Oscillatory dynamics and recursive monotonicity of positive solutions to second-order neutral delay differential equations: Analytical framework and applications

Maged Alkilayh; Nedhal Almohammed; Maged Alkilayh; Nedhal Almohammed

doi:10.3934/math.2025960

AIMS Mathematics

2025, Volume 10, Issue 9: 21595-21616. doi: 10.3934/math.2025960

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Oscillatory dynamics and recursive monotonicity of positive solutions to second-order neutral delay differential equations: Analytical framework and applications

Maged Alkilayh ^,,
Nedhal Almohammed

Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia

Received: 08 May 2025 Revised: 20 July 2025 Accepted: 25 July 2025 Published: 18 September 2025

This paper studies the oscillatory behavior of a class of second-order neutral delay differential equations (NDDEs). Using comparison principles and Riccati transformation techniques, we derived new sufficient conditions for oscillation. A recursive framework was introduced to enhance the monotonic properties of positive solutions, leading to sharper criteria. Auxiliary inequalities, exponential estimates, and integral averaging methods support the analysis. Illustrative examples demonstrate the applicability and improvements over existing results, positioning our findings within a broader context relevant to engineering and biological models.
- differential equations,
- neutral delay differential equations,
- oscillation,
- positive solutions,
- monotonic properties,
- recursive dynamics,
- numerical validation,
- applied modeling
Citation: Maged Alkilayh, Nedhal Almohammed. Oscillatory dynamics and recursive monotonicity of positive solutions to second-order neutral delay differential equations: Analytical framework and applications[J]. AIMS Mathematics, 2025, 10(9): 21595-21616. doi: 10.3934/math.2025960

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Abstract

This paper studies the oscillatory behavior of a class of second-order neutral delay differential equations (NDDEs). Using comparison principles and Riccati transformation techniques, we derived new sufficient conditions for oscillation. A recursive framework was introduced to enhance the monotonic properties of positive solutions, leading to sharper criteria. Auxiliary inequalities, exponential estimates, and integral averaging methods support the analysis. Illustrative examples demonstrate the applicability and improvements over existing results, positioning our findings within a broader context relevant to engineering and biological models.

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