Abstract
New nonoscillation and oscillation criteria are derived for scalar delay differential equations and and in the critical case including equations with several unbounded delays, without the usual assumption that the parameters and of the equations are continuous functions. These conditions improve and extend some known oscillation results in the critical case for delay differential equations.
Highlights
It is well known that a scalar linear equation with delay xt 1 e x t −11.1 has a nonoscillatory solution as t → ∞
To the best of our knowledge, we are the first to investigate the critical case of such equations
In conclusion we note that there exist numerous results on nonoscillation for various classes of delay differential equations in a noncritical case
Summary
1.1 has a nonoscillatory solution as t → ∞. This means that there exists an eventually positive solution. The coefficient 1/e is called critical with the following meaning: for any α > 1/e, all solutions of the equation xt αx t − 1 0. 1.2 are oscillatory while, for α ≤ 1/e, there exists an eventually positive solution
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