*(English)*Zbl 1169.34050

This paper is devoted to the problem of the existence of two classes of asymptotically different positive solutions of the delay equation

as $t\to \infty ,$ where $f:{\Omega}\to \mathbb{R}$ is a continuous quasi-bounded functional that satisfies a local Lipschitz condition with respect to the second argument and ${\Omega}$ is an open subset in $\mathbb{R}\times C\left(\right[-r,0],\mathbb{R})$. Two approaches are used. One is the method of monotone sequences and the other is the retract method combined with Razumikhin’s technique. By means of linear estimates of the right-hand side of the equation considered, inequalities for both types of positive solutions are given as well. Finally, the authors give an illustrative example and formulate some open problems.

##### MSC:

34K25 | Asymptotic theory of functional-differential equations |