## Abstract

Consider the second-order linear delay differential equation ${x}^{\prime \prime}\left(t\right)+p\left(t\right)x\left(\tau \left(t\right)\right)=0$, $t\ge {t}_{0}$, where $p\in C\left(\left[{t}_{0},\infty \right),{\mathbb{R}}^{+}\right)$, $\tau \in C\left(\left[{t}_{0},\infty \right),\mathbb{R}\right)$, $\tau \left(t\right)$ is nondecreasing, $\tau \left(t\right)\le t$ for $t\ge {t}_{0}$ and ${\mathrm{lim}}_{t\to \infty}\tau \left(t\right)=\infty $, the (discrete analogue) second-order difference equation ${\Delta}^{2}x\left(n\right)+p\left(n\right)x\left(\tau \left(n\right)\right)=0$, where $\Delta x\left(n\right)=x\left(n+1\right)-x\left(n\right)$, ${\Delta}^{2}=\Delta \circ \Delta $, $p:\mathbb{N}\to {\mathbb{R}}^{+}$, $\tau :\mathbb{N}\to \mathbb{N}$, $\tau \left(n\right)\le n-1$, and ${\mathrm{lim}}_{n\to \infty}\tau \left(n\right)=+\infty $, and the second-order functional equation $x\left(g\left(t\right)\right)=P\left(t\right)x\left(t\right)+Q\left(t\right)x\left({g}^{2}\left(t\right)\right)$, $t\ge {t}_{0}$, where the functions $P$, $Q\in C\left(\left[{t}_{0},\infty \right),{\mathbb{R}}^{+}\right)$, $g\in C\left(\left[{t}_{0},\infty \right),\mathbb{R}\right)$, $g\left(t\right)\not\equiv t$ for $t\ge {t}_{0}$, ${\mathrm{lim}}_{t\to \infty}g\left(t\right)=\infty $, and ${g}^{2}$ denotes the 2th iterate of the function $g$, that is, ${g}^{0}\left(t\right)=t$, ${g}^{2}\left(t\right)=g\left(g\left(t\right)\right)$, $t\ge {t}_{0}$. The most interesting oscillation criteria for the second-order linear delay differential equation, the second-order difference equation and the second-order functional equation, especially in the case where $\mathrm{lim}{\mathrm{inf}}_{t\to \infty}{\displaystyle {\int}_{\tau \left(t\right)}^{t}\tau \left(s\right)}p\left(s\right)ds\le 1/e$ and $$ for the second-order linear delay differential equation, and $$ and $$, for the second-order functional equation, are presented.

## Citation

L. K. Kikina. I. P. Stavroulakis. "Oscillation Criteria for Second-Order Delay, Difference, and Functional Equations." Int. J. Differ. Equ. 2010 (SI2) 1 - 14, 2010. https://doi.org/10.1155/2010/598068

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