Asymptotic performance of nonoscillatory solutions of functional differential equations involving a delayed damping term

Osama Moaaz; Asma Al-Jaser; Osama Moaaz; Asma Al-Jaser

doi:10.3934/math.20251151

AIMS Mathematics

2025, Volume 10, Issue 11: 26153-26167. doi: 10.3934/math.20251151

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Asymptotic performance of nonoscillatory solutions of functional differential equations involving a delayed damping term

Osama Moaaz ^{1,2
,
,},
Asma Al-Jaser ^{3
,
,}

1.
Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Mansoura University, 35516 Mansoura, Egypt
3.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

Received: 02 August 2025 Revised: 17 October 2025 Accepted: 30 October 2025 Published: 12 November 2025
MSC : 34C10, 34K11

This work investigated the asymptotic performance of nonoscillatory solutions to the functional differential equation (FDE) $ \left(\alpha \left(u\right) \left\vert y^{\prime }\left(u\right) \right\vert ^{\kappa -1}y^{\prime }\left(u\right) \right) ^{\prime } \; + \; p\left(u\right) F\left(y^{\prime }\left(\delta \left(u\right) \right) \right) \; + \; q\left(u\right) G\left(y\left(\rho \left(u\right) \right) \right) \; = \; 0 $, which involves a delayed damping term. Using Riccati and comparison methods, we extended the previous results to the nonlinear case of the considered equation. Furthermore, the new criteria improved upon the previous ones by removing some constraints on the delay functions. Then, for the linear case, we derived new criteria that take into account all parameters of the equation. The examples and comparisons provided illustrate the importance and novelty of our results.
- differential equations,
- second-order equation,
- delayed damping term,
- asymptotic performance of nonoscillatory solutions,
- oscillatory properties
Citation: Osama Moaaz, Asma Al-Jaser. Asymptotic performance of nonoscillatory solutions of functional differential equations involving a delayed damping term[J]. AIMS Mathematics, 2025, 10(11): 26153-26167. doi: 10.3934/math.20251151

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Abstract

This work investigated the asymptotic performance of nonoscillatory solutions to the functional differential equation (FDE) $ \left(\alpha \left(u\right) \left\vert y^{\prime }\left(u\right) \right\vert ^{\kappa -1}y^{\prime }\left(u\right) \right) ^{\prime } \; + \; p\left(u\right) F\left(y^{\prime }\left(\delta \left(u\right) \right) \right) \; + \; q\left(u\right) G\left(y\left(\rho \left(u\right) \right) \right) \; = \; 0 $, which involves a delayed damping term. Using Riccati and comparison methods, we extended the previous results to the nonlinear case of the considered equation. Furthermore, the new criteria improved upon the previous ones by removing some constraints on the delay functions. Then, for the linear case, we derived new criteria that take into account all parameters of the equation. The examples and comparisons provided illustrate the importance and novelty of our results.

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