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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

Igor D. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems," AKTA, Kharkiv, 2002. Google Scholar

[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.  Google Scholar

[8]

Thai S. Doan and Stefan Siegmund, Differential equations with random delay,, Infinite dimensional dynamical systems., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

Ludwig Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/BFb0095238.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

Igor D. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems," AKTA, Kharkiv, 2002. Google Scholar

[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.  Google Scholar

[8]

Thai S. Doan and Stefan Siegmund, Differential equations with random delay,, Infinite dimensional dynamical systems., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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