By Jack K. Hale

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9. THEOREM 3 . 6 . 4 . If X is a Banach space, the u-periodic process is acondensing and affine, and there is a bounded trajectory of the process, then there is an u-periodic trajectory. 7. Skew product flows. In the previous section, we have discussed the u;-periodic process which includes certain types of evolutionary equations for which the vector field is uperiodic in the independent variable. There is another way to treat this problem which allows one to generalize some of the results to other types of nonautonomous process.

1) is compact. PROOF. If j = 7 + ( T ( £ n J ) ) , then J = T{B H J ) U T(J) and a ( J ) = a ( J ) = max[a(T' fl J ) , #(«/)]. If a ( J ) > 0, then T being a-condensing implies a ( T J ) < a ( J ) . In this case, a ( J ) = a(T(J3 fl J ) ) . If a(T{B D J J) > 0, then a(J) = OL{J) < a(B fl J ) < a(J) and this is a contradiction. Thus, a(J) = a( J) = 0. Since J is closed, this implies J is compact and the lemma is proved. PROOF OF THEOREM 2 . 6 . 1 . 2 and let K = co A. ) is bounded and K attracts B.

Since TJ = J and J is bounded in X 2 , it follows that U{J) is bounded in X\. Let p be chosen so that rU(J) C B = Blp. We show that J C C l £ M n X 2 . Let d,2(x,B) = i n f ^ B \x - 3/I2. Let rj = swp{d,2(x,B),x e J}. If we show that rj — 0, then J C CI B in X2. If x e J, y e B, z = Sy + Ux, then z e B since |z|i < k\y\i + |C/x|i < kp + pr~l = p. Furthermore, |Tx — z\2 < k\x — y\2, which implies that d2(Ta:, B) = inf |Tx - y\2 < inf {|T:r - *| 2 , * = Sy + t/x} 2/GB 1/6B < k inf |x — 2/I2 = kd2{x, B).

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