Oscillation conditions for first order neutral difference equations with positive and negative variable co-efficients
Authors :
A. Murugesan 1,∗ and K. Shanmugavalli 2
Author Address :
1 Department of Mathematics, Government Arts College (Autonomous), Salem-636007, Tamil Nadu, India.
2 Department of Mathematics, Government Arts College for Women, Salem-636008, Tamil Nadu, India.
*Corresponding author.
Abstract :
In this article, we analysis the oscillatory properties of first order neutral difference equations with positive and negative variable coefficients of the forms \begin{equation*} \Delta [x(n)+p(n)x(n-\tau)]+\sum_{i=1}^{m}q_i(n) x(n-\sigma_i)-\sum_{j=1}^{k}r_j(n) x(n-\rho_j)=0;\quad n=0,1,2,..., \quad \quad (*)\end{equation*} and \begin{equation*} \Delta [x(n)+p(n)x(n+\tau)]+\sum_{i=1}^{m}q_i(n) x(n+\sigma_i)-\sum_{j=1}^{k}r_j(n) x(n+\rho_j)=0;\quad n=0,1,2,..., \quad \quad (**)\end{equation*} where $\left\{p(n)\right\}$ 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 $i=1,2,...,m$ and $j=1,2,...,k$. We established sufficient conditions for oscillation of solutions to $(*)$ and $(**)$.
Keywords :
Oscillatory properties, neutral, delay, advanced, difference equation, positive and negative coefficients.
DOI :
Article Info :
Received : December 23, 2016; Accepted : March 01, 2017.
