Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 87, pp. 1-14.

Title: Oscillatory behavior for nonlinear homogeneous neutral difference equations of
      second order with coefficient changing sign
       
Authors: Ajit Kumar Bhuyan (Sai international School, Bhubaneswar, Odisha, India) 
         Laxmi Narayan Padhy (Konark Inst. of Science and Tech., Bhubaneswar, Odisha, India)
         Radhanath Rath (Principalhallikote Autonomous College, Berhampur, India)

Abstract:
 In this article,  we obtain sufficient conditions  so that all solutions of the neutral
 difference equation
 $$
 \Delta^{2}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k})=0,
 $$
 and all unbounded solutions of the neutral difference equation
 $$
 \Delta^{2}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k}) -u_nH(y_{\alpha(n)})=0
 $$
 are oscillatory,
 where  $\Delta y_n = y_{n+1}-y_n$, $\Delta^2 y_n =\Delta(\Delta y_n)$.
 Different types of super linear and sub linear conditions are imposed on $G$ to
 prevent the solution approaching zero or $\pm \infty$.
 
Submitted June 3, 2020. Published August 12, 2020.
Math Subject Classifications: 39A10, 39A12.
Key Words: Oscillatory solution; nonoscillatory solution; asymptotic behavior;
           difference equation.