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
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
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
M. Bartha, On stability properties for neutral differential equations with state-dependent delay. Diff. Equat. Dyn. Syst. 7, 197–220 (1999)
P. Brunovský, A. Erdélyi, H.O. Walther, On a model of a currency exchange rate - local stability and periodic solutions. J. Dyn. Diff. Equat. 16, 393–432 (2004)
O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel, H.O. Walther, Delay Equations: Functional-, Complex- and Nonlinear Analysis (Springer, New York, 1995)
R.D. Driver, A two-body problem of classical electrodynamics: the one-dimensional case. Ann. Phys. 21, 122–142 (1963)
R.D. Driver, in A Functional-Differential System of Neutral Type Arising in a Two-Body Problem of Classical Electrodynamics, ed. by J. LaSalle, S. Lefschetz. International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics (Academic, New York, 1963), pp. 474–484
R.D. Driver, Existence and continuous dependence of solutions of a neutral functional differential equation. Arch. Rat. Mech. Anal. 19, 149–166 (1965)
R.D. Driver, A neutral system with state-dependent delay. J. Diff. Equat. 54, 73–86 (1984)
L.J. Grimm, Existence and continuous dependence for a class of nonlinear neutral-differential equations. Proc. Amer. Math. Soc. 29, 467–473 (1971)
J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations (Springer, New York 1993)
F. Hartung, T.L. Herdman, J. Turi, On existence, uniqueness and numerical approximation for neutral equations with state-dependent delays. Appl. Numer. Math. 24, 393–409 (1997)
F. Hartung, T.L. Herdman, J. Turi, Parameter identifications in classes of neutral differential equations with state-dependent delays. Nonlinear Anal. TMA 39, 305–325 (2000)
F. Hartung, T. Krisztin, H.O. Walther, J. Wu, in Functional Differential Equations with State-Dependent Delay: Theory and Applications, ed. by A. Canada, P. Drabek, A. Fonda. Handbook Of Differential Equations, Ordinary Differential Equations, vol 3 (Elsevier Science B. V., North Holland, Amsterdam, 2006), pp. 435–545
Z. Jackiewicz, Existence and uniqueness of solutions of neutral delay-differential equations with state-dependent delays. Funkcialaj Ekvacioj 30, 9–17 (1987)
T. Krisztin, A local unstable manifold for differential equations with state-dependent delay. Discrete Continuous Dyn. Syst. 9, 993–1028 (2003)
T. Krisztin, J. Wu, Monotone semiflows generated by neutral equations with different delays in neutral and retarded parts. Acta Mathematicae Universitatis Comenianae 63, 207–220 (1994)
J. Mallet-Paret, R.D. Nussbaum, P. Paraskevopoulos, Periodic solutions for functional differential equations with multiple state-dependent time lags. Topological Methods in Nonlinear Analysis 3, 101–162 (1994)
R.D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations. Annali di Matematica Pura ed Applicata IV Ser. 101, 263–306 (1974)
M. Preisenberger, Periodische Lösungen Neutraler Differenzen-Differentialgleichungen. Doctoral dissertation, München, 1984
A. Rezounenko, Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions. Nonlinear Anal. 70, 3978–3986 (2009)
E. Stumpf, On a Differential Equation with State-Dependent Delay: A Global Center-Unstable Manifold Bordered by a Periodic Orbit. Doctoral dissertation, Hamburg, 2010
H.O. Walther, Stable periodic motion of a system with state-dependent delay. Diff. Integral Equat. 15, 923–944 (2002)
H.O. Walther, The solution manifold and C1-smoothness of solution operators for differential equations with state dependent delay. J. Diff. Equat. 195, 46–65 (2003)
H.O. Walther, Smoothness Properties of Semiflows for Differential Equations with State Dependent Delay. Russian, in Proceedings of the International Conference on Differential and Functional Differential Equations, Moscow, 2002, vol 1, pp. 40–55, Moscow State Aviation Institute (MAI), Moscow 2003. English version: J. Mathematical Sciences 124 (2004), 5193–5207
H.O. Walther, Convergence to square waves in a price model with delay. Discrete Continuous Dyn. Syst. 13, 1325–1342 (2005)
H.O. Walther, Bifurcation of periodic solutions with large periods for a delay differential equation. Annali di Matematica Pura ed Applicata 185, 577–611 (2006)
H.O. Walther, On a model for soft landing with state-dependent delay. J. Dyn. Diff. Equat. 19, 593–622 (2007)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-1-4614-4523-4_9
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-4522-7
Online ISBN: 978-1-4614-4523-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)