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Exponential attractors in Banach spaces. (English) Zbl 1040.37069
Let E be a Banach space, UE an open set and S:UE a C 1 -map. The authors consider the discrete dynamical system (DS) {S n } n=1 generated by S, extending the theory of exponential attractors from such DS in Hilbert space [A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for dissipative evolution equations, Research in Applied Mathematics 37, Chichester: Wiley, Paris: Masson (1994; Zbl 0842.58056)] on Banach spaces. The following requirements are postulated: 1. the semiflow is C 1 in some absorbing ball, and 2. the linearized semiflow at every point inside the absorbing ball is splitting into the sum of a compact operator plus a contraction.
MSC:
37L30Attractors and their dimensions, Lyapunov exponents
35B41Attractors (PDE)
35Q30Stokes and Navier-Stokes equations
47H20Semigroups of nonlinear operators