@Article{Qian2019,
author = "Qian, Chuanxi and Smith, Justin",
title = "On existence and global attractivity of periodic solutions of nonlinear delay differential equations",
journal = "Opuscula Math.",
year = "2019",
volume = "39",
number = "6",
pages = "839--862",
doi = "https://doi.org/10.7494/OpMath.2019.39.6.839",
url = "https://doi.org/10.7494/OpMath.2019.39.6.839",
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 \(\omega\)-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. (\(\ast\)) \[\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 (\(\ast\)) converges to this nonnegative periodic solution \(\tilde{x}(t)\) of the associated unforced equation(\(\ast\ast\)). Applications from mathematical biology and numerical examples are also given.",
keywords = "delay differential equation, periodic solution, global attractivity",
issn = "1232-9274"
}