# 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 - A, 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.  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.  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, (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.  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.  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, 3 (2006), 435.  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.   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.  doi: 10.1142/S0218202507002145.  Google Scholar [9] K. Korvasová, Linearized Stability in Case of State-Dependent Delay: A Simple Test Example,, Master's thesis, (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.  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.  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, (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, (2008), 561.  doi: 10.1007/978-3-7643-7794-6_34.  Google Scholar

show all references

##### 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.  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.  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, (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.  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.  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, 3 (2006), 435.  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.   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.  doi: 10.1142/S0218202507002145.  Google Scholar [9] K. Korvasová, Linearized Stability in Case of State-Dependent Delay: A Simple Test Example,, Master's thesis, (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.  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.  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, (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, (2008), 561.  doi: 10.1007/978-3-7643-7794-6_34.  Google Scholar
 [1] Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103 [2] Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074 [3] Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110 [4] Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048 [5] Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107 [6] Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305 [7] Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321 [8] Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial & Management Optimization, 2021, 17 (1) : 317-337. doi: 10.3934/jimo.2019113 [9] Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341 [10] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468 [11] Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461 [12] Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467 [13] Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454 [14] Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443 [15] Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158 [16] Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107 [17] Tien-Yu Lin, Bhaba R. Sarker, Chien-Jui Lin. An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts. Journal of Industrial & Management Optimization, 2021, 17 (1) : 467-484. doi: 10.3934/jimo.2020043 [18] Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249 [19] Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462 [20] Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, 2021, 20 (1) : 359-368. doi: 10.3934/cpaa.2020270

2019 Impact Factor: 1.338

## Metrics

• PDF downloads (69)
• HTML views (0)
• Cited by (5)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]