Nonlinear Analysis 71 (2009) 3948–3955 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Positive periodic solution of second-order neutral functional differential equations Wing-Sum Cheung a,∗ , Jingli Ren b , Weiwei Han b a Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, China b Department of Mathematics, Zhengzhou University, Zhengzhou 450001, China article info Article history: Received 18 January 2008 Accepted 13 February 2009 Keywords: Neutral functional differential equation Positive periodic solutions Fixed point Second order Operator abstract In this paper, we consider two types of second-order neutral functional differential equations. By choosing available operators and applying Krasnoselskii’s fixed point theorem, we obtain sufficient conditions for the existence of periodic solutions to such equations. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction Neutral functional differential equations manifest themselves in many fields including Biology, Mechanics and Economics [1–3]. For example, in population dynamics, since a growing population consumes more (or less) food than a matured one, depending on individual species, this leads to neutral functional equations [2]. These equations also arise in classical ‘‘cobweb’’ models in Economics where current demand depends on price but supply depends on the previous period [4]. The study on neutral functional differential equations is more intricate than ordinary delay differential equations, that is why there are plenty of results on the existence of positive periodic solutions for various types of first-order or second- order ordinary delay differential equations [5–11], while studies on positive periodic solutions for neutral differential equations are rather infrequent, and most of them are confined to first-order neutral differential equations, see, e.g., [12,13]. Recently, in [14], Wu and Wang discussed the second-order neutral delay differential equation (x(t ) − cx(t − δ)) ′′ + a(t )x(t ) = λb(t )f (x(t − τ(t ))), (1.1) where λ is a positive parameter, δ and c are constants with |c | = 1, a(t ), b(t ) ∈ C (R,(0, ∞)), f ∈ C ([0, ∞), [0, ∞)), and a(t ), b(t ), τ(t ) are ω-periodic functions. The key step in [14] is the application of a theorem of Zhang in [15] for the neutral operator (Ax)(t ) = x(t ) −cx(t −δ), and the celebrated fixed point index theorem, to obtain the existence of positive periodic solutions for (1.1) with c < 0. In this paper, we consider the following two types of second-order neutral functional differential equations (x(t ) − cx(t − τ(t ))) ′′ = a(t )x(t ) − f (t , x(t − τ(t ))), (1.2) and (x(t ) − cx(t − τ(t ))) ′′ =−a(t )x(t ) + f (t , x(t − τ(t ))), (1.3) ∗ Corresponding author. Tel.: +852 28591996; fax: +852 25592225. E-mail addresses: wscheung@hkucc.hku.hk, wscheung@hku.hk (W.-S. Cheung), renjl@zzu.edu.cn (J. Ren). 0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.02.064