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July 2017, 37(7): 3939-3961. doi: 10.3934/dcds.2017167

Dynamical properties of nonautonomous functional differential equations with state-dependent delay

Departamento de Matemática Aplicada, Universidad de Valladolid, Paseo del Cauce 59, 47011 Valladolid, Spain

Received  June 2016 Revised  March 2017 Published  April 2017

A type of nonautonomous n-dimensional state-dependent delay differential equation (SDDE) is studied. The evolution law is supposed to satisfy standard conditions ensuring that it can be imbedded, via the Bebutov hull construction, in a new map which determines a family of SDDEs when it is evaluated along the orbits of a flow on a compact metric space. Additional conditions on the initial equation, inherited by those of the family, ensure the existence and uniqueness of the maximal solution of each initial value problem. The solutions give rise to a skew-product semiflow which may be noncontinuous, but which satisfies strong continuity properties. In addition, the solutions of the variational equation associated to the SDDE determine the Fréchet differential with respect to the initial state of the orbits of the semiflow at the compatibility points. These results are key points to start using topological tools in the analysis of the long-term behavior of the solution of this type of nonautonomous SDDEs.

Citation: Ismael Maroto, Carmen NÚÑez, Rafael Obaya. Dynamical properties of nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3939-3961. doi: 10.3934/dcds.2017167
References:
[1]

M. V. Barbarossa and H. O. Walther, Linearized stability for a new class of neutral equations with state-dependent delay, Differ. Equ. Dyn. Syst., 24 (2016), 63-79. doi: 10.1007/s12591-014-0204-z.

[2]

Y. ChenQ. Hu and J. Wu, Second-order differentiability with respect to parameters for differential equations with adaptative delays, Front. Math. China, 5 (2010), 221-286. doi: 10.1007/s11464-010-0005-9.

[3]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr. , 25, Amer. Math. Soc. , 1988.

[4]

J. K. Hale and S. M. Verdyun Lunel, Introduction to Functional Differential Equations Appl. Math. Sciences, 99, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7.

[5]

F. Hartung, On differentiability of solutions with respect to parameters in a class of functional differential equations, Funct. Differ. Equ., 4 (1997), 65-79.

[6]

F. Hartung, Differentiability of solutions with respect to the initial data in differential equations with state-dependent delays, J. Dynam. Differential Equations, 23 (2011), 843-884. doi: 10.1007/s10884-011-9218-1.

[7]

F. HartungT. KrisztinH. O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, Handbook of Differential Equations: Ordinary Differential Equations, Elsevier, North-Holland, 3 (2006), 435-545. doi: 10.1016/S1874-5725(06)80009-X.

[8]

X. He and R. de la Llave, Construction of quasi-periodic solutions of state-dependent delay differential equations by the parameterization method Ⅱ: Analytic case, J. Differential Equations, 261 (2016), 2068-2108. doi: 10.1016/j.jde.2016.04.024.

[9]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay Lecture Notes in Math. , 1473, Springer-Verlag, 1991. doi: 10.1007/BFb0084432.

[10]

Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with state-dependent delay, J. Differential Equations, 248 (2010), 2801-2840. doi: 10.1016/j.jde.2010.03.020.

[11]

Q. HuJ. Wu and X. Zou, Estimates of periods and global continua of periodic solutions for state-dependent delay equations, SIAM J. Math. Anal., 44 (2012), 2401-2427. doi: 10.1137/100793712.

[12]

T. Krisztin and A. Rezounenko, Parabolic partial differential equations with discrete state-dependent delay: Classical solutions and solution manifold, J. Differential Equations, 260 (2016), 4454-4472. doi: 10.1016/j.jde.2015.11.018.

[13]

J. Mallet-Paret and R. D. Nussbaum, Stability of periodic solutions of state-dependent delay-differential equations, J. Differential Equations, 250 (2011), 4085-4103. doi: 10.1016/j.jde.2010.10.023.

[14]

I. Maroto, C. Núñez and R. Obaya, Exponential stability for nonautonomous functional differential equations with state-dependent delay, to appear in Discrete Contin. Dyn. Syst. , Ser. B 2016.

[15]

I. Maroto, C. Núñez and R. Obaya, Existence of global attractor for a biological neural network modellized by a nonautonomous state-dependent delay differential equation, submitted, 2016.

[16]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations Appl. Math. Sci. , 143, Springer-Verlag, 2002. doi: 10.1007/978-1-4757-5037-9.

[17]

H. O. Walther, The solution manifold and $C^1$-smoothness for differential equations with state-dependent delay, J. Differential Equations, 195 (2003), 46-65. doi: 10.1016/j.jde.2003.07.001.

[18]

H. O. Walther, Smoothness of semiflows for differential equations with state-dependent delays, J. Math. Sci., 124 (2004), 5193-5207. doi: 10.1016/j.jde.2003.07.001.

[19]

J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay Series in Nonlinear Analysis and Applications 6, Walter de Gruyter, Berlin, New York, 2001. doi: 10.1515/9783110879971.

show all references

References:
[1]

M. V. Barbarossa and H. O. Walther, Linearized stability for a new class of neutral equations with state-dependent delay, Differ. Equ. Dyn. Syst., 24 (2016), 63-79. doi: 10.1007/s12591-014-0204-z.

[2]

Y. ChenQ. Hu and J. Wu, Second-order differentiability with respect to parameters for differential equations with adaptative delays, Front. Math. China, 5 (2010), 221-286. doi: 10.1007/s11464-010-0005-9.

[3]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr. , 25, Amer. Math. Soc. , 1988.

[4]

J. K. Hale and S. M. Verdyun Lunel, Introduction to Functional Differential Equations Appl. Math. Sciences, 99, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7.

[5]

F. Hartung, On differentiability of solutions with respect to parameters in a class of functional differential equations, Funct. Differ. Equ., 4 (1997), 65-79.

[6]

F. Hartung, Differentiability of solutions with respect to the initial data in differential equations with state-dependent delays, J. Dynam. Differential Equations, 23 (2011), 843-884. doi: 10.1007/s10884-011-9218-1.

[7]

F. HartungT. KrisztinH. O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, Handbook of Differential Equations: Ordinary Differential Equations, Elsevier, North-Holland, 3 (2006), 435-545. doi: 10.1016/S1874-5725(06)80009-X.

[8]

X. He and R. de la Llave, Construction of quasi-periodic solutions of state-dependent delay differential equations by the parameterization method Ⅱ: Analytic case, J. Differential Equations, 261 (2016), 2068-2108. doi: 10.1016/j.jde.2016.04.024.

[9]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay Lecture Notes in Math. , 1473, Springer-Verlag, 1991. doi: 10.1007/BFb0084432.

[10]

Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with state-dependent delay, J. Differential Equations, 248 (2010), 2801-2840. doi: 10.1016/j.jde.2010.03.020.

[11]

Q. HuJ. Wu and X. Zou, Estimates of periods and global continua of periodic solutions for state-dependent delay equations, SIAM J. Math. Anal., 44 (2012), 2401-2427. doi: 10.1137/100793712.

[12]

T. Krisztin and A. Rezounenko, Parabolic partial differential equations with discrete state-dependent delay: Classical solutions and solution manifold, J. Differential Equations, 260 (2016), 4454-4472. doi: 10.1016/j.jde.2015.11.018.

[13]

J. Mallet-Paret and R. D. Nussbaum, Stability of periodic solutions of state-dependent delay-differential equations, J. Differential Equations, 250 (2011), 4085-4103. doi: 10.1016/j.jde.2010.10.023.

[14]

I. Maroto, C. Núñez and R. Obaya, Exponential stability for nonautonomous functional differential equations with state-dependent delay, to appear in Discrete Contin. Dyn. Syst. , Ser. B 2016.

[15]

I. Maroto, C. Núñez and R. Obaya, Existence of global attractor for a biological neural network modellized by a nonautonomous state-dependent delay differential equation, submitted, 2016.

[16]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations Appl. Math. Sci. , 143, Springer-Verlag, 2002. doi: 10.1007/978-1-4757-5037-9.

[17]

H. O. Walther, The solution manifold and $C^1$-smoothness for differential equations with state-dependent delay, J. Differential Equations, 195 (2003), 46-65. doi: 10.1016/j.jde.2003.07.001.

[18]

H. O. Walther, Smoothness of semiflows for differential equations with state-dependent delays, J. Math. Sci., 124 (2004), 5193-5207. doi: 10.1016/j.jde.2003.07.001.

[19]

J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay Series in Nonlinear Analysis and Applications 6, Walter de Gruyter, Berlin, New York, 2001. doi: 10.1515/9783110879971.

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