Existence of multiple periodic solutions for first order functional differential equations

We obtain sufficient conditions for the existence of three T -periodic solutions of the first order functional differential equation u(t) = a(t)g(u(t))u(t) − b(t)f (u(t − τ(t))), where a, b, τ ∈ C(R, R) are T -periodic functions, f , g ∈ C(R, R), and g is not necessarily bounded. As an application of our theorem, we also derived criteria for the existence of three T -periodic solutions of the eigenvalue problem u(t) = a(t)g(u(t))u(t) − λb(t)f (u(t − τ(t))), where λ is a positive parameter. Our analysis mainly relies on the lower and upper solution method and the topological degree theory. Examples are provided to apply our results. © 2011 Elsevier Ltd. All rights reserved.