Some oscillation theorems for second order nonlinear neutral type difference equations

E. Thandapania,* and V. Balasubramanianb

Some oscillation theorems for second order nonlinear neutral type difference equations

Authors :

E. Thandapani^a,* and V. Balasubramanian^b

Author Address :

^aRamanujan Institute For Advanced Study in Mathematics, University of Madras, Chennai-600005, Tamil Nadu, India.
^bDepartment of Mathematics, Periyar University, Salem-636 011, Tamil Nadu, India.

*Corresponding author.

Abstract :

In this paper some new sufficient conditions for the oscillatory behavior of second order nonlinear neutral type difference equation of the form
$$\Delta \Big(a_n \Delta (x_n +p_n x_{n-k}) \Big) + q_n f(x_{\sigma(n+1)})=0$$
where $\{a_n\},~\{p_n\}$ and $\{q_n\}$ are real sequences, $\{\sigma(n)\}$ is a sequence of integers, $k$ is a positive integer and $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous with
$uf(u)>0$ for $u\neq 0$ are established. Examples are provided to illustrate the main results.