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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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
K. Korvasová, Linearized Stability in Case of State-Dependent Delay: A Simple Test Example, Master's thesis, Universiteit Utrecht, 2011. |
[10] |
N. Kosovalic, Y. Chen and J. Wu, Algebraic-delay differential systems: Age structured population modeling, $C^0$-extendable submanifolds and linearization, Preprint. |
[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. |
[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. |
[13] |
A. M. de Roos and L. Persson, Population and Community Ecology of Ontogenetic Development, Princeton University Press, Princeton, 2013. |
[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. |
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), 1023-1069.
doi: 10.1137/060659211. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
K. Korvasová, Linearized Stability in Case of State-Dependent Delay: A Simple Test Example, Master's thesis, Universiteit Utrecht, 2011. |
[10] |
N. Kosovalic, Y. Chen and J. Wu, Algebraic-delay differential systems: Age structured population modeling, $C^0$-extendable submanifolds and linearization, Preprint. |
[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. |
[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. |
[13] |
A. M. de Roos and L. Persson, Population and Community Ecology of Ontogenetic Development, Princeton University Press, Princeton, 2013. |
[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. |
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