Oscillation conditions for first order neutral difference equations with positive and negative variable co-efficients

Abstract

In this article, we analysis the oscillatory properties of first order neutral difference equations with positive and negative variable coefficients of the forms
$$
\Delta[x(n)+p(n) x(n-\tau)]+\sum_{i=1}^m q_i(n) x\left(n-\sigma_i\right)-\sum_{j=1}^k r_j(n) x\left(n-\rho_j\right)=0 ; \quad n=0,1,2, \ldots,
$$
and
$$
\Delta[x(n)+p(n) x(n+\tau)]+\sum_{i=1}^m q_i(n) x\left(n+\sigma_i\right)-\sum_{j=1}^k r_j(n) x\left(n+\rho_j\right)=0 ; \quad n=0,1,2, \ldots,
$$
where $\{p(n)\}$ is a sequence of real numbers, $\left\{q_i(n)\right\}$ and $\left\{r_j(n)\right\}$ are sequences of positive real numbers, $\tau$ is a positive integer, $\sigma_i$ and $\rho_j$ are nonnegative integers, for $i=1,2, \ldots, m$ and $j=1,2, \ldots, k$. We established sufficient conditions for oscillation of solutions to above systems.