Random differential equations with random delays

Stochastics and Dynamics

Volumn 11, Issue 2-3, 2011, Pages 369-388

  Garrido Atienza, M J ^a   Ogrowsky, A ^b   Schmalfuss, B ^b  

^a UNIVERSITY OF SEVILLE   (Spain)

^b UNIVERSITY OF PADERBORN   (Germany)

Author keywords

random attractor; random delay; Random differential equations; random dynamical system; random fixed point

Indexed keywords

EID: 80051695353     PISSN: 02194937     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0219493711003358     Document Type: Article

References (28)

1. L. Arnold, Random Dynamical Systems (Springer, 1998).
2. P. R. Beesack, Gronwall Inequalities (Carleton Mathematical Lecture Notes, 1975).
3. T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Disc. Contin. Dyn. Syst. 21 (2008) 415-443.
4. T. Caraballo, M. J. Garrido-Atienza and B. Schmalfuss, Existence of exponentially attracting stationary solutions for delay evolution equations, Disc. Contin. Dyn. Syst. 18 (2007) 271-293.
5. T. Caraballo, P. E. Kloeden and J. Real, Discretization of asymptotically stable stationary solutions of delay differential equations with a randon stationary delay, J. Dynam. Differential Equations 18 (2006) 863-880. (Pubitemid 44569565)
6. T. Caraballo, P. Marín-Rubio and J. Valero, Attractors for differential equations with unbounded delays, J. Differential Equations 239 (2007) 311-342.
7. C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions (Springer, 1977).
8. I. D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dynam. Differential Equations 13 (2001) 355-380. (Pubitemid 33808765)
9. D. Cheban and B. Schmalfuss, Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations, J. Math. Anal. Appl. 340 (2008) 374-393.
10. I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems (AKTA, 2002).
11. K. L. Cooke, Y. Kuang and B. Li, Analyses of an antiviral immune response model with time delays, Canad. Appl. Math. Quart. 6 (1998) 321-354.
12. H. Crauel, T. S. Doan and S. Siegmund, Difference equations with random delay, J. Difference Equ. Appl. 15 (2009) 627-647.
13. T. S. Doan and S. Siegmund, Differential equations with random delay, in preparation.
14. F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stoch. Stoch. Rep. 59 (1996) 21-45.
15. M. J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Anal. 64 (2006) 1100-1118. (Pubitemid 43052539)
16. M. J. Garrido-Atienza and J. Real, Existence and uniqueness of solutions for delay evolution equations of second order in time, J. Math. Anal. Appl. 283 (2003) 582-609. (Pubitemid 36989383)
17. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations (Springer-Verlag, 1995).
18. Y. Hino, S. Murakami and T. Naito, Functional-differential Equations with Infinite Delay, Lecture Notes in Mathematics, Vol. 1473 (Springer, 1991).
19. S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis (Kluwer, 1997).
20. P. Imkeller and C. Lederer, On the cohomology of flows of stochastic and random differential equations, Probab. Theory Relat. Fields 120 (2001) 209-235.
21. P. Imkeller and B. Schmalfuss, The conjugacy of stochastic and random differential equations and the existence of global attractors, J. Dynam. Differential Equations 13 (2001) 215-249. (Pubitemid 33808761)
22. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics (Academic Press, 1993).
23. W. Mather, M. R. Bennett, J. Hasty and L. S. Tsimring, Delay-induced degradeand-fire oscillations in small genetics circuits, Phys. Rev. Lett. 102 (2009) 1-4.
24. J. D. Murray, Mathematical Biology (Springer, 1993).
25. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer, 1983).
26. B. Schmalfuss, A random fixed point theorem and the random graph transformation, J. Math. Anal. Appl. 225 (1998) 91-113. (Pubitemid 128344868)
27. B. Schmalfuss, Attractors for the non-autonomous dynamical systems, Int. Conf. Differential Equations 1-2 (1999) 684-689.
28. E. Zeidler, Nonlinear Functional Analysis and Its Applications I (Springer, 1986).