Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays

This study elucidates the sufficient conditions for the first-order nonlinear differential equations with periodic coefficients and time-varying delays to have positive periodic solutions. Our results are proved using the Krasnosel'skii fixed point theorem. In this article, we have identified two sets Delta and del and proved that at least one positive periodic solution exists in the interval between the point belonging to Delta and the point belonging to del. We propose simple conditions that guarantee the existence of sets Delta and del. In addition, we obtain the necessary conditions for the existence of positive periodic solutions of the first-order nonlinear differential equations when the periodic coefficients satisfy certain conditions. Finally, examples and numerical simulations are used to illustrate the validity of our results.