Almost periodicity of mild solutions of inhomogeneous periodic cauchy problems

Journal article

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. © 1999 Academic Press.

Authors: Batty, C; Hutter, W; Rabiger, F

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: JOURNAL OF DIFFERENTIAL EQUATIONS
• Volume: 156
• Issue: 2
• Pages: 309-327
• Publication date: 1999-08-10
• DOI: 10.1006/jdeq.1998.3610
• ISSN: 0022-0396
• Language: English
• Keywords: periodic; countable; inhomogeneous; evolution Family; Cauchy problem; monodromy operator; almost periodic; spectrum; totally ergodic