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An improved oscillation criterion for first order difference equations. (English) Zbl 1363.39015

Summary: This paper is concerned with the oscillatory behavior of first order difference equation with general argument \[ \Delta x(n)+p(n)x(\tau(n))=0,\qquad n=0,1,\dots\tag{\(\ast\)} \] where \((p(n))\) is a sequence of nonnegative real numbers and \((\tau(n))\) is a sequence of integers. Let the number \(m\) be defined by \[ m=\liminf_{n\to\infty}\sum_{j=\tau(n)}^{n-1}p(j)\left(\frac{j-\tau(j)+1}{j-\tau(j)}\right) ^{j-\tau(j)+1}. \] It is proved that, all solutions of equation (\(\ast\)) oscillate if the condition \(m>1\) is satisfied.

MSC:

39A21 Oscillation theory for difference equations
39A10 Additive difference equations
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