By Cheng S.-Y., Choi H., Greene R.E. (eds.)
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Example text
The equality holds for all 11. 8) (but now using the continuity assumption on Ak, 1 :$ k :$ n - 1, instead of the strong wellposedness). Therefore, for each 1 :$ k :$ n - 1, 11. (t):= i 0 t (t - s)n-l (n _ 1)! 1 Basic properties = with u(k)(O) 0, 0 ~ k ~ n - 1. Analogously, we can verify that for each 1 ~ k ~ n - 1, u E E, v(t):= i t (t - s)n-l (n -1)! Sk_l(S)uds 0 satisfies r w(t):= 10 (t - s)n-2 (n _ 2)! 9) with x(k)(O) = 0, 0 ~ k ~ n - 1. In conclusion, u(·) = x(·); differentiating (n - l)-times yields Sk(t)U = lt [Sk-l(S) - Sn_l(s)Ak]uds, 1 ~k~n- 1.
E E, 1 :$ k :$ n - 2, Sk(·)11. E Ck(R+, E) (in the case of n ~ 3), then (ACPn ) is strongly wellposed. Proof. 7) n-times yields that for 11. E D n - 1, Ao i 0 t (t - s)n-l (n _ 1)! ds t; t n- 1 (n - 1)! 11. - Sn_l(t)11. - n-l Aj t Jo (t _ s)n-j-l (n _ j _ 1)! ds, t ~ o. The equality holds for all 11. 8) (but now using the continuity assumption on Ak, 1 :$ k :$ n - 1, instead of the strong wellposedness). Therefore, for each 1 :$ k :$ n - 1, 11. (t):= i 0 t (t - s)n-l (n _ 1)! 1 Basic properties = with u(k)(O) 0, 0 ~ k ~ n - 1.
So (i) is true. £, Uj = 0 (j i= k). £ E 1J(Ao), Therefore we obtain (iii). Immediately, (iv) follows from (iii). £ E E be arbitrary. We take a sequence {