By Jack K. Hale

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1 . / / the family {TA,A G A} is collectively (3-condensing, then it is collectively asymptotically smooth. PROOF. 5, each T\ is asymptotically smooth. Suppose B is a nonempty closed bounded set, TxB C B for each A G A, and let Kx be a compact set in B that attracts B under TA. 6 to the space B and obtain a compact attractor in B which attracts B under T\. Thus, we can take Kx to be this compact attractor. In particular, T A ^ A = K\. If V = U A G A ^ ' t n e n ^ *s bounded. Also Kx invariant under Tx implies 0(V) = 0 ((J TXKX ] < (5 ((J Txv) .

DISSIPATIVENESS IN TWO SPACES 29 COROLLARY 2 . 9 . 2 . 1 is satisfied, and T is an ex-contraction in X\, then there exists a connected global attractor A in X\. Furthermore, there is a fixed point of T in Xi. Under a slightly stronger hypothesis on the map T, we can prove the following result. THEOREM 2 . 9 . 3 . Suppose (Hi),(H2), and (H3) are satisfied and S is also a contraction on X2. / / either (i) X\ is dense in X2 or (ii) U: X2 —> X\ is such that, if B and U(B) are bounded in X2, then U(B) is bounded in X±, then U is conditionally completely continuous on X2 and T is a conditional acontraction on X2.

2 for continuous semigroups is LEMMA 3 . 2 . 6 . Suppose T(i) = S{t) + U(t) where U(t) is completely continuous for t > 0 and there is a continuous function k: i ? + x i ? + —• R+ such that k(t,r) -> 0 as t - • oo and \S{t)x\ < k(t,r) for t > 0, \x\ < r. Then T(t) is asymptotically smooth. 3. Stability of invariant sets. Suppose T(t): X —• X is a C r -semigroup for some r > 0 and J is an invariant set. The set J is stable if, for any neighborhood V of J , there is a neighborhood U of J such that T(<)(7 C V for all t > 0.