Forum Math. 13 (2001), 581±588 Forum Mathematicum ( de Gruyter 2001 Almost automorphy, almost periodicity and stability of motions in Banach spaces Gaston M. N'guerekata (Communicated by Michael Brin) Abstract. We present su½cient conditions for ensuring almost automorphy of bounded solu- tions of di¨erential equations of the form x 0 t Axt f t in Banach spaces. We also give a generalization to Banach spaces of a result by A. M. Fink on almost periodicity of strongly stable motions. 1991 Mathematics Subject Classi®cation: 34G10; 47D05. 1 Introduction Let X be a Banach space, f A L y R; X  and A a (generally unbounded) linear operator acting in X. We are concerned in this paper with the important problem of the structure of bounded solutions of the abstract di¨erential equation x 0 t Axt f t. This problem was initially raised and solved by Bohr and Neugebauer in 1926 (cf [1] for instance) in a ®nite dimensional space, assuming that the function f t is almost periodic. Then several authors (Amerio, Corduneanu, Goldstein, Zaidman, N'Guerekata, . . .) have extended this result to abstract di¨erential equations and other situations in in®nite dimensional spaces. In this article we consider a larger class of functions called almost automorphic. This notion was introduced in the literature by S. Bochner (1962) and extensively studied by Zaidman and Zaki for functions with values in a Banach space. Section 1 provides a generalization to Banach spaces of a result by A. M. Fink [1] Theorem 11.5 page 181 on almost periodic strongly stable motions. In section 2 we discuss some conditions under which solutions of the above equations approach almost automorphic functions at in®nity. Brought to you by | Rutgers University Authenticated Download Date | 6/1/15 12:44 AM