On Poisson's state-dependent delay

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January 2013, 33(1): 365-379. doi: 10.3934/dcds.2013.33.365

On Poisson's state-dependent delay

Hans-Otto Walther ^1,

1.	Mathematisches Institut, Universität Gießen, Arndtstr. 2, D 35392 Gießen, Germany

Received July 2011 Revised October 2011 Published September 2012

Abstract
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In 1806 Poisson published one of the first papers on functional differential equations. Among others he studied an example with a state-dependent delay, which is motivated by a geometric problem. This example is not covered by recent results on initial value problems for differential equations with state-dependent delay. We show that the example generates a semiflow of differentiable solution operators, on a manifold of differentiable functions and away from a singular set. Initial data in the singular set produce multiple solutions.

Keywords: Functional differential equation, nonuniqueness., state-dependent delay.

Mathematics Subject Classification: Primary: 34K05; Secondary: 37L9.

Citation: Hans-Otto Walther. On Poisson's state-dependent delay. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 365-379. doi: 10.3934/dcds.2013.33.365

References:

[1]	O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H. O. Walther, "Delay Equations: Functional-, Complex- and Nonlinear Analysis," Springer, New York, 1995.
[2]	J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer, New York, 1993.
[3]	F. Hartung, Linearized stability for a class of neutral functional differential equations with state-dependent delays, Journal of Nonlinear Analysis: Theory, Methods and Applications, 69 (2008), 1629-1643.
[4]	F. Hartung, T. Krisztin, H. O. Walther and J. H. Wu, Functional differential equations with state-dependent delays: Theory and applications, in "Handbook of Differential Equations: Ordinary Differential Equations," Vol. III, 435-545, Elsevier/North-Holland, Amsterdam, 2006.
[5]	M. C. Irwin, "Smooth Dynamical Systems," Academic Press, London, 1980.
[6]	M. C. Mackey, Commodity price fluctuations:price-dependent delays and nonlinearities as explanatory factors, J. Economic Theory, 48 (1989), 497-509.
[7]	M. C. Mackey, personal communication, 2011.
[8]	S. D. Poisson, Sur les équations auxdifférences melées, Journal de l'Ecole Polytechnique, Tome, VI (1806), 126-147.
[9]	H. O. Walther, The solution manifold and $C^1$-smoothness of solution operators for differential equations with statedependent delay, J. Differential Eqs., 195 (2003), 46-65. doi: 10.1016/j.jde.2003.07.001.
[10]	H. O. Walther, Smoothness properties of semiflows for differential equations with state dependent delay, in "Proceedings of the International Conference on Differential and Functional Differential Equations, Moscow, 2002," 1 (2002), 40-55, Moscow State Aviation Institute (MAI), Moscow 2003, English version: Journal of the Mathematical Sciences, 124 (2004), 5193-5207. doi: 10.1023/B:JOTH.0000047253.23098.12.
[11]	H. O. Walther, Algebraic-delay differential systems, state-dependent delay, and temporal orderof reactions, J. Dynamics and Differential Eqs., 21 (2009), 195-232. doi: 10.1007/s10884-009-9129-6.
[12]	H. O. Walther, Semiflows for neutral equations with state-dependent delays,, Fields Inst. Communications, ().
[13]	H. O. Walther, Linearized stability for semiflows generated by a class of neutral equations, with applications to state-dependent delays, Journal of Dynamics and Differential Equations, 22 (2010), 439-462. doi: 10.1007/s10884-010-9168-z.
[14]	H. O. Walther, Differential equations with locally bounded delay, J. Differential Equations, 252 (2012), 3001-3039.

show all references

References:

[1]	O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H. O. Walther, "Delay Equations: Functional-, Complex- and Nonlinear Analysis," Springer, New York, 1995.
[2]	J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer, New York, 1993.
[3]	F. Hartung, Linearized stability for a class of neutral functional differential equations with state-dependent delays, Journal of Nonlinear Analysis: Theory, Methods and Applications, 69 (2008), 1629-1643.
[4]	F. Hartung, T. Krisztin, H. O. Walther and J. H. Wu, Functional differential equations with state-dependent delays: Theory and applications, in "Handbook of Differential Equations: Ordinary Differential Equations," Vol. III, 435-545, Elsevier/North-Holland, Amsterdam, 2006.
[5]	M. C. Irwin, "Smooth Dynamical Systems," Academic Press, London, 1980.
[6]	M. C. Mackey, Commodity price fluctuations:price-dependent delays and nonlinearities as explanatory factors, J. Economic Theory, 48 (1989), 497-509.
[7]	M. C. Mackey, personal communication, 2011.
[8]	S. D. Poisson, Sur les équations auxdifférences melées, Journal de l'Ecole Polytechnique, Tome, VI (1806), 126-147.
[9]	H. O. Walther, The solution manifold and $C^1$-smoothness of solution operators for differential equations with statedependent delay, J. Differential Eqs., 195 (2003), 46-65. doi: 10.1016/j.jde.2003.07.001.
[10]	H. O. Walther, Smoothness properties of semiflows for differential equations with state dependent delay, in "Proceedings of the International Conference on Differential and Functional Differential Equations, Moscow, 2002," 1 (2002), 40-55, Moscow State Aviation Institute (MAI), Moscow 2003, English version: Journal of the Mathematical Sciences, 124 (2004), 5193-5207. doi: 10.1023/B:JOTH.0000047253.23098.12.
[11]	H. O. Walther, Algebraic-delay differential systems, state-dependent delay, and temporal orderof reactions, J. Dynamics and Differential Eqs., 21 (2009), 195-232. doi: 10.1007/s10884-009-9129-6.
[12]	H. O. Walther, Semiflows for neutral equations with state-dependent delays,, Fields Inst. Communications, ().
[13]	H. O. Walther, Linearized stability for semiflows generated by a class of neutral equations, with applications to state-dependent delays, Journal of Dynamics and Differential Equations, 22 (2010), 439-462. doi: 10.1007/s10884-010-9168-z.
[14]	H. O. Walther, Differential equations with locally bounded delay, J. Differential Equations, 252 (2012), 3001-3039.

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