Title: Discrete convergence and the equivalence of equi-attraction and the continuous convergence of attractors

Authors: Peter Kloeden, Sergey Piskarev

Addresses: FB Mathematik, Johann Wolfgang Goethe Universitat, D-60054 Frankfurt am Main, Germany. ' Scientific Research Computer Center, Moscow State University, Vorobjevy Gory, Moscow 119899, Russia

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.

Keywords: discrete convergence; dynamical systems; abstract parabolic equations; general approximation schemes; compact convergence; resolvents; upper semicontinuity; lower semicontinuity; attractors.

DOI: 10.1504/IJDSDE.2007.013743

International Journal of Dynamical Systems and Differential Equations, 2007 Vol.1 No.1, pp.38 - 43

Published online: 23 May 2007 *

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