Existence of multiple periodic solutions for first order functional differential equations

We obtain sufficient conditions for the existence of three TT-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))),u′(t)=a(t)g(u(t))u(t)−b(t)f(u(t−τ(t))), where a,b,τ∈C(R,R)a,b,τ∈C(R,R) are TT-periodic functions, f,g∈C(R,R)f,g∈C(R,R), and gg is not necessarily bounded. As an application of our theorem, we also derived criteria for the existence of three TT-periodic solutions of the eigenvalue problem u′(t)=a(t)g(u(t))u(t)−λb(t)f(u(t−τ(t))),u′(t)=a(t)g(u(t))u(t)−λb(t)f(u(t−τ(t))), where λλ 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.