Oscillations of Difference Equations with Several Oscillating Coefficients

Abstract

We study the oscillatory behavior of the solutions of the difference equation $\mathrm{\Delta }x(n)+{\sum }_{i=1}^m{p}_i(\mathrm{n})x({\tau }_i(n))=0,n\in {\mathbb{N}}_0[\nabla x\left(n\right)-{\sum }_{i=1}^m{p}_i\left(n\right)x\left({\sigma }_i\left(n\right)\right)=0,\mathrm{\hspace{0.17em}n}\in \mathbb{N}]$ where $(p_i(n))$, $1\le i\le m$ are real sequences with oscillating terms, ${\tau }_i(n)[{\sigma }_i(n)]$, $1\le i\le m$ are general retarded (advanced) arguments, and $\mathrm{\Delta }[\nabla ]$ denotes the forward (backward) difference operator $\mathrm{\Delta }x(n)=x(n+1)-x(n)[\nabla x(n)=x(n)-x(n-1)]$. Examples illustrating the results are also given.