By Jack K. Hale, Luis T. Magalhães, Waldyr M. Oliva (auth.)
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Extra info for An Introduction to Infinite Dimensional Dynamical Systems — Geometric Theory
N sufficiently large one may define a function G(I5 Nxt ) = y'(t) yet) = g(x t ) •.. '~J for some e lR} by as a Ck g ejV. 1-4)). 4, taking has finite codimension in CO. be a basis for a linear complement of range(A - I). Let By the argument of the preceding paragraph, one can get the value of the solution i y t* of yet) = f'(xt)Yt + gi(xt ), yo = 0, arbitrarily close to choosing the i gl'· •. ,gJ appropriately in ~ can be made so close that Yt* J in the definition of f' (xt)Y t + g(x t ). as ro Thus, for oi ro i = Yti *, i form a basis for a r = 0nx*t* (tn* ....
The principal results are applicable not only to RFDE's but to the abstract dynamical systems considered in Section 1. K be a topological space. Let if there exists an integer n We say that such that, for every open covering K, there exists another open covering point of K is finite dimensional n+l K belongs to at most refining ~, sets of ~'. mens ion of K, dim K, is defined as the minimum n perty. Then =n dimmn and, if ~ of ~ such that every In this case, the disatisfying this pro- K is a compact finite dimensional space, it is homeomorphic to a subset of mn with n =2 dim K + 1.
B*. The set K is compact. It i ~i i=O 1r is then easy to show that ~ B* c K for n ~ NCB*) and ~tH c ~tHO c K nr -48- for t ~ (N(B*) N(H))r. + Consequently, the compact set K attracts all o compact sets of C Applying the above argument to the compact set ~tK c K for t ~ (N(K) + N(B*))r. Therefore Clearly, ,/ is compact and ,/ c w(K). t. there are sequences as ger ... Since 00 i J as j... w(K) c K. and 00 one can find a subsequence of proves w(K) ~ir1/li = 1/1 Let,/ = nn>O ~nrK. On the other hand, i f 1/1 E w(K) (j).