Two classes of asymptotically different positive solutions of the equation View the MathML source

The equation View the MathML sourceẏ(t)=−f(t,yt) is considered for t→∞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.