By Jack K. Hale, Luis T. Magalhães, Waldyr M. Oliva (auth.)
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Extra resources for An Introduction to Infinite Dimensional Dynamical Systems — Geometric Theory
5), jective. 13) -40- for ~ ° given. 14) e< 0. 15 ) is any function of period t*. 16) t*-periodic continuC([-t*,O], mn) the constant function with value to each element e< 1/11 (0). assign Then t*-periodic} is equal to Since (A-L)I [-t*,O] t* > 1) and is compact (for the same reason that A is when (A-I)I [-t*,O] it follows that is an isomorphism identifying V has finite codimension in CO' we can proceed as for t* > 1 to get &! is surjective, implying that p) f--:;oo (/I -I)1ji + fa Cp and Noting that ~l(-t*) gl"" ,gJ E:r such that the map surjective and finishing the proof of the lemma.
Is the £-neighborhood of J. If T: X + X is continuous and there is a compact set which attracts compact sets of X and J = nn K TnK, then (i) J is independent of K; (ii) J is maximal, cOmpact, invariant; (iii) J is stable and attracts compact sets of X. If, in addition, T is asymptotically smooth, then (iv) for any compact set H =X, there is a neighborhood H such that is bounded and J particular, J is uniformly asymptotically stable. 8 and, in addition, gives more information about the strong attractivity properties of the set J.
If then fl(F) fl(F) is invariant. 1. 7. If FEEt closed and, moreover, if ~r is an RFDE on a compact manifold M,~ fI(F)!!. is one-to-one on A(F), then fI(F) is invariant. Most of the results in this section are valid in a more abstract setting. We state the results without proof, for maps, and the extension to flows is easy to accomplish. Throughout the discussion is continuous. 8. is the £-neighborhood of J. If T: X + X is continuous and there is a compact set which attracts compact sets of X and J = nn K TnK, then (i) J is independent of K; (ii) J is maximal, cOmpact, invariant; (iii) J is stable and attracts compact sets of X.