Nonlinear Neutral Differential Equations with Multiple Delays: An Oscillation Analysis
- 1Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
Abstract
This paper investigates the oscillatory behavior of the second-order nonlinear neutral differential equation rsω′sα′+∑i=1nφi(s)xγσi(s)=0, ωs:=xs+ϑsxτs, in the noncanonical setting ∫∞r−1/α(ξ)dξ<∞, under the large neutral coefficient assumption ϑs≥ℓ>1. The aim is to establish new sufficient conditions for the oscillation of all solutions within a unified analytical framework that covers several delayed arguments simultaneously. The analysis combines the eventual monotonic properties of the neutral function with linearization and comparison techniques and treats separately the two parameter regimes α≥γ and α<γ, which exhibit different asymptotic behavior. Six oscillation criteria are obtained, each expressed as a single integral condition, and these criteria are shown to form a hierarchy in which one criterion in each regime is the least restrictive of the six; this is a comparison among sufficient conditions and not a sharpness (necessity) statement. Two refinements are also established: a delay-selection version of the criteria, which treats the delays separately and removes the need for the ordering σi≤τ for all but one delay, and a sharper lower bound for the constant produced by the neutral relation. Five examples are presented covering both parameter regimes, including one with a non-constant neutral coefficient and distinct forcing coefficients, and a linear Euler-type example for which the oscillation threshold can be bracketed rigorously so that the distance of the criteria from optimality can be measured, together with a quantitative comparison of the resulting oscillation thresholds and their dependence on the delay ratio and on the neutral bound. They illustrate the applicability of the criteria in a parameter range not covered by several closely related studies, which assume 0≤ϑ(s)<1.
Related
- Oscillation of Noncanonical Neutral-Type Delay Differential Equations
- Improved Solution–Function Inequalities and Kneser-Type Oscillation Criteria for Noncanonical Neutral Differential Equations with Continuous Delay
- New Oscillation Results for Second-Order Advanced Differential Equations with Mixed Neutral Nonlinearities
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- Numerical methods for nonlinear stochastic differential equations driven by subdiffusion
BibTeX
@article{alkilayh2026nonlinear,
title = {{Nonlinear Neutral Differential Equations with Multiple Delays: An Oscillation Analysis}},
author = {Maged Alkilayh and Bader Alshaqqawi},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203741},
url = {https://www.mdpi.com/2227-7390/14/20/3741},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Nonlinear Neutral Differential Equations with Multiple Delays: An Oscillation Analysis
AU - Maged Alkilayh
AU - Bader Alshaqqawi
PY - 2026
DA - 2026/10/14
JO - Mathematics
DO - 10.3390/math14203741
UR - https://www.mdpi.com/2227-7390/14/20/3741
AB - This paper investigates the oscillatory behavior of the second-order nonlinear neutral differential equation rsω′sα′+∑i=1nφi(s)xγσi(s)=0, ωs:=xs+ϑsxτs, in the noncanonical setting ∫∞r−1/α(ξ)dξ<∞, under the large neutral coefficient assumption ϑs≥ℓ>1. The aim is to establish new sufficient conditions for the oscillation of all solutions within a unified analytical framework that covers several delayed arguments simultaneously. The analysis combines the eventual monotonic properties of the neutral function with linearization and comparison techniques and treats separately the two parameter regimes α≥γ and α<γ, which exhibit different asymptotic behavior. Six oscillation criteria are obtained, each expressed as a single integral condition, and these criteria are shown to form a hierarchy in which one criterion in each regime is the least restrictive of the six; this is a comparison among sufficient conditions and not a sharpness (necessity) statement. Two refinements are also established: a delay-selection version of the criteria, which treats the delays separately and removes the need for the ordering σi≤τ for all but one delay, and a sharper lower bound for the constant produced by the neutral relation. Five examples are presented covering both parameter regimes, including one with a non-constant neutral coefficient and distinct forcing coefficients, and a linear Euler-type example for which the oscillation threshold can be bracketed rigorously so that the distance of the criteria from optimality can be measured, together with a quantitative comparison of the resulting oscillation thresholds and their dependence on the delay ratio and on the neutral bound. They illustrate the applicability of the criteria in a parameter range not covered by several closely related studies, which assume 0≤ϑ(s)<1.
SN - 2227-7390
ER -