On the Critical Case in Oscillation for Differential Equations with a Single Delay and with Several Delays

Abstract

New nonoscillation and oscillation criteria are derived for scalar delay differential equations $\dot{x}\left(t\right)+a\left(t\right)x\left(h\left(t\right)\right)=0,a\left(t\right)\ge 0,h\left(t\right)\le t,t\ge t_0,$ and $\dot{x}\left(t\right)+{\displaystyle {\sum }_{k=1}^m{a}_k\left(t\right)x\left(h_k\left(t\right)\right)=0},a_k\left(t\right)\ge 0,h_k\left(t\right)\le t,$ and $t\ge t_0,$ in the critical case including equations with several unbounded delays, without the usual assumption that the parameters $a,h,a_k$, and $h_k$ of the equations are continuous functions. These conditions improve and extend some known oscillation results in the critical case for delay differential equations.