Positive Periodic Solutions of Nonautonomous Functional Differential Equations Depending on a Parameter

where a = a(t), h = h(t), and τ = τ(t) are continuous T-periodic functions. We will also assume that T > 0, λ > 0, f = f (t) as well as h = h(t) are positive, ∫T 0 a(t)dt > 0. Functional differential equations with periodic delays appear in a number of ecological models. In particular, our equation can be interpreted as the standard Malthus population model y′ = −a(t)y subject to perturbation with periodical delay. One important question is whether these equations can support positive periodic solutions. Such questions have been studied extensively by a number of authors (cf. [1, 2, 3, 4, 6, 7] and the references therein). In this paper, we are concerned with the existence and nonexistence of periodic solutions when the parameter λ varies. For this purpose, we call a continuously differentiable and T-periodic function a periodic solution of (1) associated with λ∗ if it satisfies (1) when λ = λ∗. We show that there exists λ∗ > 0 such that (1) has at least one positive T-periodic solution for λ ∈ (0,λ∗] and does not have any T-periodic positive solutions for λ > λ∗. Our technique is based on the well-known upper and lower solutions method (cf. [5]). We proceed from (1) and obtain [ y(t)exp (∫ t