Nonlinear Analysis 67 (2007) 1412–1418 www.elsevier.com/locate/na Pseudo-almost periodicity of some nonautonomous evolution equations with delay ✩ Hui-Sheng Ding a , Jin Liang a , Gaston M. N’Gu´ er´ ekata b , Ti-Jun Xiao a,∗ a Department of Mathematics, University of Science and Technology of China, Hefei 230026, People’s Republic of China b Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, USA Received 6 May 2006; accepted 18 July 2006 Abstract This paper is concerned with pseudo-almost periodicity of the solutions to the nonautonomous evolution equation with delay u ′ (t ) = A(t )u(t ) + f (t , u(t − h)). Some sufficient conditions which ensure the existence and uniqueness of pseudo-almost periodic mild solutions to the evolution equation with delay are given. An example is shown to illustrate our results. c  2006 Elsevier Ltd. All rights reserved. Keywords: Pseudo-almost periodic; Almost periodic; Evolution family; Exponential dichotomy 1. Preliminaries In this paper, we investigate the pseudo-almost periodicity of the solutions to the following nonautonomous evolution equation with delay: u ′ (t ) = A(t )u (t ) + f (t , u (t − h )), t ∈ R (1.1) in a Banach space X , where h ≥ 0 is a fixed constant, and A(t ) and f (t , u ) satisfy the hypotheses (H1)–(H4) recalled in Section 2. Recently, the existence of pseudo-almost periodic solutions to various differential equations has been of great interest for many researchers (cf. [3,8–10,16,21] and references therein). Many authors have studied the pseudo- almost periodicity of the solutions to Eq. (1.1) in the case where A(t ) = A and h = 0 (see, e.g., [3,8,10,16]). More precisely, in [3,8] the existence and uniqueness of pseudo-almost periodic solutions to some semilinear differential equations has been considered in the case where A is a so-called Hille–Yosida operator. In [16], such a problem has been studied when A is the infinitesimal generator of a compact semigroup. The problem has also been investigated in the case of − A generating an analytic semigroup in [10]. ✩ The work was supported partly by the National Natural Science Foundation of China 10571165, the NCET-04-0572 and the Specialized Research Fund for the Doctoral Program of Higher Education of China 20030358033. ∗ Corresponding author. E-mail address: xiaotj@ustc.edu.cn (T.-J. Xiao). 0362-546X/$ - see front matter c  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2006.07.026