• OpenAccess
  • The Existence and Stability of Synchronizing Solution of Non-Autonomous Equations with Multiple Delays  [ICPDE 2016]
  • DOI: 10.4236/jamp.2016.47136   PP.1294 - 1299
  • Author(s)
  • Jinying Wei, Yongjun Li, Xiaohua Zhuo
  • In this paper, we consider an abstract non-autonomous evolution equation with multiple delays in a Hilbert space H:  u'(t) + Au(t) = F(u(t-r1),...,u((t-rn)) + g(t), where A: D(A)?HH is a positive definite selfadjoint operator,  F: Hna H is a nonlinear mapping,  r1,...,rn are nonnegative constants, and  g(t) C(;H) is bounded. Motivated by [1] [2], we obtain the existence and stability of synchronizing solution under some convergence condition. By this result, we provide a general approach for guaranteeing the existence and stability of periodic, quasiperiodic or almost periodic solution of the equation.

  • Pullback Attractor, Cocycle System, Stability, Synchronizing Solution
  • References
  • [1]
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    Li, D. and Duan, J. (2009) Structure of the Set of Bounded Solutions for a Class of Nonautonomous Second-Order Differential Equations. Journal of Differential Equations, 246, 1754-1773.
    Li, D., Wei, J. and Wang, J. (2013) On the Dynamics of Abstract Retarded Evolution Equations. Abstract and Applied Analysis, 2013, 9 p.
    Li, Y. (2011) Existence and Asymptotic Stability of Periodic Solution for Evolution Equations with Delays. Journal of Functional Analysis, 261, 1309-1324.
    Zhu, J., Liu, Y. and Li, Z. (2008) The Existence and Attractivity of Time Periodic Solutions for Evolution Equations with Delays. Nonlinear Analysis: Real World Applications, 9, 842-851.
    Cheban, D., Kloeden, P.E. and Schmalfuss, B. (2002) The Relationship between Pullback, Forwards and Global Attractors of Nonautonomous Dynamical Systems. Nonlinear Dynamics and Systems Theory, 2, 9-28.

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