ANNALES POLONICI MATHEMATICI 96.3 (2009) Existence and uniqueness of positive periodic solutions for a class of integral equations with parameters by Shu-Gui Kang (Datong), Bao Shi (Yantai) and Sui Sun Cheng (Hsinchu) Abstract. Existence of periodic solutions of functional differential equations with parameters such as Nicholson’s blowflies model call for the investigation of integral equa- tions with parameters defined over spaces with periodic structures. In this paper, we study one such equation φ(x)= λ  [x,x+ω]∩Ω K(x, y)h(y)f (y,φ(y - τ (y))) dy, x ∈ Ω, by means of the proper value theory of operators in Banach spaces with cones. Existence, uniqueness and continuous dependence of proper solutions are established. 1. Introduction. Existence of solutions of differential equations is often established by means of fixed point theorems for integral equations. Such an approach naturally calls for the investigation of integral equations and operator equations. Recent investigations (see e.g. [1, 5–8] and the references cited in [7]) of the existence of periodic solutions of functional differential equations such as (1) φ ′ (x)= −a(x)φ(x)+ f (φ(x)), x ∈ R, where a = a(x) is a positive continuous 2π-periodic function defined on R, show that fixed points techniques applied to integral equations of the form (2) φ(x)= x+2π  x K(x, y)f (φ(y)) dy, x ∈ R, where K(x, y)= exp  y x a(t) dt exp  2π 0 a(t) dt − 1 , x, y ∈ R, 2000 Mathematics Subject Classification : Primary 45M15; Secondary 45M05. Key words and phrases : integral equations, positive periodic solution, cone, eigenvalue. DOI: 10.4064/ap96-3-3 [227] c  Instytut Matematyczny PAN, 2009