Second-order advanced dynamic equations on time scales: Oscillation analysis via monotonicity properties

Samy E. Affan; Elmetwally M. Elabbasy; Bassant M. El-Matary; Taher S. Hassan; Ahmed M. Hassan; Samy E. Affan; Elmetwally M. Elabbasy; Bassant M. El-Matary; Taher S. Hassan; Ahmed M. Hassan

doi:10.3934/math.2025206

AIMS Mathematics

2025, Volume 10, Issue 2: 4473-4491. doi: 10.3934/math.2025206

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Second-order advanced dynamic equations on time scales: Oscillation analysis via monotonicity properties

1.
Department of Mathematics, Faculty of Science, Benha University, Benha-Kalubia 13518, Egypt
2.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
3.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
4.
Department of Mathematics, College of Science, University of Hail, Hail 2440, Saudi Arabia
5.
Jadara University Research Center, Jadara University, Jordan

Received: 28 November 2024 Revised: 26 January 2025 Accepted: 17 February 2025 Published: 28 February 2025
MSC : 26E70, 34C10, 34K11, 34K42, 34N05

This paper derives new oscillation criteria for a class of second-order non-canonical advanced dynamic equations of the form
$ \begin{equation*} \left(\zeta(\ell) \varkappa^{\Delta}(\ell)\right)^{\Delta} + q (\ell) \varkappa(\wp(\ell)) = 0. \end{equation*} $
The derived results are based on establishing dynamic inequalities, which lead to novel monotonicity properties of the solutions. These properties are then used to derive new oscillatory conditions. This approach has been successfully applied to difference and differential equations due to the sharpness of its criteria. However, no analogous studies have adopted a similar methodology for dynamic equations on time scales. Furthermore, this study includes examples to illustrate the importance and sharpness of the main results.
- Kneser-type,
- sharp,
- oscillation,
- non-canonical,
- advanced,
- dynamic equations,
- differential equations,
- monotonicity properties
Citation: Samy E. Affan, Elmetwally M. Elabbasy, Bassant M. El-Matary, Taher S. Hassan, Ahmed M. Hassan. Second-order advanced dynamic equations on time scales: Oscillation analysis via monotonicity properties[J]. AIMS Mathematics, 2025, 10(2): 4473-4491. doi: 10.3934/math.2025206

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Abstract

This paper derives new oscillation criteria for a class of second-order non-canonical advanced dynamic equations of the form

$ \begin{equation*} \left(\zeta(\ell) \varkappa^{\Delta}(\ell)\right)^{\Delta} + q (\ell) \varkappa(\wp(\ell)) = 0. \end{equation*} $

The derived results are based on establishing dynamic inequalities, which lead to novel monotonicity properties of the solutions. These properties are then used to derive new oscillatory conditions. This approach has been successfully applied to difference and differential equations due to the sharpness of its criteria. However, no analogous studies have adopted a similar methodology for dynamic equations on time scales. Furthermore, this study includes examples to illustrate the importance and sharpness of the main results.

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