Consider the delay difference equation

$$\begin{aligned} \Delta {}x(n)+\sum _{k=1}^{m}p_{k}(n)x(n-\tau _{k})=0 \quad \text {for}\ n=0,1,\ldots , \end{aligned}$$

where \(\Delta \) is the forward difference operator, i.e., \(\Delta {}x(n):=x(n+1)-x(n)\), \(\tau _{k}\) is a nonnegative integer and \(\{p_{k}(n)\}_{n=0}^{\infty }\) is a nonnegative sequence of reals for \(k=1,2,\ldots ,m\). New oscillation and nonoscillation results, which essentially improve known results in the literature, are established. These results are extended to the more general difference equation

$$\begin{aligned} \Delta {}x(n)+\sum _{k=1}^{m}p_{k}(n)x(\sigma _{k}(n))=0 \quad \text {for}\ n=0,1,\ldots . \end{aligned}$$

Examples illustrating the significance of the results are given.

中文翻译：

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具有多个时滞的差分方程的振动性和非振动性

考虑延迟差方程

$$ \ begin {aligned} \ Delta {} x（n）+ \ sum _ {k = 1} ^ {m} p_ {k}（n）x（n- \ tau _ {k}）= 0 \ quad \ text {for} \ n = 0,1，\ ldots，\ end {aligned} $$

其中\（\ Delta \）是前向差分算子，即\（\ Delta {} x（n）：= x（n + 1）-x（n）\），\（\ tau _ {k} \ ）是一个非负整数，而\（\ {p_ {k}（n）\} _ {n = 0} ^ {\ infty} \）是\（k = 1,2，\ ldots， m \）。建立了新的振荡和非振荡结果，这些结果基本上改善了文献中的已知结果。这些结果扩展到更一般的差分方程

$$ \ begin {aligned} \ Delta {} x（n）+ \ sum _ {k = 1} ^ {m} p_ {k}（n）x（\ sigma _ {k}（n））= 0 \四\ text {for} \ n = 0,1，\ ldots。\ end {aligned} $$

给出了说明结果重要性的示例。