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

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.