Two classes of asymptotically different positive solutions of the equation [formula omitted

Abstract

The equation y ̇ ( t ) = − f ( t , y t ) 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.