On oscillation of differential and difference equations with non-monotone delays
Abstract For the delay differential equation x ′ ( t ) + p ( t ) x ( h ( t ) ) = 0 , p ( t ) ⩾ 0 , h ( t ) ⩽ t , t ⩾ 0 , lim t → ∞ h ( t ) = ∞ , we demonstrate that the inequality limsup t → ∞ ∫ h ( t ) t p ( u ) d u > 1 is not sufficient for oscillation. Moreover, for any A > 0 the relation limsup t → ∞ ∫ h ( t ) t p ( u ) d u > A , generally, does not imply oscillation. A similar result is obtained for difference equations. In terms of the maximal argument, new sufficient oscillation conditions are presented for differential and difference equations.