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 & Continuous Dynamical Systems - A, 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. Google Scholar

[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. Google Scholar

[3]

Tomás Caraballo, José A. Langa and James C. Robinson, Attractors for differential equations with variable delays,, J. Math. Anal. Appl., 260 (2001), 421. Google Scholar

[4]

Huabin Chen, Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays,, Statist. Probab. Lett., 80 (2010), 50. Google Scholar

[5]

Hans Crauel and Franco Flandoli, Attractors for random dynamical systems,, Probab. Theory Related Fields, 100 (1994), 365. Google Scholar

[6]

Jack K. Hale, "Asymptotic Behavior of Dissipative Systems,", Mathematical Surveys and Monographs, 25 (1988). Google Scholar

[7]

Jack K. Hale and Sjoerd M. Verduyn Lunel, "Introduction to Functional-Differential Equations,", volume 99 of Applied Mathematical Sciences. Springer-Verlag, (1993). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems,, Stoch. Dyn., 3 (2003), 101. Google Scholar

[12]

Tibor Krisztin, Global dynamics of delay differential equations,, Periodica Mathematica Hungarica, 56 (2008), 83. Google Scholar

[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, (1999). Google Scholar

[14]

Michal Křížek, Numerical experience with the finite speed of gravitational interaction,, Math. Comput. Simulation, 50 (1999), 237. Google Scholar

[15]

Yang Kuang, "Delay Differential Equations with Applications in Population Dynamics,", volume 191 of Mathematics in Science and Engineering. Academic Press Inc., (1993). Google Scholar

[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. Google Scholar

[17]

Roger D. Nussbaum, Differential-delay equations with two time lags,, Mem. Amer. Math. Soc., 16 (1978). Google Scholar

[18]

Roger D. Nussbaum, Functional differential equations,, in, 2 (2002), 461. Google Scholar

[19]

Martin Rasmussen, "Attractivity and Bifurcation for Nonautonomous Dynamical Systems,", volume 1907 of Lecture Notes in Mathematics. Springer, (1907). Google Scholar

[20]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations,, in, (1992), 185. Google Scholar

[21]

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

[22]

George R. Sell, Nonautonomous differential equations and topological dynamics. II. Limiting equations,, Trans. Amer. Math. Soc., 127 (1967), 263. Google Scholar

[23]

H. O. Walther, Dynamics of delay differential equations,, in, 205 (2006), 411. Google Scholar

show all references

References:
[1]

T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems,, Nonlinear Anal., 64 (2006), 484. Google Scholar

[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. Google Scholar

[3]

Tomás Caraballo, José A. Langa and James C. Robinson, Attractors for differential equations with variable delays,, J. Math. Anal. Appl., 260 (2001), 421. Google Scholar

[4]

Huabin Chen, Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays,, Statist. Probab. Lett., 80 (2010), 50. Google Scholar

[5]

Hans Crauel and Franco Flandoli, Attractors for random dynamical systems,, Probab. Theory Related Fields, 100 (1994), 365. Google Scholar

[6]

Jack K. Hale, "Asymptotic Behavior of Dissipative Systems,", Mathematical Surveys and Monographs, 25 (1988). Google Scholar

[7]

Jack K. Hale and Sjoerd M. Verduyn Lunel, "Introduction to Functional-Differential Equations,", volume 99 of Applied Mathematical Sciences. Springer-Verlag, (1993). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems,, Stoch. Dyn., 3 (2003), 101. Google Scholar

[12]

Tibor Krisztin, Global dynamics of delay differential equations,, Periodica Mathematica Hungarica, 56 (2008), 83. Google Scholar

[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, (1999). Google Scholar

[14]

Michal Křížek, Numerical experience with the finite speed of gravitational interaction,, Math. Comput. Simulation, 50 (1999), 237. Google Scholar

[15]

Yang Kuang, "Delay Differential Equations with Applications in Population Dynamics,", volume 191 of Mathematics in Science and Engineering. Academic Press Inc., (1993). Google Scholar

[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. Google Scholar

[17]

Roger D. Nussbaum, Differential-delay equations with two time lags,, Mem. Amer. Math. Soc., 16 (1978). Google Scholar

[18]

Roger D. Nussbaum, Functional differential equations,, in, 2 (2002), 461. Google Scholar

[19]

Martin Rasmussen, "Attractivity and Bifurcation for Nonautonomous Dynamical Systems,", volume 1907 of Lecture Notes in Mathematics. Springer, (1907). Google Scholar

[20]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations,, in, (1992), 185. Google Scholar

[21]

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

[22]

George R. Sell, Nonautonomous differential equations and topological dynamics. II. Limiting equations,, Trans. Amer. Math. Soc., 127 (1967), 263. Google Scholar

[23]

H. O. Walther, Dynamics of delay differential equations,, in, 205 (2006), 411. Google Scholar

[1]

Xue-Li Song, Yan-Ren Hou. Pullback $\mathcal{D}$-attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 991-1009. doi: 10.3934/dcds.2012.32.991

[2]

Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934/dcdsb.2014.19.1801

[3]

Peter E. Kloeden, Jacson Simsen. Pullback attractors for non-autonomous evolution equations with spatially variable exponents. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2543-2557. doi: 10.3934/cpaa.2014.13.2543

[4]

Flank D. M. Bezerra, Vera L. Carbone, Marcelo J. D. Nascimento, Karina Schiabel. Pullback attractors for a class of non-autonomous thermoelastic plate systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3553-3571. doi: 10.3934/dcdsb.2017214

[5]

Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4899-4912. doi: 10.3934/dcdsb.2019036

[6]

Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195

[7]

T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037

[8]

Iacopo P. Longo, Sylvia Novo, Rafael Obaya. Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5491-5520. doi: 10.3934/dcds.2019224

[9]

Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17

[10]

Jong Yeoul Park, Jae Ug Jeong. Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2687-2702. doi: 10.3934/dcdsb.2016068

[11]

Zhijian Yang, Yanan Li. Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous kirchhoff wave models. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2629-2653. doi: 10.3934/dcds.2018111

[12]

Julia García-Luengo, Pedro Marín-Rubio, José Real, James C. Robinson. Pullback attractors for the non-autonomous 2D Navier--Stokes equations for minimally regular forcing. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 203-227. doi: 10.3934/dcds.2014.34.203

[13]

Bo You, Chengkui Zhong, Fang Li. Pullback attractors for three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1213-1226. doi: 10.3934/dcdsb.2014.19.1213

[14]

Ting Li. Pullback attractors for asymptotically upper semicompact non-autonomous multi-valued semiflows. Communications on Pure & Applied Analysis, 2007, 6 (1) : 279-285. doi: 10.3934/cpaa.2007.6.279

[15]

Lu Yang, Meihua Yang, Peter E. Kloeden. Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2635-2651. doi: 10.3934/dcdsb.2012.17.2635

[16]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194

[17]

María Anguiano, Tomás Caraballo, José Real, José Valero. Pullback attractors for reaction-diffusion equations in some unbounded domains with an $H^{-1}$-valued non-autonomous forcing term and without uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 307-326. doi: 10.3934/dcdsb.2010.14.307

[18]

Sylvia Novo, Carmen Núñez, Rafael Obaya, Ana M. Sanz. Skew-product semiflows for non-autonomous partial functional differential equations with delay. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4291-4321. doi: 10.3934/dcds.2014.34.4291

[19]

Rafael Obaya, Ana M. Sanz. Persistence in non-autonomous quasimonotone parabolic partial functional differential equations with delay. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3947-3970. doi: 10.3934/dcdsb.2018338

[20]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]