Nonlinear Analysis 71 (2009) e432–e435 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Almost automorphic solutions to second-order semilinear evolution equations Gaston M. N’Guérékata ∗ Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, USA article info MSC: 44A35 42A85 42A75 Keywords: Almost automorphic function Uniform spectrum Semilinear evolution equations abstract In this paper we give some sufficient conditions for ensuring the existence and uniqueness of a mild almost automorphic solution to a second-order semilinear evolution equation in a Banach space. We also present some properties of the (new) notion of a uniform spectrum of bounded functions. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction In this paper we are concerned with the existence and uniqueness of an almost automorphic solution to the semilinear equation d 2 dt x(t ) = Au(t ) + f (t , x(t )), t ∈ R, (1) where A : D(A) ⊂ X  → X is a densely defined and closed linear operator, which is also the infinitesimal generator of a holomorphic semigroup, and f : R × X  → X is almost automorphic and jointly continuous. The existence of almost automorphic, almost periodic, asymptotically almost periodic, and pseudo-almost periodic solutions is one of most attractive topics in the qualitative theory of differential equations due to their significance and applications in physical sciences. The concept of almost automorphy (a.a. for short), which is the central issue in this paper, was first introduced in the literature by Bochner in the earlier sixties; it is a natural generalization of the notion of almost periodicity (see [7,8]). In the last decade, several authors including Nguyen van Minh, J. Liang, Ti-Jun Xiao, T. Diagana, D. Bugajewski, K. Ezzinbi, L. Maniar, B. Basit, J. A. Goldstein, A. Pankov, J. Liu, H. S. Ding, G. M. N’Guérékata and others, have produced extensive literature on the theory of almost automorphy and its applications to differential equations. We would like to mention particularly the papers [2–6,1], related to the present work. The main result in this paper is a generalization of Theorem 3.3 in [5] to the case where f is not necessarily lipschitzian. We start the paper with a presentation of the notion of a uniform spectrum of bounded functions introduced recently in [2] and used as a tool alternative to the Carleman spectrum to prove the existence of almost automorphic mild solutions to linear evolution equations with almost automorphic forcing term. We point out the fact that although the concept of a uniform spectrum of bounded functions was introduced in the framework of almost automorphy, it has been used in [6] to prove the existence and uniqueness of bounded continuous (not necessarily almost automorphic) mild solutions to the first-order equation ˙ x(t ) = Ax(t ) + f (t ) ∗ Tel.: +1 443 885 3965; fax: +1 443 885 8216. E-mail address: gaston.n’guerekata@morgan.edu. 0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.11.004