On oscillatory first order nonautonomous functional difference systems

Year 2025, Volume: 54 Issue: 5, 1758 - 1773, 29.10.2025

Sunita Das , Arun Kumar Tripathy

https://doi.org/10.15672/hujms.1563103

Abstract

In this work, an illustrative discussion has been made on sufficient conditions under which all vector solutions of first order 2-dim nonautonomous neutral delay difference systems of the form
$$\Delta \left[%
\begin{array}{c}
u(\theta)+b(\theta)u(\theta-\kappa)\\
v(\theta)+b(\theta)v(\theta-\kappa) \\
\end{array}%
\right]=
\begin{bmatrix}
{ a_{1}(\theta)} \quad a_{2}(\theta) \quad\\
a_{3}(\theta) \quad a_{4}(\theta)\quad \\
\end{bmatrix}
\left[%
\begin{array}{c}
g_1(u(\theta-\gamma))\quad\\
g_2(v(\theta-\eta)) \quad\\
\end{array}%
\right]+\left[%
\begin{array}{cc}
\varphi_1(\theta)\quad\\
\varphi_2(\theta) \quad\\
\end{array}%
\right], \theta\geq\rho$$
are oscillatory, where $\kappa>0,$ $\gamma\geq 0, \eta\geq 0$ are integers, $a_{j}(\theta), j=1,2,3,4, b(\theta), \varphi_{1}(\theta),$ $\varphi_{2}(\theta)$ are sequences of real numbers for $\theta\in\mathbb{N}(\theta_{0})$ and $g_1, g_2\in\mathcal{C}(\mathbb{R}, \mathbb{R})$ are nondecreasing with the properties $\phi g_1(\phi)>0, \psi g_2(\psi)>0$ for $\phi\neq 0, \psi\neq 0.$ We verify our results with the examples.

Keywords

oscillation , nonoscillation , nonautonomous system of neutral equations

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