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Oscillation criteria for second-order delay, difference, and functional equations. (English) Zbl 1207.34082

Consider the second-order linear delay differential equation
\[ x''(t)+p(t)x(\tau(t))=0, \quad t\geq t_0, \]
where \(p\in C([t_0,\infty),\mathbb R^+)\), \(\tau\in C([t_0,\infty),\mathbb R)\), \(\tau(t)\) is nondecreasing, \(\tau(t)\leq t\) for \(t\geq t_0\) 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 \(\Delta x(n)=x(n+1)-x(n)\), \(\Delta^2=\Delta\circ\Delta\), \(p:\mathbb N\to\mathbb R^+\), \(\tau:\mathbb N\to\mathbb N\), \(\tau(n)\leq n-1\), and \(\lim_{n\to\infty}\tau(n)=+\infty\), and the second-order functional equation \[ x(g(t))=P(t)x(t)+Q(t)x(g^2(t)),\quad t\geq t_0, \]
where the functions \(P,Q\in C([t_0,\infty),\mathbb R^+)\), \(g\in C([t_0,\infty),\mathbb R)\), \(g(t)\not\equiv t\) for \(t\geq t_0\), \(\lim_{t\to\infty}g(t)=\infty\), and \(g^2\) denotes the second iterate of the function \(g\), that is, \(g^0(t)=t\), \(g^2(t)=g(g(t))\), \(t\geq t_0\). 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 \(\liminf_{t\to\infty}\int^t_{\tau(t)}\tau(s)p(s)\,ds\leq 1/e\) and \(\limsup_{t\to\infty}\int^t_{\tau(t)}\tau(s)p(s)\,ds<1\) for the second-order linear delay differential equation, and \(0<\liminf_{t\to\infty}\{Q(t)P(g(t))\}\leq 1/4\) and \(\limsup_{t\to\infty}\{Q(t)P(g(t))\}<1\), for the second-order functional equation, are presented.

MSC:

34K11 Oscillation theory of functional-differential equations
39A21 Oscillation theory for difference equations
39B22 Functional equations for real functions
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References:

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