# American Institute of Mathematical Sciences

January  2016, 36(1): 137-149. doi: 10.3934/dcds.2016.36.137

## Linearization of solution operators for state-dependent delay equations: A simple example

 1 Department of Mathematics, Utrecht University , Budapestlaan 6, 3584 CD Utrecht, Netherlands, Netherlands

Received  September 2014 Revised  April 2015 Published  June 2015

For state-dependent delay equations, it may easily happen that the equation is not differentiable. This hampers the formulation and the proof of the Principle of Linearized Stability. The fact that an equation is not differentiable does not, by itself, imply that the solution operators are not differentiable. And indeed, the aim of this paper is to present a simple example with differentiable solution operators despite of lack of differentiability of the equation. The example takes the form of a renewal equation and is motivated by a population dynamical model.
Citation: Odo Diekmann, Karolína Korvasová. Linearization of solution operators for state-dependent delay equations: A simple example. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 137-149. doi: 10.3934/dcds.2016.36.137
##### References:
 [1] O. Diekmann, P. Getto and M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM Journal on Mathematical Analysis, 39 (2008), 1023-1069. doi: 10.1137/060659211.  Google Scholar [2] O. Diekmann, M. Gyllenberg, J. A. J. Metz, S. Nakaoka and A. M. de Roos, Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example, Journal of Mathematical Biology, 61 (2010), 277-318. doi: 10.1007/s00285-009-0299-y.  Google Scholar [3] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walter, Delay Equations. Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.  Google Scholar [4] F. Hartung, On second-order differentiability with respect to parameters for differential equations with state-dependent delays, Journal of Dynamics and Differential Equations, 25 (2013), 1089-1138. doi: 10.1007/s10884-013-9330-5.  Google Scholar [5] I. Győri and F. Hartung, Exponential stability of a state-dependent delay system, Discrete and Continuous Dynamical Systems - Series A, 18 (2007), 773-791. doi: 10.3934/dcds.2007.18.773.  Google Scholar [6] F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional Differential Equations with State-Dependent Delays: Theory and Applications, in Handbook of Differential Equations: Ordinary Differential Equations (eds. A. Cañada, P. Drábek and A. Fonda), North-Holland, 3 (2006), 435-545. doi: 10.1016/S1874-5725(06)80009-X.  Google Scholar [7] F. Hartung and J. Turi, Linearized stability in functional differential equations with state-dependent delays, Discrete and Continuous Dynamical Systems Supplements, Special (2001), 416-425. Google Scholar [8] M. L. Hbid, E. Sánchez and R. Bravo de la Parra, State-dependent delays associated to threshold phenomena in structured population dynamics, Mathematical Models and Methods in Applied Sciences, 17 (2007), 877-900. doi: 10.1142/S0218202507002145.  Google Scholar [9] K. Korvasová, Linearized Stability in Case of State-Dependent Delay: A Simple Test Example, Master's thesis, Universiteit Utrecht, 2011. Google Scholar [10] N. Kosovalic, Y. Chen and J. Wu, Algebraic-delay differential systems: Age structured population modeling, $C^0$-extendable submanifolds and linearization,, Preprint., ().   Google Scholar [11] N. Kosovalic, F. M. G. Magpantay, Y. Chen and J. Wu, Abstract algebraic-delay differential systems and age structured population dynamics, Journal of Differential Equations, 255 (2013), 593-609. doi: 10.1016/j.jde.2013.04.025.  Google Scholar [12] S. Mirrahimi, B. Perthame and J. Y. Wakano, Direct competition results from strong competition for limited resource, Journal of Mathematical Biology, 68 (2014), 931-949. doi: 10.1007/s00285-013-0659-5.  Google Scholar [13] A. M. de Roos and L. Persson, Population and Community Ecology of Ontogenetic Development, Princeton University Press, Princeton, 2013. Google Scholar [14] W. M. Ruess, Linearized stability and regularity for nonlinear age-dependent population models, in Functional Analysis and Evolution Equations (eds. H. Amann, W. Arendt, M. Hieber, F. M. Neubrander, S. Nicaise and J. Below), Birkhäuser Basel, (2008), 561-576. doi: 10.1007/978-3-7643-7794-6_34.  Google Scholar

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##### References:
 [1] O. Diekmann, P. Getto and M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM Journal on Mathematical Analysis, 39 (2008), 1023-1069. doi: 10.1137/060659211.  Google Scholar [2] O. Diekmann, M. Gyllenberg, J. A. J. Metz, S. Nakaoka and A. M. de Roos, Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example, Journal of Mathematical Biology, 61 (2010), 277-318. doi: 10.1007/s00285-009-0299-y.  Google Scholar [3] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walter, Delay Equations. Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.  Google Scholar [4] F. Hartung, On second-order differentiability with respect to parameters for differential equations with state-dependent delays, Journal of Dynamics and Differential Equations, 25 (2013), 1089-1138. doi: 10.1007/s10884-013-9330-5.  Google Scholar [5] I. Győri and F. Hartung, Exponential stability of a state-dependent delay system, Discrete and Continuous Dynamical Systems - Series A, 18 (2007), 773-791. doi: 10.3934/dcds.2007.18.773.  Google Scholar [6] F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional Differential Equations with State-Dependent Delays: Theory and Applications, in Handbook of Differential Equations: Ordinary Differential Equations (eds. A. Cañada, P. Drábek and A. Fonda), North-Holland, 3 (2006), 435-545. doi: 10.1016/S1874-5725(06)80009-X.  Google Scholar [7] F. Hartung and J. Turi, Linearized stability in functional differential equations with state-dependent delays, Discrete and Continuous Dynamical Systems Supplements, Special (2001), 416-425. Google Scholar [8] M. L. Hbid, E. Sánchez and R. Bravo de la Parra, State-dependent delays associated to threshold phenomena in structured population dynamics, Mathematical Models and Methods in Applied Sciences, 17 (2007), 877-900. doi: 10.1142/S0218202507002145.  Google Scholar [9] K. Korvasová, Linearized Stability in Case of State-Dependent Delay: A Simple Test Example, Master's thesis, Universiteit Utrecht, 2011. Google Scholar [10] N. Kosovalic, Y. Chen and J. Wu, Algebraic-delay differential systems: Age structured population modeling, $C^0$-extendable submanifolds and linearization,, Preprint., ().   Google Scholar [11] N. Kosovalic, F. M. G. Magpantay, Y. Chen and J. Wu, Abstract algebraic-delay differential systems and age structured population dynamics, Journal of Differential Equations, 255 (2013), 593-609. doi: 10.1016/j.jde.2013.04.025.  Google Scholar [12] S. Mirrahimi, B. Perthame and J. Y. Wakano, Direct competition results from strong competition for limited resource, Journal of Mathematical Biology, 68 (2014), 931-949. doi: 10.1007/s00285-013-0659-5.  Google Scholar [13] A. M. de Roos and L. Persson, Population and Community Ecology of Ontogenetic Development, Princeton University Press, Princeton, 2013. Google Scholar [14] W. M. Ruess, Linearized stability and regularity for nonlinear age-dependent population models, in Functional Analysis and Evolution Equations (eds. H. Amann, W. Arendt, M. Hieber, F. M. Neubrander, S. Nicaise and J. Below), Birkhäuser Basel, (2008), 561-576. doi: 10.1007/978-3-7643-7794-6_34.  Google Scholar
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