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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.

34K25Asymptotic theory of functional-differential equations