Leighton-Hille-Kneser-type oscillation criteria for Emden-Fowler neutral delay differential equations

Yingzhu Wu; Shizheng Li; Jinsen Xiao; Yuanhong Yu; Yingzhu Wu; Shizheng Li; Jinsen Xiao; Yuanhong Yu

doi:10.3934/math.20251312

AIMS Mathematics

2025, Volume 10, Issue 12: 29873-29891. doi: 10.3934/math.20251312

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Leighton-Hille-Kneser-type oscillation criteria for Emden-Fowler neutral delay differential equations

1.
Department of Mathematics, Guangdong University of Petrochemical Technology, Maoming 525000, China
2.
College of Mathematics and Computer Science, Hebei University, Baoding 071002, China
3.
School of Information Science and Engineering, Linyi University, Linyi 276000, China
4.
School of Sciences, Guangdong University of Petrochemical Technology, Maoming 525000, China
5.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Received: 07 April 2025 Revised: 06 November 2025 Accepted: 04 December 2025 Published: 18 December 2025
MSC : 34C10, 34K11

Emden-Fowler differential equations have numerous applications in physics and engineering. In this article, we extended the classical Leighton, Hille, and Kneser types oscillation criteria for second-order linear differential equations to the second-order Emden-Fowler neutral delay differential equation
$ \left(r(t)\left|z^\prime(t)\right|^{\alpha-1}z^\prime(t)\right)^\prime+q(t)\left|y(\sigma(t))\right|^{\beta}\hbox{sgn} y(\sigma(t)) = 0, \; \; t\geq t_{0}, $
where $ z(t) = y(t)+p(t)y(\tau(t)). $ The criteria obtained extended and improved some well-known results reported in the literature. Moreover, several examples are provided to show the significance of the new findings.
- Emden-Fowler differential equation,
- neutral differential equation,
- delayed argument,
- second-order,
- Riccati method,
- Leighton-Hille-Kneser-type oscillation criteria
Citation: Yingzhu Wu, Shizheng Li, Jinsen Xiao, Yuanhong Yu. Leighton-Hille-Kneser-type oscillation criteria for Emden-Fowler neutral delay differential equations[J]. AIMS Mathematics, 2025, 10(12): 29873-29891. doi: 10.3934/math.20251312

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Abstract

Emden-Fowler differential equations have numerous applications in physics and engineering. In this article, we extended the classical Leighton, Hille, and Kneser types oscillation criteria for second-order linear differential equations to the second-order Emden-Fowler neutral delay differential equation

$ \left(r(t)\left|z^\prime(t)\right|^{\alpha-1}z^\prime(t)\right)^\prime+q(t)\left|y(\sigma(t))\right|^{\beta}\hbox{sgn} y(\sigma(t)) = 0, \; \; t\geq t_{0}, $

where $ z(t) = y(t)+p(t)y(\tau(t)). $ The criteria obtained extended and improved some well-known results reported in the literature. Moreover, several examples are provided to show the significance of the new findings.

References

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