Entire Solutions of Higher Order Abstract Cauchy Problems*

We consider the abstract Cauchy problem for the higher order differential equation ny1 u n. t.q Aiu i. t.s0, tG0 is0 ). u j. 0.suj, 0FjFny1, where A0, . . . , Any1 are closed linear operators in a complex Banach space E. It is shown that if there exist u g 0,pr2., fg yp,px, r)0 and a positive integer h such that lnRl and lyhAkRl 0FkFny1.are bounded in l gC;