New oscillation criteria for first-order difference equations with several not necessarily monotonic delays

In this research, the oscillation property of first-order difference equations with several not necessarily monotonic delays is studied. We obtain sufficient oscillation criteria for both the limit inferior (liminf) and the limit superior (limsup) types. Most of the limsup conditions proposed in the previous works were dependent on the nondecreasing sequence \(\rho (l)=\underset{1 \le j \le m}{\max }\ \{\underset{0 \le j_1 \le l}{\sup }\ \sigma _j(j_1)\}\) $\rho (l)=\underset{1\le j\le m}{max}\{\underset{0\le j_1\le l}{sup}{\sigma }_j(j_1)\}$ , where \(\sigma _j(l)\) ${\sigma }_j(l)$ , \(j=1,2,\ldots ,m\) $j=1,2,\dots ,m$ , are the delays. However, these results cannot be applied to many classes of first-order difference equations with several delays, especially when \(l-\sigma _j(l)\) $l-{\sigma }_j(l)$ , for some \(j=1,2,\ldots ,m\) $j=1,2,\dots ,m$ , is small compared to the other delays. To overcome these challenges, new methods are proposed. Furthermore, the delays and the coefficient sequences are rearranged for the purpose of achieving the best results. Finally, we use several illustrative examples to demonstrate the efficiency and reliability of our results.