This paper is devoted to the problem of the existence of two classes of asymptotically different positive solutions of the delay equation
as where is a continuous quasi-bounded functional that satisfies a local Lipschitz condition with respect to the second argument and is an open subset in . 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.