Oscillatory and asymptotic behavior of solutions to second-order non-linear neutral difference equations of advanced type

Abstract

In this paper, necessary and sufficient conditions are obtained for oscillatory and asymptotic behavior of solutions to second-order non-linear neutral advanced difference equations of the form
$$
\Delta\left[\varphi(\zeta)(\Delta y(\zeta))^\lambda\right]+\sum_{i=1}^m \mu_i(\zeta) f\left(x\left(\zeta+\eta_i\right)\right)=0 ; \quad \zeta \geq \zeta_0,
$$
where $y(\zeta)=x(\zeta)+p(\zeta) x(\zeta-\kappa)$, under the assumption $\sum_{\zeta=\zeta_0}^{\infty} \frac{1}{\varphi^{\frac{1}{\zeta}}(\zeta)}=\infty$. Our main tool is Lebesgue's dominated convergence theorem. Further, some illustrate examples showing the applicability of the new results are included.