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Two classes of asymptotically different positive solutions of the equation y ˙(t)=-f(t,y t ). (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

y ˙(t)=-f(t,y t )

as t, where f:Ω is a continuous quasi-bounded functional that satisfies a local Lipschitz condition with respect to the second argument and Ω is an open subset in ×C([-r,0],). 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:
34K25Asymptotic theory of functional-differential equations