Comparison theorems on the oscillation of a class of neutral difference equations with continuous variables
Bull. Korean Math. Soc. 2010 Vol. 47, No. 2, 401-409
Printed March 1, 2010
Ba\c{s}ak Karpuz and \"{O}zkan \"{O}calan
Faculty of Sciences and Arts and Faculty of Sciences and Arts
Abstract : In this paper, we introduce an iterative method to study oscillatory properties of delay difference equations of the following form $$\nabla_{\alpha}\left[x\left(t\right)-r\left(t\right)x\left(t-\kappa\right)\right]+p\left(t\right)x\left(t-\tau\right)-q\left(t\right)x\left(t-\sigma\right)=0,\quad t\geq t_{0},$$ where $t_{0}\in\mathbb{R}$, $t$ varies in the real interval $\left[t_{0},\infty\right)$, $\alpha>0$, $\kappa,\tau,\sigma\geq0$, $r\in C\left(\left[t_{0}-\alpha,\infty\right),\mathbb{R}^{+}\right)$, $p,q\in C\left(\left[t_{0},\infty\right),\mathbb{R}^{+}\right)$ and $\nabla_{\alpha}x\left(t\right)=x\left(t\right)-x\left(t-\alpha\right)$ for $t\geq t_{0}$.
Keywords : continuous variable, neutral difference equations, oscillation, positive and negative coefficients
MSC numbers : 39A10
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