Existence of periodic solutions of functional differential equations with parameters such as Nicholson’s blowflies model call for the investigation of integral equations with parameters defined over spaces with periodic structures. In this paper, we study one such equation $\varphi \left(x\right)=\lambda {\int }_{[x,x+\omega ]\cap \Omega }K(x,y)h\left(y\right)f(y,\varphi (y-\tau \left(y\right)))dy$, 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.