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<p>We consider the first-order neutral h-difference equations ∆(<i>x</i><sub><i>n</i></sub> + <i>p</i><sub><i>n</i></sub><i>x</i><sub><i>n</i>−<i>k</i></sub>) + <i>q</i><sub><i>n</i></sub><i>x</i><sub><i>n</i>−l</sub> = 0, where ∆<i>x</i><sub><i>n</i></sub> = <i>x</i><sub><i>n+h</i></sub> − <i>x</i><sub><i>n</i></sub>, <i>h, k</i> and <i>l</i> are positive integers, <i>p</i><sub><i>n</i></sub> is an oscilatory periodic sequence with period <i>h</i> and <i>q</i><sub><i>n</i></sub> is an increasing positive sequence. The fact that the sequence <i>p</i> is an oscillatory periodic sequence is considered for the first time in this study. Moreover, we consider <i>p</i><sub><i>n</i></sub> 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.</p>

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Oscillation criteria for first-order neutral h-difference equations with oscillatory periodic coefficients

  • 期刊论文

  • 发表日期：

    2026年4月27日

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  • Reyhan Gökteke,
  • Yaşar Bolat

摘要

We consider the first-order neutral h-difference equations ∆(xn + pnxn−k) + qnxn−l = 0, where ∆xn = xn+h − xn, h, k and l are positive integers, pn is an oscilatory periodic sequence with period h and qn 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 pn 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.