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Zeitschrift für Analysis und ihre Anwendungen

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Volume 21, Issue 3, 2002, pp. 803–816
DOI: 10.4171/ZAA/1109

Published online: 2002-09-30

On Oscillation of a Differential Equation with Infinite Number of Delays

Leonid Berezansky[1] and Elena Braverman[2]

(1) Ben Gurion University of the Negev, Beer-Sheba, Israel
(2) Technion - Israel Institute of Technology, Haifa, Israel

For a scalar delay differential equation $$\dot{x}(t) + \sum^\infty_{k=1} a_k(t)x(h_k(t)) = 0 \; \; \; (H_k(t) ≤ t)$$ a connection between the following four properties is established:
- non-oscillation of this equation
- non-oscillation of the corresponding differential inequality
- positiveness of the fundamental function - existence of a non-negative solution for a certain explicitly constructed nonlinear integral inequality.
Explicit non-oscillation and oscillation conditions, comparison theorems and a criterion of the existence of a positive solution are presented for this equation.

Keywords: Delay differential equations, infinite number of delays, oscillation, non-oscillation

Berezansky Leonid, Braverman Elena: On Oscillation of a Differential Equation with Infinite Number of Delays. Z. Anal. Anwend. 21 (2002), 803-816. doi: 10.4171/ZAA/1109