Discrete convergence and the equivalence of equi-attraction and the continuous convergence of attractors
by Peter Kloeden, Sergey Piskarev
International Journal of Dynamical Systems and Differential Equations (IJDSDE), Vol. 1, No. 1, 2007

Abstract: The equi-attraction of global attractors A_n of dynamical systems S_n(•) in Banach spaces (E_n, ''•''E_n) as n → ∞ is shown to be equivalent to the continuous convergence of the A_n defined in terms of the concept of discrete convergence. The results are applied to general approximation schemes for abstract semilinear parabolic problems of the form u′(t) = Au(t) + f(u(t)); where A generates an exponentially decaying compact analytic C₀-semigroup in a Banach space E and f(•): E^α → E is globally Lipschitz and bounded, E^α being a subspace of E associated with a fractional power of the operator A. The main assumption on the approximation is the compact convergence of resolvents.

Online publication date: Wed, 23-May-2007

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