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September 10, 2025Open Access

A study on the oscillation of second-order neutral differential equations in noncanonical form

ETErcan TunçTokat Gaziosmanpaşa Üniversitesi

Key Points

New conditions for oscillation were derived, simplifying the understanding of second-order neutral differential equations.
The analysis employs the linearization technique and monotonicity properties, enhancing existing oscillation criteria.
Results are validated through examples, demonstrating superiority over previously known criteria in certain cases.
The study covers linear, sublinear, superlinear, and half-linear equations under a unified condition for oscillation.

Abstract

This paper analyzes the oscillation of the second-order neutral differential equation in noncanonical form r ( t ) ( ( x ( t ) + p ( t ) x ( π ( t ) ) ) ′ ) α ′ + k ( t ) x β ( η ( t ) ) = 0. Using a combination of the linearization technique and the monotonicity properties of the neutral term, we derive new conditions for the studied equation to be oscillatory. Our findings provide new results applicable to linear, sublinear, superlinear, half-linear and other nonlinear forms of the studied equation, and moreover, for all these cases, the oscillation of the studied equation is attained via only one condition. Further, we illustrate the significance of our results with various examples, highlighting their superiority over known criteria in the existing literature.

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