Autonomous and non-autonomous attractors for differential equations with delays

Journal of Differential Equations

Volumn 208, Issue 1 SPEC.ISS., 2005, Pages 9-41

  Caraballo, T ^a   Marin Rubio P ^a   Valero, J ^b  

^a UNIVERSITY OF SEVILLE   (Spain)

^b UNIVERSIDAD CEU CARDENAL HERRERA   (Spain)

Author keywords

Autonomous and non autonomous (pullback) attractors; Delay differential equations; Integro differential equations; Multi valued processes; Multi valued semiflows; Non uniqueness of solutions

Indexed keywords

EID: 10944259804     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jde.2003.09.008     Document Type: Article

References (42)

1. Atkinson F.V. Haddock J.R. On determining phase spaces for functional differential equations Funkcial. Ekvac. 31 1988 331-347
2. Babin A.V. Attractor of the generalized semigroup generated by an elliptic equation in a cylindrical domain Russian Acad. Sci. Izv. Math. 44 1995 207-223 (Transl. Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994))
3. A.V. Babin, M.I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, Vol. 25, North-Holland, Amsterdam, 1992.
4. Ball J.M. Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations J. Nonlinear Sci. 7 5 1997 475-502
5. Beretta E. Kuang Y. Convergence results in well-known delayed predator-prey system J. Math. Anal. Appl. 204 1996 840-853
6. Caraballo T. Langa J.A. Melnik V.S. Valero J. Pullback attractors of non-autonomous and stochastic multi-valued dynamical systems Set-Valued Anal. 11 2 2003 153-201
7. Caraballo T. Langa J.A. Robinson J. Attractors for differential equations with variable delays J. Math. Anal. Appl. 260 2 2001 421-438
8. Caraballo T. Langa J.A. Valero J. Global attractors for multi-valued random dynamical systems generated by random differential inclusions with multiplicative noise J. Math. Anal. Appl. 260 2001 602-622
9. Caraballo T. Langa J.A. Valero J. Global attractors for multi-valued random dynamical systems Nonlinear Anal. 48 2002 805-829
10. Caraballo T. Marín-Rubio P. Robinson J. A comparison between two theories for multi-valued semiflows and their asymptotic behaviour Set-Valued Anal. 11 2003 297-322
11. Chauvet G. Vielle B. Delay equation analysis of human respiratory stability Math. Biosci. 152 1998 105-122
12. Cheban D.N. Dissipative functional differential equations Bull. Acad. Sci. Rep. Moldova Mat. 2 5 1991 3-12
13. Cheban D.N. Fakeeh D.S. Global Attractors of Disperse Dynamical Systems 1994 Sigma Chişinǎu
14. Cheban D.N. Fakeeh D.S. Global attractors of infinite-dimensional dynamical systems III Bull. Acad. Sci. Rep. Moldova Mat. 18-19 2-3 1995 3-13
15. Cheban D.N. Schmalfuss B. Global attractors of nonautonomous disperse dynamical systems and differential inclusions Bull. Acad. Sci. Rep. Moldova Mat. 29 1 1999 3-22
16. Chepyzhov V.V. Vishik M.I. A Hausdorff dimension estimate for kernel sections of non-autonomous evolution equations Indiana Univ. Math. J. 42 3 1993 1057-1076
17. Chepyzhov V.V. Vishik M.I. Evolution equations and their trajectory attractors J. Math. Pures Appl. 76 1997 913-964
18. Crauel H. Debussche A. Flandoli F. Random attractors J. Dynamics Differential Equations 9 2 1997 307-341
19. Crauel H. Flandoli F. Attractors for random dynamical systems Probab. Theory Related Fields 100 3 1994 365-393
20. J.M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomathematics, Vol. 20, Springer, Heidelberg, 1979.
21. O. Elmounir, F. Simondon, Attracteurs compacts pur des problèmes d'evolution sans unicité, Ann. Fac. Sci. Toulouse Math. Sér. 6 IX, Fasc. 4 (2000) 631-654.
22. Gurney W.S. Blythr S.P. Nisbet R.M. Nicholson's blowflies revisited Nature 287 1980 17-21
23. J.K. Hale, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Springer, New York, 1977.
24. J.K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, Vol. 25, American Mathematical Society, Providence, RI, 1988.
25. Hale J.K. Kato J. Phase space for retarded equations with infinite delay Funkcial. Ekvac. 21 1978 11-41
26. J.K. Hale, S.M.V. Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, Springer, New York, 1993.
27. Kapustyan A.V. An attractor of a semiflow generated by a system of phase-field equations without uniqueness of solutions Ukrain. Mat. Zh. 51 1999 1006-1009
28. Kapustyan A.V. Melnik V.S. Valero J. Attractors of multi-valued dynamical processes generated by phase-field equations Internat. J. Bifur. Chaos 13 2003 1969-1983
29. Kloeden P.E. Schmalfuß B. Non-autonomous systems, cocycle attractors and variable time-step discretization Numer. Algorithms 14 1-3 1997 141-152
30. Kloeden P.E. Stonier D.J. Cocycle attractors in non-autonomously perturbed differential equations Dynamic Contin. Discrete Impuls Systems 4 2 1998 211-226
31. Kuang Y. Delay Differential Equations with Applications in Population Dynamics 1993 Academic Press New York
32. O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Lincei Lectures, Vol. 25, Cambridge University Press, Cambridge, 1991.
33. Lasota A. Wazewska-Czyzewska M. Mathematical problems of the red-blood cell system Ann. Polish Math. Soc. Ser. III, Appl. Math. 6 1976 23-40
34. Mart́nez Álvarez C. Estabilidad global y acotación en ecuaciones diferenciales funcionales con retraso 2001 Tesis Doctoral Universidad de Granada
35. Mackey M.C. Glass L. Oscillation and chaos in physiological control systems Science 197 1977 287-289
36. Melnik V.S. Valero J. On attractors of multi-valued semi-flows and differential inclusions Set-Valued Anal. 6 1998 83-111
37. Murray J.D. Mathematical Biology 1993 Springer Berlin
38. Saker S.H. Oscillation and global attractivity of hematopoiesis model with delay time Appl. Math. Comput. 136 2003 27-36
39. iSchmalfuß B. Backward cocycles and attractors of stochastic differential equations in: Reitmann V. Redrich T. Kosch N.J. (Eds.) International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behaviour 1992 185-192 Technische Universität Dresden
40. Sell G. Non-autonomous differential equations and dynamical systems Amer. Math. Soc. 127 1967 241-283
41. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, Vol. 68, Springer, New York, 1997.
42. Valero J. Attractors of parabolic equations without uniqueness J. Dynamics Differential Equations 13 2001 711-744