Nonlinear Analysis 71 (2009) 3705–3714 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Weighted pseudo-almost periodic solutions of a class of abstract differential equations ✩ Liuwei Zhang ∗ , Yuantong Xu Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, PR China article info Article history: Received 22 May 2008 Accepted 6 February 2009 Keywords: Weighted pseudo-almost periodicity Hille–Yosida condition Integral solution Spectral analysis Partial functional differential equations abstract For abstract linear functional differential equations with a weighted pseudo-almost periodic forcing term, we prove that the existence of a bounded solution on R + implies the existence of a weighted pseudo-almost periodic solution. Our results extend the classical theorem due to Massera on the existence of periodic solutions for linear periodic ordinary differential equations. To illustrate the results, we consider the Lotka–Volterra model with diffusion. Crown Copyright © 2009 Published by Elsevier Ltd. All rights reserved. 1. Introduction In this paper, we consider the existence of weighted pseudo-almost periodic solutions to the following partial functional differential equation d dt x(t ) = Ax(t ) + Lx t + f (t ), t ∈ R, (1.1) where A : D(A) → X is not necessary densely defined linear operator on a Banach space X, and satisfies the Hille–Yosida condition: there exist M ≥ 0, ω ∈ R such that (ω, ∞) ⊂ ρ(A) and |((λ − A) −n )|≤ M (λ − ω) n , for n ∈ N,λ>ω, where ρ(A) is the resolvent set of A and C = C ([−r , 0], X) is the space of continuous functions from [−r , 0] to X endowed with the uniform norm topology. L is a bounded linear operator from C to X and f is a weighted pseudo-almost periodic function from R to X. The function x t ∈ C is defined by x t (θ) = x(t + θ), for θ ∈ [−r , 0]. We employ the variation of constants formula obtained in [1] and new fundamental results about the spectral analysis of the solutions to establish a new principle reduction obtained in [2], We prove that the existence of a bounded solution on R + implies the existence of a weighted pseudo-almost periodic solution of Eq. (1.1). The existence and uniqueness of almost periodic type solutions of differential equations are always among the most attractive topics due to their significance and application in areas such as physics, control theory, biology, and so on. In [3–5], ✩ Supported by the National Natural Science Foundation of China (No. 10471155). ∗ Corresponding author. E-mail addresses: zhangliuwei020@yahoo.com.cn (L. Zhang), xyt@mail.zsu.edu.cn, xyt@mail.sysu.edu.cn (Y. Xu). 0362-546X/$ – see front matter Crown Copyright © 2009 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.02.032