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oscillation; difference equation; mixed type; asymptotic behavior
The aim of this work is to study oscillation properties for a scalar linear difference equation of mixed type $$ \Delta x(n)+\sum _{k=-p}^{q}a_{k}(n)x(n+k)=0,\quad n>n_{0}, $$ where $\Delta x(n)=x(n+1)-x(n)$ is the difference operator and $\{a_{k}(n)\}$ are sequences of real numbers for $k=-p,\ldots ,q$, and $p>0$, $q\geq 0$. We obtain sufficient conditions for the existence of oscillatory and nonoscillatory solutions. Some asymptotic properties are introduced.
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