Oscillation of solutions for second-order neutral multi-delay differential equations

Lanshuo Hua; Jiaxuan Sun; Wenjin Li; Yanni Pang; Lanshuo Hua; Jiaxuan Sun; Wenjin Li; Yanni Pang

doi:10.3934/math.2025830

AIMS Mathematics

2025, Volume 10, Issue 8: 18586-18602. doi: 10.3934/math.2025830

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Research article

Oscillation of solutions for second-order neutral multi-delay differential equations

1.
School of Statistics and Data Science, Jilin University of Finance and Economics, Changchun 130117, China
2.
School of Mathematics, Jilin University, Changchun 130021, China

Received: 21 June 2025 Revised: 01 August 2025 Accepted: 08 August 2025 Published: 18 August 2025
MSC : 34K11

This paper investigates the oscillatory behavior of solutions for a class of second-order neutral differential equations with multiple delays. By employing rigorous analytical methods, we establish sufficient oscillation criteria that guarantee solutions exhibit oscillatory characteristics under prescribed conditions. The main results extend existing theory by incorporating multiple delay terms and nonlinear effects, thereby broadening the applicability of oscillation theory to more complex neutral delay differential equations. A detailed mathematical analysis is conducted to explore the influence of nonlinear terms and various delay functions on the solutions. It is shown that, despite these factors influencing the specific trajectory and amplitude of oscillations, the global oscillatory nature of the solutions remains intact provided that the proposed conditions are met. Numerical experiments illustrate interaction effects among multiple delays; in some settings, the amplitude envelope may show transitions in decay rate. This research thus provides both theoretical insights and practical implications, forming a solid foundation for future studies into more general nonlinear equations, higher-order neutral equations, and systems involving neutral differential equations with delays.
- neutral differential equations,
- oscillation theory,
- delay function,
- nonlinear terms
Citation: Lanshuo Hua, Jiaxuan Sun, Wenjin Li, Yanni Pang. Oscillation of solutions for second-order neutral multi-delay differential equations[J]. AIMS Mathematics, 2025, 10(8): 18586-18602. doi: 10.3934/math.2025830

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Abstract

This paper investigates the oscillatory behavior of solutions for a class of second-order neutral differential equations with multiple delays. By employing rigorous analytical methods, we establish sufficient oscillation criteria that guarantee solutions exhibit oscillatory characteristics under prescribed conditions. The main results extend existing theory by incorporating multiple delay terms and nonlinear effects, thereby broadening the applicability of oscillation theory to more complex neutral delay differential equations. A detailed mathematical analysis is conducted to explore the influence of nonlinear terms and various delay functions on the solutions. It is shown that, despite these factors influencing the specific trajectory and amplitude of oscillations, the global oscillatory nature of the solutions remains intact provided that the proposed conditions are met. Numerical experiments illustrate interaction effects among multiple delays; in some settings, the amplitude envelope may show transitions in decay rate. This research thus provides both theoretical insights and practical implications, forming a solid foundation for future studies into more general nonlinear equations, higher-order neutral equations, and systems involving neutral differential equations with delays.

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