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An oscillation criterion for linear difference equations with general delay argument. (English) Zbl 1186.39010
Consider the delay difference equation $$x(n+1)-x(n)+p(n)x(\tau (n))=0,\tag$*$ $$ where $\{p(n)\}_{n\geq 0}$ is a sequence of integers such that $\tau (n)\leq n-1$ for all $n\geq 0$ and $\lim_{n\to \infty}\tau (n)=\infty$. The authors establish the following sufficient condition for the oscillation of all solutions of ($*$): Theorem. Assume that the sequence $\{\tau (n)\}_{n\geq 0}$ is increasing, $0<\alpha \leq -1+\sqrt{2}$, where $\alpha =\lim \inf_{n\to \infty}\sum_{j=\tau (n)}^{n-1}p(j)$. If $\lim \sup_{n\to \infty}\sum_{j=\tau (n)}^{n}p(j)>1-\frac{1}{2}(1-\alpha -\sqrt{1-2\alpha -\alpha ^{2}})$, then all solutions of ($*$) are oscillatory.

MSC:
39A21Oscillation theory (difference equations)
39A06Linear equations (difference equations)
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Full Text: DOI
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