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Zbl 1169.34050
Dibl{\'\i}k, Josef; Kúdelč{\'\i}ková, Mária
Two classes of asymptotically different positive solutions of the equation $\dot y (t)= -f(t,y_t)$. (English)
[J] Nonlinear Anal., Theory Methods Appl. 70, No. 10, A, 3702-3714 (2009). ISSN 0362-546X

This paper is devoted to the problem of the existence of two classes of asymptotically different positive solutions of the delay equation $$\dot y (t)= -f(t,y_t)$$ as $t\rightarrow \infty,$ where $f:\Omega\to \mathbb{R}$ is a continuous quasi-bounded functional that satisfies a local Lipschitz condition with respect to the second argument and $\Omega$ is an open subset in $\mathbb{R}\times C([-r,0],\mathbb{R})$. 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.
[Shangjiang Guo (Hunan)]

MSC 2000:: *34K25 Asymptotic theory of functional-differential equations

Keywords: delayed differential equation; monotone iterative method; retract method; Razumikhin's approach; asymptotic behavior

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