Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. II

• Published in: Journal of Differential Equations 2009-02, Vol.246 (3), p.1164-1186
• Main Authors: Caraballo, Tomás, Cheban, David
• Format: Article
• Language: English
• Subjects: Almost automorphic solutions; Almost periodic solution; Cocycle; Favard's condition; Non-autonomous dynamical systems
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2008.07.025

In this paper we continue the research started in a previous paper, where we proved that the linear differential equation (1) x ′ = A ( t ) x + f ( t ) with Levitan almost periodic coefficients has a unique Levitan almost periodic solution, if it has at least one bounded solution and the bounded solutions of the homogeneous equation (2) x ′ = A ( t ) x are homoclinic to zero (i.e. lim | t | → + ∞ | φ ( t ) | = 0 for all bounded solutions φ of (2)). If the coefficients of (1) are Bohr almost periodic and all bounded solutions of Eq. (2) are homoclinic to zero, then Eq. (1) admits a unique almost automorphic solution. In this second part we first generalise this result for linear functional differential equations (FDEs) of the form (3) x ′ = A ( t ) x t + f ( t ) , as well as for neutral FDEs. Analogous results for functional difference equations with finite delay and some classes of partial differential equations are also given. We study the problem of existence of Bohr–Levitan almost periodic solutions of differential equations of type (3) in the context of general semi-group non-autonomous dynamical systems (cocycles), in contrast with the group non-autonomous dynamical systems framework considered in the first part.