Asymptotic Behavior of Dissipative Systems by Jack K. Hale PDF

By Jack K. Hale

ISBN-10: 0821849344

ISBN-13: 9780821849347

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Example text

Since k(t) G [0,1) for each t > 0, the map T(si) is an a-contraction which is compact dissipative. 1, there is a fixed point X\ of T{s\). Thus, there is a complete orbit through X\ given by (j)(nsi + s) = T(s)xi,0 < s < Si,n — 0, ± 1 , ± 2 , This orbit is periodic of period Si, compact, and invariant and, therefore, must belong to the compact attractor A. Thus, x\ G A. Choose a sequence sm —> 0 as m —• oo. By the same reasoning as above, the map T(sm) has a fixed point xm G A corresponding to a periodic orbit of T(t) of period sm for all m.

For A > 0, the subspace of C(E) consisting of all maps L = L\ -f L2 with L\ compact and ||Z,21| < A is denoted by £,\{E). Given a map L G Z\(E) we define Ls = L restricted to 5 and v\{L) — min{dim5: 5 is a linear subspace of E and ||Ls|| < A}. It is easy to prove that v\{L) is finite for L G £\/2{E). The basic result for the estimate of c(K) is contained in THEOREM 2 . 8 . 2 . Let X be a Banach space, U C E an open set, T: U -+ E a C1 map, and K C U a compact set such that T(K) D K. T,|j, 0 < A < 1/2, 0 < a < (1/2A) - 1, v = SMVx^Kvx{pxT2).

Then 30 DISCRETE DYNAMICAL SYSTEMS (iv) T is point dissipative in X2. Then there is a global attractor A in X2 and T(t) is bounded dissipative in X\. If, in addition, (v) U is conditionally completely continuous on Xj, j = 1,2, then A C Xi and is a global attractor in X\. 4. As notation, let B\ = {x G Xi: \x\i < r}. We need the following lemmas. LEMMA 2 . 9 . 6 . i/(Hi),(H2), (H3) are satisfied, then, for any constant L > 0 and any A G X\ such that A and U(A) are bounded in X2, there is a constant K(L,A) such that, for anyno, 0 < n 0 < 00, and for any B C B\, ifTm(B) C A forO

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Asymptotic Behavior of Dissipative Systems by Jack K. Hale


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