In this study, we investigate the oscillation behavior of second-order self-adjoint $ q $-difference equations, focusing on the renowned Leighton oscillation theorem. Through an example, we demonstrate that the $ q $-version of Leighton's classical oscillation theorem does not hold and requires refinement. To address this, we introduce an oscillation-preserving transformation and establish alternative theorems to the ones existing in the literature. The strength of our work lies in the absence of any sign condition on the potential function. We also provide illustrative examples to support our findings and mention directions for future research.
Citation: Aǧacık Zafer, Zeynep Nilhan Gürkan. Oscillation behavior of second-order self-adjoint $ q $-difference equations[J]. AIMS Mathematics, 2024, 9(7): 16876-16884. doi: 10.3934/math.2024819
In this study, we investigate the oscillation behavior of second-order self-adjoint $ q $-difference equations, focusing on the renowned Leighton oscillation theorem. Through an example, we demonstrate that the $ q $-version of Leighton's classical oscillation theorem does not hold and requires refinement. To address this, we introduce an oscillation-preserving transformation and establish alternative theorems to the ones existing in the literature. The strength of our work lies in the absence of any sign condition on the potential function. We also provide illustrative examples to support our findings and mention directions for future research.
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Aǧacık Zafer, Zeynep Nilhan Gürkan. Oscillation behavior of second-order self-adjoint $ q $-difference equations[J]. AIMS Mathematics, 2024, 9(7): 16876-16884. doi: 10.3934/math.2024819
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