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.
📌 3 Key Takeaways
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Objective & Hypothesis: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.
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Key Findings & Data: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.
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Conclusion & Impact: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.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:oscillation · nonoscillation · nonautonomous system of neutral equations
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.
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