On the Critical Case in Oscillation for Differential Equations with a Single Delay and with Several Delays - Jaromír Baštinec; Leonid Berezansky; Josef Diblík; Zdeněk Šmarda (Abstract and Applied Analysis)
New nonoscillation and oscillation criteria are derived for scalar delay differential equations ̇𝑥(𝑡)+𝑎(𝑡)𝑥(ℎ(𝑡))=0,𝑎(𝑡)≥0,ℎ(𝑡)≤𝑡,𝑡≥𝑡0, and ∑̇𝑥(𝑡)+𝑚𝑘=1𝑎𝑘(𝑡)𝑥(ℎ𝑘(𝑡))=0,𝑎𝑘(𝑡)≥0,ℎ𝑘(𝑡)≤𝑡, and 𝑡≥𝑡0, in the critical case including equations with several unbounded delays, without the usual assumption that the parameters 𝑎,ℎ,𝑎𝑘, and ℎ𝑘 of the equations are continuous functions. These conditions improve and extend some known oscillation results in the critical case for delay differential equations.