Oscillatory and asymptotic behavior of vector solutions of first-order 2-D neutral difference systems

Abstract

The aim of this paper is to obtain the sufficient conditions for oscillation and nonoscillation of vector solutions of a class of first-order two-dimensional nonautonomous neutral delay difference systems of the form:\[\Delta\left[\begin{array}{c} \alpha(\nu)+q(\nu)\alpha(\nu-p)\\ \beta(\nu)+q(\nu)\beta(\nu-p) \\ \end{array}\right]= \left[\begin{array}{cc} a_{1}(\nu) & a_{2}(\nu) \\ a_{3}(\nu) & a_{4}(\nu) \\\end{array}\right]\left[\begin{array}{c} \phi(\alpha(\nu-l)) \\ \psi(\beta(\nu-m)) \\ \end{array}\right]+\left[\begin{array}{c} \omega_1(\nu) \\ \omega_2(\nu) \\ \end{array}\right],\]where $p>0$, $m\geq 0$, $l\geq 0$ are integers, $a_{j}(\nu)$, $j=1,2,3,4$, $q(\nu), \omega_{1}(\nu),\omega_{2}(\nu)$ are real valued sequences for $\nu\in\mathbb{N}(\nu_{0})$, and $\phi, \psi\in\mathcal{C}(\mathbb{R}, \mathbb{R})$ are bounded functions with the properties $u\phi(u)>0$, $v\psi(v)>0$ for $u\neq 0$, $v\neq 0$. We verify some of our results with illustrative examples.