International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 72, Pages 3959-3964
doi:10.1155/S0161171204306277
On a higher-order evolution equation with a Stepanov-bounded solution
Department of Computer Science, Vanier College, 821 Avenue Ste Croix, St. Laurent H4L 3X9, Quebec, Canada
Received 12 June 2003; Revised 5 August 2004
Copyright © 2004 Aribindi Satyanarayan Rao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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).