Abstract
Sufficient oscillation conditions involving $\limsup $ and $\liminf $ for first-order differential equations with several non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative application of the Gr\"{o}nwall inequality. Examples illustrating the significance of the results are also given.
Highlights
In this paper we consider the differential equation with several variable deviating arguments of either delay m x (t) + pi(t)x τi(t) =, t ≥, ( . ) i=or advanced type m x (t) – pi(t)x σi(t) =, t ≥ . ∀t ≥ and lim t→∞ τi(t) = ∞,≤ i ≤ m, and σi(t) ≥ t, t ≥, ≤ i ≤ m, Braverman et al Advances in Difference Equations (2016) 2016:87 respectively
An equation is oscillatory if all its solutions oscillate
The following lemma provides an estimation for a rate of decay for a positive solution
Summary
If there exists an eventually positive or an eventually negative solution, the equation is nonoscillatory. An equation is oscillatory if all its solutions oscillate. ) oscillate, while if σmin(t) m lim inf t→∞ t pi(s) ds > e , i= ) with one delay, in Braverman and Karpuz [ ] established the following theorem in the case that the argument τ (t) is non-monotone and g(t) is defined as g(t) = sup τ (s), t ≥ .
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