One-Signed Periodic Solutions of First-Order Functional Differential Equations with a Parameter

We study one-signed periodic solutions of the first-order functional differential equation u t −a t u t λb t f u t − τ t , t ∈ R by using global bifurcation techniques. Where a, b ∈ C R, 0, ∞ are ω−periodic functions with ω 0 a t dt > 0, ω 0 b t dt > 0, τ is a continuous ω-periodic function, and λ > 0 is a parameter. f ∈ C R,R and there exist two constants s 2 < 0 < s 1 such that f s 2 f 0 f s 1 0, f s > 0 for s ∈ 0, s 1 ∪ s 1 , ∞ and f s < 0 for s ∈ −∞, s 2 ∪ s 2 , 0

Existence results on positive periodic solutions for impulsive functional differential equations

A class of first order nonlinear functional differential equations with impulses is studied. It is shown that there exist one or two positive T-periodic solutions under certain assumptions, and no positive T-periodic solution under some other assumptions. Applications to some impulsive biological models and an example, which can not be covered by known results, are given to illustrate the main results