# American Institute of Mathematical Sciences

September  2013, 18(7): 1793-1804. doi: 10.3934/dcdsb.2013.18.1793

## Attractivity for neutral functional differential equations

 1 Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla 2 Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, United Kingdom

Received  February 2013 Revised  March 2013 Published  May 2013

We study the long term dynamics of non-autonomous functional differential equations. Namely, we establish existence results on pullback attractors for non-linear neutral functional differential equations with time varying delays. The two main results differ in smoothness properties of delay functions.
Citation: Tomás Caraballo, Gábor Kiss. Attractivity for neutral functional differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1793-1804. doi: 10.3934/dcdsb.2013.18.1793
##### References:
 [1] T. Caraballo and G. Kiss, Attractors for differential equations with multiple variable delay, Discrete Contin. Dyn. Syst., 33 (2013), 1365-1374. doi: 10.3934/dcds.2013.33.1365. [2] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis, 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111. [3] T. Caraballo, P. Marín-Rubio and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 9-41. doi: 10.1016/j.jde.2003.09.008. [4] Tomás Caraballo, José A. Langa and James C. Robinson, Attractors for differential equations with variable delays, J. Math. Anal. Appl., 260 (2001), 421-438. doi: 10.1006/jmaa.2000.7464. [5] T. Caraballo, J. Real and T. Taniguchi, The exponential stability of neutral stochastic delay partial differential equations, Discrete Contin. Dyn. Syst., 18 (2007), 295-313. doi: 10.3934/dcds.2007.18.295. [6] H. Chen, Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays, Statist. Probab. Lett., 80 (2010), 50-56. doi: 10.1016/j.spl.2009.09.011. [7] J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. [8] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," 99 of Applied Mathematical Sciences. Springer-Verlag, New York, 1993. [9] G. Kiss and B. Krauskopf, Stability implications of delay distribution for first-order and second-order systems, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 327-345. doi: 10.3934/dcdsb.2010.13.327. [10] G. Kiss and B. Krauskopf, Stabilizing effect of delay distribution for a class of second-order systems without instantaneous feedback, Dynamical Systems: An International Journal, 26 (2011), 85-101. doi: 10.1080/14689367.2010.523889. [11] G. Kiss and J.-P. Lessard, Computational fixed point theory for differential delay equations with multiple time lags, Journal of Differential Equations, 252 (2012), 3093-3115. doi: 10.1016/j.jde.2011.11.020. [12] P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112. doi: 10.1142/S0219493703000632. [13] P. Kloeden and M. Rasmussen, "Nonautonomous Dynamical Systems," Mathematical Surveys and Monographs, 176, American Mathematical Society, Providence, RI, 2011. [14] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," 191 of Mathematics in Science and Engineering. Academic Press Inc., Boston, MA, 1993. [15] P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Anal., 71 (2009), 3956-3963. doi: 10.1016/j.na.2009.02.065. [16] R. D. Nussbaum, Functional differential equations, in "Handbook of Dynamical Systems," 2, 461-499. North-Holland, Amsterdam, (2002). doi: 10.1016/S1874-575X(02)80031-5. [17] M. Rasmussen, "Attractivity and Bifurcation for Nonautonomous Dynamical Systems," 1907 of Lecture Notes in Mathematics. Springer, Berlin, 2007. doi: 10.1007/978-3-540-71225-1. [18] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, in "International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behaviour," 185-192. Dresden, (1992). [19] G. R. Sell, Nonautonomous differential equations and topological dynamics. I. The basic theory, Trans. Amer. Math. Soc., 127 (1967), 241-262. [20] G. R. Sell, Nonautonomous differential equations and topological dynamics. II. Limiting equations, Trans. Amer. Math. Soc., 127 (1967), 263-283. doi: 10.1090/S0002-9947-1967-0212314-4. [21] H. O. Walther, Dynamics of delay differential equations, in "Delay Differential Equations and Applications," 205 of NATO Sci. Ser. II Math. Phys. Chem., 411-476. Springer, Dordrecht, (2006). doi: 10.1007/1-4020-3647-7_10. [22] J. Wu, H. Xia and B. Zhang, Topological transversality and periodic solutions of neutral functional-differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 199-220. doi: 10.1017/S0308210500027530.

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##### References:
 [1] T. Caraballo and G. Kiss, Attractors for differential equations with multiple variable delay, Discrete Contin. Dyn. Syst., 33 (2013), 1365-1374. doi: 10.3934/dcds.2013.33.1365. [2] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis, 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111. [3] T. Caraballo, P. Marín-Rubio and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 9-41. doi: 10.1016/j.jde.2003.09.008. [4] Tomás Caraballo, José A. Langa and James C. Robinson, Attractors for differential equations with variable delays, J. Math. Anal. Appl., 260 (2001), 421-438. doi: 10.1006/jmaa.2000.7464. [5] T. Caraballo, J. Real and T. Taniguchi, The exponential stability of neutral stochastic delay partial differential equations, Discrete Contin. Dyn. Syst., 18 (2007), 295-313. doi: 10.3934/dcds.2007.18.295. [6] H. Chen, Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays, Statist. Probab. Lett., 80 (2010), 50-56. doi: 10.1016/j.spl.2009.09.011. [7] J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. [8] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," 99 of Applied Mathematical Sciences. Springer-Verlag, New York, 1993. [9] G. Kiss and B. Krauskopf, Stability implications of delay distribution for first-order and second-order systems, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 327-345. doi: 10.3934/dcdsb.2010.13.327. [10] G. Kiss and B. Krauskopf, Stabilizing effect of delay distribution for a class of second-order systems without instantaneous feedback, Dynamical Systems: An International Journal, 26 (2011), 85-101. doi: 10.1080/14689367.2010.523889. [11] G. Kiss and J.-P. Lessard, Computational fixed point theory for differential delay equations with multiple time lags, Journal of Differential Equations, 252 (2012), 3093-3115. doi: 10.1016/j.jde.2011.11.020. [12] P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112. doi: 10.1142/S0219493703000632. [13] P. Kloeden and M. Rasmussen, "Nonautonomous Dynamical Systems," Mathematical Surveys and Monographs, 176, American Mathematical Society, Providence, RI, 2011. [14] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," 191 of Mathematics in Science and Engineering. Academic Press Inc., Boston, MA, 1993. [15] P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Anal., 71 (2009), 3956-3963. doi: 10.1016/j.na.2009.02.065. [16] R. D. Nussbaum, Functional differential equations, in "Handbook of Dynamical Systems," 2, 461-499. North-Holland, Amsterdam, (2002). doi: 10.1016/S1874-575X(02)80031-5. [17] M. Rasmussen, "Attractivity and Bifurcation for Nonautonomous Dynamical Systems," 1907 of Lecture Notes in Mathematics. Springer, Berlin, 2007. doi: 10.1007/978-3-540-71225-1. [18] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, in "International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behaviour," 185-192. Dresden, (1992). [19] G. R. Sell, Nonautonomous differential equations and topological dynamics. I. The basic theory, Trans. Amer. Math. Soc., 127 (1967), 241-262. [20] G. R. Sell, Nonautonomous differential equations and topological dynamics. II. Limiting equations, Trans. Amer. Math. Soc., 127 (1967), 263-283. doi: 10.1090/S0002-9947-1967-0212314-4. [21] H. O. Walther, Dynamics of delay differential equations, in "Delay Differential Equations and Applications," 205 of NATO Sci. Ser. II Math. Phys. Chem., 411-476. Springer, Dordrecht, (2006). doi: 10.1007/1-4020-3647-7_10. [22] J. Wu, H. Xia and B. Zhang, Topological transversality and periodic solutions of neutral functional-differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 199-220. doi: 10.1017/S0308210500027530.
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