© Hindawi Publishing Corp. ON A HIGHER-ORDER EVOLUTION EQUATION WITH A STEPANOV-BOUNDED SOLUTION

We study strong solutions u: R → X, a Banach space X, of the nth-order evolution equa-tion u(n) −Au(n−1) = f, an infinitesimal generator of a strongly continuous group A: D(A) ⊆ X → X, and a given forcing term f: R → X. It is shown that if X is reflexive, u and u(n−1) are Stepanov-bounded, and f is Stepanov almost periodic, then u and all deriva-tives u′,...,u(n−1) are strongly almost periodic. In the case of a general Banach space X, a corresponding result is obtained, proving weak almost periodicity of u, u′,...,u(n−1). 2000 Mathematics Subject Classification: 34G10, 34C27, 47D03