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Zbl 0890.34059
Berezansky, Leonid; Braverman, Elena
On non-oscillation of a scalar delay differential equation. (English)
[J] Dyn. Syst. Appl. 6, No.4, 567-580 (1997). ISSN 1056-2176

Summary: For the scalar delay differential equation $$\dot x(t)+ \sum^m_{k=1} A_k(t)x \bigl( h_k(t) \bigr)=0, \quad h_k(t) \le t,$$ a connection between the following properties is established: non-oscillation, positiveness of the fundamental function and existence of a nonnegative solution for a certain explicitly constructed nonlinear integral inequality. Explicit non-oscillation and oscillation conditions, a comparison theorem and a criterion for existence of a positive solution are presented. Some of the results are generalized to an impulsive delay differential equation.

MSC 2000:: *34K11 Oscillation theory of functional-differential equations
34A37 Differential equations with impulses
34K25 Asymptotic theory of functional-differential equations

Keywords: non-oscillation; positiveness; fundamental function; nonnegative solution; nonlinear integral inequality; impulsive delay differential equation

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