Oscillation Criteria for Second-Order Delay, Difference, and Functional Equations

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 $p\in C\left(\left[t_0,\infty \right),{ℝ}^{+}\right)$, $\tau \in C\left(\left[t_0,\infty \right),ℝ\right)$, $\tau \left(t\right)$ is nondecreasing, $\tau \left(t\right)\le t$ for $t\ge t_0$ and ${\lim }_{t\to \infty }\tau \left(t\right)=\infty$, 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 $\Delta x\left(n\right)=x\left(n+1\right)-x\left(n\right)$, ${\Delta }^2=\Delta \circ \Delta$, $p:\mathbb{N}\to {ℝ}^{+}$, $\tau :\mathbb{N}\to \mathbb{N}$, $\tau \left(n\right)\le n-1$, and ${\lim }_{n\to \infty }\tau \left(n\right)=+\infty$, 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 $P$, $Q\in C\left(\left[t_0,\infty \right),{ℝ}^{+}\right)$, $g\in C\left(\left[t_0,\infty \right),ℝ\right)$, $g\left(t\right)\not\equiv t$ for $t\ge t_0$, ${\lim }_{t\to \infty }g\left(t\right)=\infty$, 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 $\lim {\inf }_{t\to \infty }{\displaystyle {\int }_{\tau \left(t\right)}^t\tau \left(s\right)}p\left(s\right)ds\le 1/e$ and for the second-order linear delay differential equation, and and , for the second-order functional equation, are presented.