi+71. MR 2559965, DOI 10.1090/S0065-9266-09-00568-7
- Pierre Magal and Shigui Ruan, On semilinear Cauchy problems with non-dense domain, Adv. Differential Equations 14 (2009), no. 11-12, 1041–1084. MR 2560868
- F. M. G. Magpantay, N. Kosovalić, and J. Wu, An age-structured population model with state-dependent delay: derivation and numerical integration, SIAM J. Numer. Anal. 52 (2014), no. 2, 735–756. MR 3187671, DOI 10.1137/120903622
- John Mallet-Paret, Roger D. Nussbaum, and Panagiotis Paraskevopoulos, Periodic solutions for functional-differential equations with multiple state-dependent time lags, Topol. Methods Nonlinear Anal. 3 (1994), no. 1, 101–162. MR 1272890, DOI 10.12775/TMNA.1994.006
- R. M. Nisbet and W. S. C. Gurney, The systematic formulation of population models for insects with dynamically varying instar duration, Theoret. Population Biol. 23 (1983), no. 1, 114–135. MR 700823, DOI 10.1016/0040-5809(83)90008-4
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
- Alexander V. Rezounenko, Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions, Nonlinear Anal. 70 (2009), no. 11, 3978–3986. MR 2515314, DOI 10.1016/j.na.2008.08.006
- Alexander V. Rezounenko, Non-linear partial differential equations with discrete state-dependent delays in a metric space, Nonlinear Anal. 73 (2010), no. 6, 1707–1714. MR 2661353, DOI 10.1016/j.na.2010.05.005
- Wolfgang M. Ruess, Flow invariance for nonlinear partial differential delay equations, Trans. Amer. Math. Soc. 361 (2009), no. 8, 4367–4403. MR 2500891, DOI 10.1090/S0002-9947-09-04833-8
- Jan Sieber, Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations, Discrete Contin. Dyn. Syst. 32 (2012), no. 8, 2607–2651. MR 2903983, DOI 10.3934/dcds.2012.32.2607
- Hal L. Smith, Reduction of structured population models to threshold-type delay equations and functional-differential equations: a case study, Math. Biosci. 113 (1993), no. 1, 1–23. MR 1201765, DOI 10.1016/0025-5564(93)90006-V
- Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. MR 1301779, DOI 10.1007/978-1-4612-0873-0
- Horst R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations 3 (1990), no. 6, 1035–1066. MR 1073056
- Hans-Otto Walther, The solution manifold and $C^1$-smoothness for differential equations with state-dependent delay, J. Differential Equations 195 (2003), no. 1, 46–65. MR 2019242, DOI 10.1016/j.jde.2003.07.001
- Hans-Otto Walther, Algebraic-delay differential systems, state-dependent delay, and temporal order of reactions, J. Dynam. Differential Equations 21 (2009), no. 1, 195–232. MR 2482014, DOI 10.1007/s10884-009-9129-6
- Hans-Otto Walther, Differential equations with locally bounded delay, J. Differential Equations 252 (2012), no. 4, 3001–3039. MR 2871791, DOI 10.1016/j.jde.2011.11.004
- Jianhong Wu, Theory and applications of partial functional-differential equations, Applied Mathematical Sciences, vol. 119, Springer-Verlag, New York, 1996. MR 1415838, DOI 10.1007/978-1-4612-4050-1
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
- MR Author ID: 998152
- Email: kosovalic@southalabama.edu
- Y. Chen
- Affiliation: Department of Mathematics, Wilfrid Laurier University, 75 University Avenue West, Waterloo, Ontario, Canada N2L 3C5
- MR Author ID: 363105
- Email: ychen@wlu.ca
- J. Wu
- Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3
- MR Author ID: 226643
- Email: wujh@mathstat.yorku.ca
- 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) - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 3387-3419
- MSC (2010): Primary 34K05; Secondary 34A09, 92D25
- DOI: https://doi.org/10.1090/tran/6760
- MathSciNet review: 3605975