Existence and Asymptotic Behavior of Positive Solutions of FunctionalDifferential Equations of Delayed Type

Abstract

Solutions of the equation $\dot{y}\left(t\right)=-f(t,y_t)$ are considered for $t\to \infty$. The existence of two classes of positive solutions which are asymptotically different is proved using the retract method combined with Razumikhin's technique. With the aid of two auxiliary linear equations, which are constructed using upper and lower linear functional estimates of the right-hand side of the equation considered, inequalities for both types of positive solutions are given as well.