An improved result on bounded/periodic solutions for some scalar delay differential equations

In [2] we established an existence theorem on bounded/periodic solutions for a class of scalar delay differential equations of the form$ \begin{equation} \frac{du}{dt} = f(t,u(t), u(t-r_1), \cdots, u(t-r_n)), \qquad t \in \mathbb R, \;\;\;\;\;(1)\end{equation} $under the assumptions that the constant delays $ r_k>0 $, $ k = 1, \cdots, n $, are 'small' and $ f $ satisfies a one-sided Lipschitz condition on the variables $ u(t), u(t-r_1), \cdots, u(t-r_n) $. In this paper, we improve this result in the case that $ f $ is strictly increasing in some variables $ u(t-r_k) $ and obtain a new result that allows larger values of $ r_k $ with which the equation (1) still has a bounded/periodic solution. We illustrate this result via some population models.