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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

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:
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##### 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.  Google Scholar [2] Y. Chen, Q. 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.  Google Scholar [3] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr. , 25, Amer. Math. Soc. , 1988.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [7] F. Hartung, T. Krisztin, H. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] Q. Hu, J. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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