In this paperwe study a non-autonomous lattice dynamical system with delay. Under rather general growth and dissipative conditions on the nonlinear term,we define a non-autonomous dynamical system and prove the existence of a pullback attractor for such system as well. Both multivalued and single-valued cases are considered.
@article{bwmeta1.element.doi-10_1515_msds-2015-0003, author = {Tom\'as Caraballo and Francisco Morillas and Jos\'e Valero}, title = {Attractors for non-autonomous retarded lattice dynamical systems}, journal = {Nonautonomous Dynamical Systems}, volume = {2}, year = {2015}, zbl = {1329.34119}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_msds-2015-0003} }
Tomás Caraballo; Francisco Morillas; José Valero. Attractors for non-autonomous retarded lattice dynamical systems. Nonautonomous Dynamical Systems, Tome 2 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_msds-2015-0003/
[1] A.Y. Abdallah, Exponential attractors for first-order lattice dynamical systems, J. Math. Anal. Appl., 339 (2008), 217-224. | Zbl 1127.37051
[2] J.M. Amigó, A. Giménez, F. Morillas and J. Valero, Attractors for a lattice dynamical system generated by non-newtonian fluids modeling suspensions, Internat. J. Bifur. Chaos, 20 (2010), 2681-2700. [WoS] | Zbl 1202.34022
[3] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti, 1976. | Zbl 0328.47035
[4] P.W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stochastics & Dynamics, 6 (2006), no.1, 1–21. | Zbl 1105.60041
[5] P.W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos, 11 (2001), 143–153. | Zbl 1091.37515
[6] T. Caraballo, M.J. Garrido-Atienza, B. Schmalfuß and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443. | Zbl 1155.60025
[7] T. Caraballo and P. E. Kloeden, Non-autonomous attractors for integro-differential evolution equations, Discrete and Continuous Dynamical Systems Series S. , 2 (2009), 17-36. | Zbl 1185.45016
[8] T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335. [WoS][Crossref] | Zbl 1155.60324
[9] T. Caraballo, T. Lukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 263-268. | Zbl 1085.37054
[10] T. Caraballo, T. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. | Zbl 1128.37019
[11] T. Caraballo, F. Morillas and J. Valero, Random Attractors for stochastic lattice systems with non-Lipschitz nonlinearity, J. Diff. Equat. App., 17 (2011), no2, 161-184. [Crossref] | Zbl 1223.39010
[12] T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), no. 2, 667-693. [WoS] | Zbl 1248.37045
[13] T. Caraballo, F. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems, Discrete Cont. Dyn. Systems, Series A, 34 (2014), 51-77. | Zbl 1323.34087
[14] A.M. Gomaa, On existence of solutions and solution sets of differential equations and differential inclusions with delay in Banach spaces, J. Egyptian Math. Soc. (2012, to appear). | Zbl 1264.34150
[15] X. Han, Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, J. Math. Anal. Appl., 376 (2011), 481-493. [WoS] | Zbl 1209.60038
[16] X. Han, W. Shen, S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266. | Zbl 1208.60063
[17] V. Lakshmikantham, A.R. Mitchell, R.W. Mitchell, On the existence of solutions of differential equations of retarde type in a Banach space, Annales Polonici Mathematici, XXXV (1978), 253-260. | Zbl 0373.34034
[18] P.Marín-Rubio, J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Anal., 71 (2009), 3956-3963. | Zbl 1174.37016
[19] F. Morillas, J. Valero, A Peano’s theorem and attractors for lattice dynamical systems, Internat. J. Bifur. Chaos, 19 (2009), 557-578. [WoS] | Zbl 1170.37337
[20] F. Morillas, J. Valero, On the connectedness of the attainability set for lattice dynamical systems, J. Diff. Equat. App., 18 (2012), 675-692. [Crossref] | Zbl 1243.34012
[21] A. Sikorska-Nowak, Retarded functional differential equations in Banach spaces and Henstock-Kurzweil-Pettis integrals, Disscusiones Mathematicae, 27 (2007), 315-327. | Zbl 1149.34053
[22] B. Wang, Dynamics of systems of infinite lattices, J. Differential Equations, 221 (2006), 224-245. | Zbl 1085.37056
[23] B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl,. 331 (2007), 121-136. | Zbl 1112.37076
[24] W. Yan, Y. Li and Sh. Ji, Random attractors for first order stochastic retarded lattice dynamical systems, J. Math. Phys., 51 (2010), 17 pages. [WoS] | Zbl 1309.37076
[25] C. Zhao and S. Zhou, Attractors of retarded first order lattice systems, Nonlinearity, 20 (2007), no. 8, 1987-2006. [Crossref][WoS] | Zbl 1130.34053
[26] C. Zhao, Sh. Zhou, Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. Math. Anal. Appl., 354 (2009), 78–95. | Zbl 1192.37106
[27] C. Zhao, S. Zhou andW.Wang, Compact kernel sections for lattice systems with delays, Nonlinear Analysis TMA, 70 (2009), 1330-1348. | Zbl 1169.34055
[28] S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368. [WoS] | Zbl 1173.37331
[29] S. Zhou, W. Shi, Attractors and dimension of dissipative lattice systems, J. Differential Equations, 224 (2006), 172-204. | Zbl 1091.37023