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We consider a mild solution u of a well-posed, inhomogeneous, Cauchy problem, u* (t)=A(t) u(t)+ f (t), on a Banach space X, where A( } ) is periodic. For a problem on R + , we show that u is asymptotically almost periodic if f is asymptotically almost periodic, u is bounded, uniformly continuous and totally ergodic, and the spectrum of the monodromy operator V contains only countably many points of the unit circle. For a problem on R, we show that a bounded, uniformly continuous solution u is almost periodic if f is almost periodic and various supplementary conditions are satisfied. We also show that there is a unique bounded solution subject to certain spectral assumptions on V, f and u. Academic Press

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We investigate the asymptotic properties of the inhomogeneous nonautonomous evo-is a Hille-Yosida operator on a Banach space X, B(t), t ∈ R, is a family of operators in ᏸ(D(A), X) satisfying certain boundedness and measurability conditions and f ∈ L 1 loc (R, X). The solutions of the corresponding homogeneous equations are represented by an evolution family (U B (t, s)) t≥s . For various function spaces Ᏺ we show conditions on (U B (t, s)) t≥s and f which ensure the existence of a unique solution contained in Ᏺ. In particular, if (U B (t, s)) t≥s is p-periodic there exists a unique bounded solution u subject to certain spectral assumptions on U B (p, 0), f and u. We apply the results to nonautonomous semilinear retarded differential equations. For certain p-periodic retarded differential equations we derive a characteristic equation which is used to determine the spectrum of (U B (t, s)) t≥s .