Eigenvalue ratios for the conformable fractional vibrating string equations with single-well densities

Mengze Gu; Mengze Gu

doi:10.3934/math.20251197

AIMS Mathematics

2025, Volume 10, Issue 11: 27235-27246. doi: 10.3934/math.20251197

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Eigenvalue ratios for the conformable fractional vibrating string equations with single-well densities

Mengze Gu ^,

School of Mathematics and Science, Changsha Normal University, Changsha 410083, China

Received: 08 September 2025 Revised: 12 November 2025 Accepted: 17 November 2025 Published: 24 November 2025
MSC : 34B24

In this paper, we investigated in a class of Sturm-Liouville vibrating string problems involving conformable fractional derivatives and studied the behavior of their eigenvalue ratios. Under the assumptions that the potential function is of single-barrier type or the density function is of single-well type, we rigorously established the optimal upper bound for the eigenvalue ratios $ \lambda_n / \lambda_m $, namely
$ \frac{\lambda_n}{\lambda_m} \le \left(\frac{n}{m}\right)^2 \quad (n>m). $
Moreover, we showed that equality holds if and only if the density or the potential is constant. Our novelty was in extending the classical Sturm–Liouville eigenvalue inequalities to the conformable fractional setting.
- eigenvalue ratio,
- conformable fractional,
- Sturm-Liouville,
- Prüfer transformation,
- single-well
Citation: Mengze Gu. Eigenvalue ratios for the conformable fractional vibrating string equations with single-well densities[J]. AIMS Mathematics, 2025, 10(11): 27235-27246. doi: 10.3934/math.20251197

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Abstract

In this paper, we investigated in a class of Sturm-Liouville vibrating string problems involving conformable fractional derivatives and studied the behavior of their eigenvalue ratios. Under the assumptions that the potential function is of single-barrier type or the density function is of single-well type, we rigorously established the optimal upper bound for the eigenvalue ratios $ \lambda_n / \lambda_m $, namely

$ \frac{\lambda_n}{\lambda_m} \le \left(\frac{n}{m}\right)^2 \quad (n>m). $

Moreover, we showed that equality holds if and only if the density or the potential is constant. Our novelty was in extending the classical Sturm–Liouville eigenvalue inequalities to the conformable fractional setting.

References

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