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 equation 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 derivatives 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).