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

Abstract

Consider the delay differential equation with a forcing term $\tag{\(\ast\)} x'(t)=-f(t,x(t))+g(t,x(t-\tau ))+r(t), \quad t \geq 0$ $(∗)$ where $f(t,x): [0,\infty) \times [0,\infty) \to \mathbb{R}$, $g(t,x): [0,\infty) \times [0,\infty) \to [0,\infty)$ are continuous functions and ω-periodic in $t$, $r(t): [0,\infty) \to\mathbb{R}$ is a continuous function and $\tau \in (0,\infty)$ is a positive constant. We first obtain a sufficient condition for the existence of a unique nonnegative periodic solution $\tilde{x}(t)$ of the associated unforced differential equation of Eq. $(∗)$ $\tag{\(\ast\ast\)} x'(t)=-f(t,x(t))+g(t,x(t-\tau)), \quad t \geq 0.$ $(∗∗)$ Then we obtain a sufficient condition so that every nonnegative solution of the forced equation $(∗)$ converges to this nonnegative periodic solution $\tilde{x}(t)$ of the associated unforced equation $(∗∗)$. Applications from mathematical biology and numerical examples are also given.