# Article

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Keywords:
difference equation; retarded argument; advanced argument; oscillatory solution; nonoscillatory solution
Summary:
Consider the difference equation $$\Delta x(n)+\sum _{i=1}^{m}p_{i}(n)x(\tau _{i}(n))=0,\quad n\geq 0\quad \bigg [\nabla x(n)-\sum _{i=1}^{m}p_{i}(n)x(\sigma _{i}(n))=0,\quad n\geq 1\bigg ],$$ where $(p_{i}(n))$, $1\leq i\leq m$ are sequences of nonnegative real numbers, $\tau _{i}(n)$ [$\sigma _{i}(n)$], $1\leq \break i\leq m$ are general retarded (advanced) arguments and $\Delta$ [$\nabla$] denotes the forward (backward) difference operator $\Delta x(n)=x(n+1)-x(n)$ [$\nabla x(n)=x(n)-x(n-1)$]. New oscillation criteria are established when the well-known oscillation conditions $$\limsup _{n\rightarrow \infty }\sum _{i=1}^{m}\sum _{j=\tau (n)}^{n}p_{i}(j)>1 \quad \bigg [\limsup _{n\rightarrow \infty }\sum _{i=1}^{m}\sum _{j=n}^{\sigma (n)}p_{i}(j)>1\bigg ]$$ and $$\liminf _{n\rightarrow \infty }\sum _{i=1}^{m}\sum _{j=\tau _{i}(n)}^{n-1}p_{i}(j)>\frac {1}{\rm e} \quad \bigg [\liminf _{n\rightarrow \infty }\sum _{i=1}^{m}\sum _{j=n+1}^{\sigma _{i}(n)}p_{i}(j)>\frac {1}{\rm e}\bigg ]$$ are not satisfied. Here $\tau (n)=\max _{1\leq i\leq m}\tau _{i}(n)$ $[ \sigma (n)=\min _{1\leq i\leq m}\sigma _{i}(n) ]$. The results obtained essentially improve known results in the literature. Examples illustrating the results are also given.
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