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Zbl 1207.34082
Kikina, L.K.; Stavroulakis, I.P.
Oscillation criteria for second-order delay, difference, and functional equations. (English)
[J] Int. J. Differ. Equ. 2010, Article ID 598068, 14 p. (2010). ISSN 1687-9643; ISSN 1687-9651/e

Consider the second-order linear delay differential equation $$x''(t)+p(t)x(\tau(t))=0, \quad t\ge t_0,$$ 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 $\lim_{t\to\infty}\tau(t)=\infty$, the (discrete analogue) second-order difference equation $$\Delta^2x(n)+p(n)x(\tau(n))=0,$$ 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 $\lim_{n\to\infty}\tau(n)=+\infty$, and the second-order functional equation $$x(g(t))=P(t)x(t)+Q(t)x(g^2(t)),\quad t\ge t_0,$$ 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 $t\ge t_0$, $\lim_{t\to\infty}g(t)=\infty$, and $g^2$ denotes the second 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 $\liminf_{t\to\infty}\int^t_{\tau(t)}\tau(s)p(s)\,ds\le 1/e$ and $\limsup_{t\to\infty}\int^t_{\tau(t)}\tau(s)p(s)\,ds<1$ for the second-order linear delay differential equation, and $0<\liminf_{t\to\infty}\{Q(t)P(g(t))\}\le 1/4$ and $\limsup_{t\to\infty}\{Q(t)P(g(t))\}<1$, for the second-order functional equation, are presented.

MSC 2000:: *34K11 Oscillation theory of functional-differential equations
39A21
39B22 Functional equations for real functions

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