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

Attractivity for neutral functional differential equations

Tomás Caraballo 1, and  Gábor Kiss 2,

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.
Keywords: pullback attractor, Non-autonomous dynamics, neutral equation..
Mathematics Subject Classification: Primary: 37C30; Secondary: 34D45, 34K2.
Citation: Tomás Caraballo, Gábor Kiss. Attractivity for neutral functional differential equations. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[8]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," 99 of Applied Mathematical Sciences. Springer-Verlag, New York, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112. doi: 10.1142/S0219493703000632.  Google Scholar

[13]

P. Kloeden and M. Rasmussen, "Nonautonomous Dynamical Systems," Mathematical Surveys and Monographs, 176, American Mathematical Society, Providence, RI, 2011.  Google Scholar

[14]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," 191 of Mathematics in Science and Engineering. Academic Press Inc., Boston, MA, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[19]

G. R. Sell, Nonautonomous differential equations and topological dynamics. I. The basic theory, Trans. Amer. Math. Soc., 127 (1967), 241-262.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[8]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," 99 of Applied Mathematical Sciences. Springer-Verlag, New York, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112. doi: 10.1142/S0219493703000632.  Google Scholar

[13]

P. Kloeden and M. Rasmussen, "Nonautonomous Dynamical Systems," Mathematical Surveys and Monographs, 176, American Mathematical Society, Providence, RI, 2011.  Google Scholar

[14]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," 191 of Mathematics in Science and Engineering. Academic Press Inc., Boston, MA, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[19]

G. R. Sell, Nonautonomous differential equations and topological dynamics. I. The basic theory, Trans. Amer. Math. Soc., 127 (1967), 241-262.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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