Two classes of asymptotically different positive solutions of the equation ẏ(t)=−f(t,yt)

The equation ẏ(t)=−f(t,yt) is considered for t→∞. The existence of two classes of positive solutions which are asymptotically different is proved using two different techniques - the method of monotone sequences and the retract method combined with Razumikhin's technique. With the aid of linear estimates of the right-hand side of the equation considered, inequalities for both types of positive solutions are given as well. The results are illustrated by an example.