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Keywords:
first-order linear difference equation; delayed difference equation; neutral difference equation; oscillation; oscillatory; periodic coefficient
first-order linear difference equation; delayed difference equation; neutral difference equation; oscillation; oscillatory; periodic coefficient
Summary:
We consider the first-order neutral h-difference equations $\Delta (x_{n}+p_{n}x_{n-k})+q_{n}x_{n-l}=0,$ where $\Delta x_{n}=x_{n+h}-x_{n}$, $h$, $k$ and $l$ are positive integers, $p_{n}$ is an oscilatory periodic sequence with period $h$ and $q_{n}$ is an increasing positive sequence. The fact that the sequence $p$ is an oscillatory periodic sequence is considered for the first time in this study. Moreover, we consider $p_{n}$ can be, quickly oscillatory sequence, and we obtain some new oscillatory criteria. Also we give some example equations, and recurrence formulas of these example equations were obtained, and calculations were made using the Python program and graphs were created to prove that our results are true.
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