Asymptotic behavior of certain delay differential equations with forcing term. (Q1873667)

The authors study the asymptotic behavior of solutions to the forced delay differential equation \[ x'(t)= -p(t) f(t-\tau))+ r(t),\quad t\geq 0, \] where \(p\in C([0,\infty), (0,\infty))\), \(r\in C([0,\infty),\mathbb{R})\), \(\tau> 0\) and \(f: \mathbb{R}\to \mathbb{R}\) is increasing. Some sufficient conditions that guarantee every solution of the equation to converge to zero are obtained. The results are applied to some well-known ecological delay differential equations.