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

## Abstract

Consider the second-order linear delay differential equation ${x}^{\prime \prime }(t)+p(t)x(\tau (t))=0$, $t\ge {t}_{0}$, where $p\in C([{t}_{0},\infty ),{\Bbb R}^{+})$, $\tau \in C([{t}_{0},\infty ),\Bbb R)$, $\tau (t)$ is nondecreasing, $\tau (t)\le t$ for $t\ge {t}_{0}$ and ${\mathrm{lim}}_{t\to \infty }\tau (t)=\infty$, the (discrete analogue) second-order difference equation ${\Delta }^{2}x(n)+p(n)x(\tau (n))=0$, where $\Delta x(n)=x(n+1)-x(n)$, ${\Delta }^{2}=\Delta \circ \Delta$, $p:\Bbb N\to {\Bbb R}^{+}$, $\tau :\Bbb N\to \Bbb N$, $\tau (n)\le n-1$, and ${\mathrm{lim}}_{n\to \infty }\tau (n)=+\infty$, and the second-order functional equation $x(g(t))=P(t)x(t)+Q(t)x({g}^{2}(t))$, $t\ge {t}_{0}$, where the functions $P$, $Q\in C([{t}_{0},\infty ),{\Bbb R}^{+})$, $g\in C([{t}_{0},\infty ),\Bbb R)$, $g(t)\not\equiv t$ for $t\ge {t}_{0}$, ${\mathrm{lim}}_{t\to \infty }g(t)=\infty$, and ${g}^{2}$ denotes the 2th iterate of the function $g$, that is, ${g}^{0}(t)=t$, ${g}^{2}(t)=g(g(t))$, $t\ge {t}_{0}$. 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 $\mathrm{lim}{\mathrm{inf}}_{t\to \infty }{\int }_{\tau (t)}^{t}\tau (s)p(s)ds\le 1/e$ and $\mathrm{lim}{\mathrm{sup}}_{t\to \infty }{\int }_{\tau (t)}^{t}\tau (s)p(s)ds<1$ for the second-order linear delay differential equation, and $0<\mathrm{lim}{\mathrm{inf}}_{t\to \infty }\{Q(t)P(g(t))\}\le 1/4$ and $\mathrm{lim}{\mathrm{sup}}_{t\to \infty }\{Q(t)P(g(t))\}<1$, 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

## Information

Received: 2 December 2009; Accepted: 9 January 2010; Published: 2010
First available in Project Euclid: 26 January 2017

zbMATH: 1207.34082
MathSciNet: MR2607727
Digital Object Identifier: 10.1155/2010/598068