Oscillatory behavior of solutions of third order semi-canonical dynamic equations on time scale

Ahmed M. Hassan; Clemente Cesarano; Sameh S. Askar; Ahmad M. Alshamrani; Ahmed M. Hassan; Clemente Cesarano; Sameh S. Askar; Ahmad M. Alshamrani

doi:10.3934/math.20241178

AIMS Mathematics

2024, Volume 9, Issue 9: 24213-24228. doi: 10.3934/math.20241178

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Oscillatory behavior of solutions of third order semi-canonical dynamic equations on time scale

1.
Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt
2.
Section of Mathematics, International Telematic University Uninettuno, CorsoVittorio Emanuele Ⅱ, 39, 00186 Roma, Italy
3.
Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 27 April 2024 Revised: 12 July 2024 Accepted: 17 July 2024 Published: 16 August 2024
MSC : 34K11, 34C10, 34N05

This paper investigates the oscillatory behavior of nonlinear third-order dynamic equations on time scales. Our main approach is to transform the equation from its semi-canonical form into a more tractable canonical form. This transformation simplifies the analysis of oscillation behavior and allows us to derive new oscillation criteria. These criteria guarantee that all solutions to the equation oscillate. Our results extend and improve upon existing findings in the literature, particularly for the special cases where $ \mathbb{T} = \mathbb{R} $ and $ \mathbb{T} = \mathbb{Z} $. Additionally, we provide illustrative examples to demonstrate the practical application of the developed criteria.
- third order,
- oscillatory,
- semi-canonical,
- non-canonical,
- canonical
Citation: Ahmed M. Hassan, Clemente Cesarano, Sameh S. Askar, Ahmad M. Alshamrani. Oscillatory behavior of solutions of third order semi-canonical dynamic equations on time scale[J]. AIMS Mathematics, 2024, 9(9): 24213-24228. doi: 10.3934/math.20241178

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Abstract

This paper investigates the oscillatory behavior of nonlinear third-order dynamic equations on time scales. Our main approach is to transform the equation from its semi-canonical form into a more tractable canonical form. This transformation simplifies the analysis of oscillation behavior and allows us to derive new oscillation criteria. These criteria guarantee that all solutions to the equation oscillate. Our results extend and improve upon existing findings in the literature, particularly for the special cases where $ \mathbb{T} = \mathbb{R} $ and $ \mathbb{T} = \mathbb{Z} $. Additionally, we provide illustrative examples to demonstrate the practical application of the developed criteria.

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