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

Abstract

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 Δ\Delta and ∇\nabla and proved that at least one positive periodic solution exists in the interval between the point belonging to Δ\Delta and the point belonging to ∇\nabla . We propose simple conditions that guarantee the existence of sets Δ\Delta and ∇\nabla . 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.

Keywords

• positive periodic solutions
• time-varying delays
• krasnosel’skii fixed point theorem
• differential equations
• 34a34
• 34k13
• 47h10
• 92d25