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Journal of Dynamics and Differential Equations

, Volume 18, Issue 4, pp 863–880 | Cite as

Discretization of Asymptotically Stable Stationary Solutions of Delay Differential Equations with a Random Stationary Delay

  • Tomás Caraballo
  • Peter E. Kloeden
  • José Real
Article

We prove the existence of a stationary random solution to a delay random ordinary differential system, which attracts all other solutions in both pullback and forwards senses. The equation contains a one-sided dissipative Lipschitz term without delay, while the random delay appears in a globally Lipschitz one. The delay function only needs to be continuous in time. Moreover, we also prove that the split implicit Euler scheme associated to the random delay differential system generates a discrete time random dynamical system, which also possesses a stochastic stationary solution with the same attracting property, and which converges to the stationary solution of the delay random differential equation pathwise as the stepsize goes to zero.

Keywords

Random delay pullback attractor stationary solution split implicit Euler scheme 

Mathematics Subject Classifications (2000)

34D45 34K20 34K25 47H20 58F39 73K70 

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Tomás Caraballo
    • 1
  • Peter E. Kloeden
    • 2
  • José Real
    • 1
  1. 1.Departamento de Ecuaciones Diferenciales y Análisis NuméricoUniversidad de SevillaSevillaSpain
  2. 2.Institut für MathematikJohann Wolfgang Goethe-UniversitätFrankfurt am MainGermany

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