# On Oscillation of a Differential Equation with Infinite Number of Delays

### Leonid Berezansky

Ben Gurion University of the Negev, Beer-Sheba, Israel### Elena Braverman

Technion - Israel Institute of Technology, Haifa, Israel

## Abstract

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.

## Cite this article

Leonid Berezansky, Elena Braverman, On Oscillation of a Differential Equation with Infinite Number of Delays. Z. Anal. Anwend. 21 (2002), no. 3, pp. 803–816

DOI 10.4171/ZAA/1109