On the upper bounds for the distance between zeros of solutions of a first-order linear neutral differential equation with several delays

Emad R. Attia; Emad R. Attia

doi:10.3934/math.20241145

AIMS Mathematics

2024, Volume 9, Issue 9: 23564-23583. doi: 10.3934/math.20241145

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Research article

On the upper bounds for the distance between zeros of solutions of a first-order linear neutral differential equation with several delays

Emad R. Attia ^{1,2
,
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1.
Department of Mathematics, College of Sciences and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt

Received: 29 April 2024 Revised: 27 June 2024 Accepted: 01 July 2024 Published: 07 August 2024
MSC : 34K11, 39A10, 39A99

This work is devoted to studying the distribution of zeros of a first-order neutral differential equation with several delays
$ \begin{equation*} \left[y(t)+a(t)y\left(t-\sigma\right)\right]'+ \sum\limits_{j = 1}^n b_j(t)y\left(t-\mu_j\right) = 0, \quad \quad \quad t \geq t_0. \end{equation*} $
New estimations for the upper bounds of the distance between successive zeros are obtained. The properties of a positive solution of a first-order differential inequality with several delays in a closed interval are studied, and many results are established. We apply these results to a first-order neutral differential equation with several delays and also to a first-order differential equation with several delays. Our results for the differential equation with several delays not only provide new estimations but also improve many previous ones. Also, the results are formulated in a general way such that they can be applied to any functional differential equation for which studying the distance between zeros is equivalent to studying this property for a first-order differential inequality with several delays. Further, new estimations of the upper bounds for certain equations are given. Finally, a comparison with all previous results is shown at the end of this paper.
- neutral differential equations,
- several delays,
- oscillation,
- distance between zeros
Citation: Emad R. Attia. On the upper bounds for the distance between zeros of solutions of a first-order linear neutral differential equation with several delays[J]. AIMS Mathematics, 2024, 9(9): 23564-23583. doi: 10.3934/math.20241145

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Abstract

This work is devoted to studying the distribution of zeros of a first-order neutral differential equation with several delays

$ \begin{equation*} \left[y(t)+a(t)y\left(t-\sigma\right)\right]'+ \sum\limits_{j = 1}^n b_j(t)y\left(t-\mu_j\right) = 0, \quad \quad \quad t \geq t_0. \end{equation*} $

New estimations for the upper bounds of the distance between successive zeros are obtained. The properties of a positive solution of a first-order differential inequality with several delays in a closed interval are studied, and many results are established. We apply these results to a first-order neutral differential equation with several delays and also to a first-order differential equation with several delays. Our results for the differential equation with several delays not only provide new estimations but also improve many previous ones. Also, the results are formulated in a general way such that they can be applied to any functional differential equation for which studying the distance between zeros is equivalent to studying this property for a first-order differential inequality with several delays. Further, new estimations of the upper bounds for certain equations are given. Finally, a comparison with all previous results is shown at the end of this paper.

References

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