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By Jack K. Hale

ISBN-10: 0821849344

ISBN-13: 9780821849347

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Without loss in generality suppose B is closed. 4 imply that each Ax exists. The hypothesis that B attracts compact sets implies that Ax C B and V = UA€A A\ is bounded. Also, the hypothesis of collectively asymptotically smooth implies that V is precompact. To prove the upper semicontinuity of Ax, we must show that if {Xj,j > 1} C A, {xj,j > 1} C X are sequences such that x3- G Axj for each j \ Xj —• A0, Xj —• XQ as j —• oo, then XQ G AX0. Since AXJ is invariant under Tx5, there is a y3 G AXJ such that Xj = Tx^j.

1 makes use of the following result of Horn [1970] which is stated without proof. LEMMA 2 . 6 . 4 . Let So C Si C 52 be convex subsets of a Banach space X with So, S2 c°mPact and S\ open in S2. Let T: S2 —• X be a continuous map such that, for some integer m > 0, T3'(Si) C S 2 , 0 < j < m - I, Tj(Si) C S 0 , m < j < 2m — 1. Then T has a fixed point. We need the following result. LEMMA 2 . 6 . 5 . Suppose K C B C S C X are convex subsets of a Banach space X with K compact, S closed and bounded, and B open in S.

The set M = {(£, x): x = u(t, 0, xo), x0 E Ao, t E R} C R x X is invariant and a global attractor. PROOF. 6. Since the process is u;-periodic, the remaining assertions will follow if we show that Aa = U(a, 0)Ao for 0 < o < UJ. Since U(U,0)AQ = Ao and U(- a, a)Ufa 0)A 0 = Ufa LU)U{UJ, 0 ) A 0 = Ufau)Ao = Ufa0)AQ and U(a,0)Ao is invariant under Ta But it is clear that £/(cr, 0)AQ attracts bounded sets under Ta since A 0 attracts bounded sets under TQ.

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Asymptotic Behavior of Dissipative Systems by Jack K. Hale


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