Unstable invariant manifolds for a nonautonomous differential equation with nonautonomous unbounded delay

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August 2013, 18(6): 1663-1681. doi: 10.3934/dcdsb.2013.18.1663

Unstable invariant manifolds for a nonautonomous differential equation with nonautonomous unbounded delay

Arne Ogrowsky ^1, and Björn Schmalfuss ^2,

1.	Institut für Mathematik, Universität Paderborn, Warburger Straße 100, 33098 Paderborn, Germany
2.	Institut für Stochastik, Friedrich Schiller Universität Jena, Ernst Abbe Platz 2, 07737 Jena, Germany

Received January 2012 Revised March 2012 Published March 2013

Abstract
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In this paper we deal with a nonautonomous differential equation with a nonautonomous delay. The aim is to establish the existence of an unstable invariant manifold to this differential equation for which we use the Lyapunov-Perron transformation. However, the delay is assumed to be unbounded which makes it necessary to use nonclassical methods.

Keywords: nonautonomous delay, cocycles., Unstable invariant manifolds, nonautonomous differential equations.

Mathematics Subject Classification: Primary: 34C45, 34F05; Secondary: 37B5.

Citation: Arne Ogrowsky, Björn Schmalfuss. Unstable invariant manifolds for a nonautonomous differential equation with nonautonomous unbounded delay. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1663-1681. doi: 10.3934/dcdsb.2013.18.1663

References:

[1]	Ludwig Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/BFb0095238.
[2]	Tomás Caraballo, Jinqiao Duan, Kening Lu and Björn Schmalfuß, Invariant manifolds for random and stochastic partial differential equations, Adv. Nonlinear Stud., 10 (2010), 23-52.
[3]	Tomás Caraballo, María J. Garrido-Atienza, Björn Schmalfuß and José Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443. doi: 10.3934/dcds.2008.21.415.
[4]	Tomás Caraballo, Peter E. Kloeden and José Real, Discretization of asymptotically stable stationary solutions of delay differential equations with a random stationary delay, J. Dynam. Differential Equations, 18 (2006), 863-880. doi: 10.1007/s10884-006-9022-5.
[5]	Carmen Chicone and Yuri Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," Mathematical Surveys and Monographs, 70, American Mathematical Society, Providence, RI, 1999.
[6]	Igor D. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems," AKTA, Kharkiv, 2002.
[7]	Igor D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dynam. Differential Equations, 13 (2001), 355-380. doi: 10.1023/A:1016684108862.
[8]	Thai S. Doan and Stefan Siegmund, Differential equations with random delay,, Infinite dimensional dynamical systems., ().
[9]	María J. Garrido-Atienza, Kening Lu and Björn Schmalfuß, Unstable invariant manifolds for stochastic PDEs driven by a fractional Brownian motion, J. Differential Equations, 248 (2010), 1637-1667. doi: 10.1016/j.jde.2009.11.006.
[10]	María J. Garrido-Atienza, Arne Ogrowsky and Björn Schmalfuß, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388. doi: 10.1142/S0219493711003358.
[11]	Salah-Eldin A. Mohammed, Tusheng Zhang and Huaizhong Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Mem. Amer. Math. Soc., 196 (2008), vi+105 pp.

show all references

References:

[1]	Ludwig Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/BFb0095238.
[2]	Tomás Caraballo, Jinqiao Duan, Kening Lu and Björn Schmalfuß, Invariant manifolds for random and stochastic partial differential equations, Adv. Nonlinear Stud., 10 (2010), 23-52.
[3]	Tomás Caraballo, María J. Garrido-Atienza, Björn Schmalfuß and José Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443. doi: 10.3934/dcds.2008.21.415.
[4]	Tomás Caraballo, Peter E. Kloeden and José Real, Discretization of asymptotically stable stationary solutions of delay differential equations with a random stationary delay, J. Dynam. Differential Equations, 18 (2006), 863-880. doi: 10.1007/s10884-006-9022-5.
[5]	Carmen Chicone and Yuri Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," Mathematical Surveys and Monographs, 70, American Mathematical Society, Providence, RI, 1999.
[6]	Igor D. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems," AKTA, Kharkiv, 2002.
[7]	Igor D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dynam. Differential Equations, 13 (2001), 355-380. doi: 10.1023/A:1016684108862.
[8]	Thai S. Doan and Stefan Siegmund, Differential equations with random delay,, Infinite dimensional dynamical systems., ().
[9]	María J. Garrido-Atienza, Kening Lu and Björn Schmalfuß, Unstable invariant manifolds for stochastic PDEs driven by a fractional Brownian motion, J. Differential Equations, 248 (2010), 1637-1667. doi: 10.1016/j.jde.2009.11.006.
[10]	María J. Garrido-Atienza, Arne Ogrowsky and Björn Schmalfuß, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388. doi: 10.1142/S0219493711003358.
[11]	Salah-Eldin A. Mohammed, Tusheng Zhang and Huaizhong Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Mem. Amer. Math. Soc., 196 (2008), vi+105 pp.

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