# American Institute of Mathematical Sciences

April  2013, 33(4): 1365-1374. doi: 10.3934/dcds.2013.33.1365

## Attractors for differential equations with multiple variable delays

 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 Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

Received  September 2011 Revised  November 2011 Published  October 2012

We establish some results on the existence of pullback attractors for non--autonomous delay differential equations with multiple delays. In particular, we generalise some recent works on the existence of pullback attractors for delay differential equations.
Citation: Tomás Caraballo, Gábor Kiss. Attractors for differential equations with multiple variable delays. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1365-1374. doi: 10.3934/dcds.2013.33.1365
##### References:
 [1] T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. [2] 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. [3] Tomás Caraballo, José A. Langa and James C. Robinson, Attractors for differential equations with variable delays, J. Math. Anal. Appl., 260 (2001), 421-438. [4] Huabin Chen, Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays, Statist. Probab. Lett., 80 (2010), 50-56. [5] Hans Crauel and Franco Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. [6] Jack K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. [7] Jack K. Hale and Sjoerd M. Verduyn Lunel, "Introduction to Functional-Differential Equations," volume 99 of Applied Mathematical Sciences. Springer-Verlag, New York, 1993. [8] Gábor Kiss and Bernd Krauskopf, Stability implications of delay distribution for first-order and second-order systems, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 327-345. [9] Gábor Kiss and Bernd 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. [10] Gábor Kiss and Jean-Philippe Lessard, Computational fixed point theory for differential delay equations with multiple time lags, J. Differential Equations, 252 (2012), 3093-3115. [11] P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112. [12] Tibor Krisztin, Global dynamics of delay differential equations, Periodica Mathematica Hungarica, 56 (2008) 83-95. [13] Tibor Krisztin, Hans-Otto Walther and Jianhong Wu, "Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback," volume 11 of Fields Institute Monographs. American Mathematical Society, Providence, RI, 1999. [14] Michal Křížek, Numerical experience with the finite speed of gravitational interaction, Math. Comput. Simulation, 50 (1999), 237-245. Modelling '98 (Prague). [15] Yang Kuang, "Delay Differential Equations with Applications in Population Dynamics," volume 191 of Mathematics in Science and Engineering. Academic Press Inc., Boston, MA, 1993. [16] Pedro Marín-Rubio and José Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Anal., 71 (2009), 3956-3963. [17] Roger D. Nussbaum, Differential-delay equations with two time lags, Mem. Amer. Math. Soc., 16 (1978), vi+62. [18] Roger D. Nussbaum, Functional differential equations, in "Handbook of Dynamical Systems," 2, 461-499. North-Holland, Amsterdam, 2002. [19] Martin Rasmussen, "Attractivity and Bifurcation for Nonautonomous Dynamical Systems," volume 1907 of Lecture Notes in Mathematics. Springer, Berlin, 2007. [20] 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. [21] George R. Sell, Nonautonomous differential equations and topological dynamics. I. The basic theory, Trans. Amer. Math. Soc., 127 (1967), 241-262. [22] George R. Sell, Nonautonomous differential equations and topological dynamics. II. Limiting equations, Trans. Amer. Math. Soc., 127 (1967), 263-283. [23] 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.

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##### References:
 [1] T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. [2] 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. [3] Tomás Caraballo, José A. Langa and James C. Robinson, Attractors for differential equations with variable delays, J. Math. Anal. Appl., 260 (2001), 421-438. [4] Huabin Chen, Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays, Statist. Probab. Lett., 80 (2010), 50-56. [5] Hans Crauel and Franco Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. [6] Jack K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. [7] Jack K. Hale and Sjoerd M. Verduyn Lunel, "Introduction to Functional-Differential Equations," volume 99 of Applied Mathematical Sciences. Springer-Verlag, New York, 1993. [8] Gábor Kiss and Bernd Krauskopf, Stability implications of delay distribution for first-order and second-order systems, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 327-345. [9] Gábor Kiss and Bernd 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. [10] Gábor Kiss and Jean-Philippe Lessard, Computational fixed point theory for differential delay equations with multiple time lags, J. Differential Equations, 252 (2012), 3093-3115. [11] P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112. [12] Tibor Krisztin, Global dynamics of delay differential equations, Periodica Mathematica Hungarica, 56 (2008) 83-95. [13] Tibor Krisztin, Hans-Otto Walther and Jianhong Wu, "Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback," volume 11 of Fields Institute Monographs. American Mathematical Society, Providence, RI, 1999. [14] Michal Křížek, Numerical experience with the finite speed of gravitational interaction, Math. Comput. Simulation, 50 (1999), 237-245. Modelling '98 (Prague). [15] Yang Kuang, "Delay Differential Equations with Applications in Population Dynamics," volume 191 of Mathematics in Science and Engineering. Academic Press Inc., Boston, MA, 1993. [16] Pedro Marín-Rubio and José Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Anal., 71 (2009), 3956-3963. [17] Roger D. Nussbaum, Differential-delay equations with two time lags, Mem. Amer. Math. Soc., 16 (1978), vi+62. [18] Roger D. Nussbaum, Functional differential equations, in "Handbook of Dynamical Systems," 2, 461-499. North-Holland, Amsterdam, 2002. [19] Martin Rasmussen, "Attractivity and Bifurcation for Nonautonomous Dynamical Systems," volume 1907 of Lecture Notes in Mathematics. Springer, Berlin, 2007. [20] 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. [21] George R. Sell, Nonautonomous differential equations and topological dynamics. I. The basic theory, Trans. Amer. Math. Soc., 127 (1967), 241-262. [22] George R. Sell, Nonautonomous differential equations and topological dynamics. II. Limiting equations, Trans. Amer. Math. Soc., 127 (1967), 263-283. [23] 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.
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