On existence and global attractivity of periodic solutions of nonlinear delay differential equations

. Consider the delay differential equation with a forcing term x′(t) = −f(t, x(t)) + g(t, x(t− τ)) + r(t), t ≥ 0 (∗) where f(t, x) : [0,∞)× [0,∞)→ R, g(t, x) : [0,∞)× [0,∞)→ [0,∞) are continuous functions and ω-periodic in t, r(t) : [0,∞)→ R is a continuous function and τ ∈ (0,∞) is a positive constant. We first obtain a sufficient condition for the existence of a unique nonnegative periodic solution x̃(t) of the associated unforced differential equation of Eq. (∗) x′(t) = −f(t, x(t)) + g(t, x(t− τ)), t ≥ 0. (∗∗) Then we obtain a sufficient condition so that every nonnegative solution of the forced equation (∗) converges to this nonnegative periodic solution x̃(t) of the associated unforced equation (∗∗). Applications from mathematical biology and numerical examples are also given.