Criteria for the existence of positive solutions to delayed functional differential equations

Abstract

The paper is concerned with the large time behavior of solutions to functional delayed differential equations $\dot y(t)=f(t,y_t)$ where $f\colon \Omega_n \mapsto {\mathbb{R}}^n$ is a continuous map satisfying a local Lipschitz condition with respect to the second argument and $\Omega_n$ is an open subset in $\mathbb{R}\times {\cal C}_n$, ${\cal C}_n:={C}_n ([-r,0],\mathbb{R}^n)$, $r>0$. Criteria on the existence of positive solutions (different from the well-known published results) and their estimates from above are derived. The results are illustrated by examples.

Keywords

• positive solution
• large time behavior
• delayed differential equation