Analysis of Boundedness and Stability in Second-Order Differential Equations with Variable Delay

Abstract

Second-order differential equations (SODE) are essential for modeling a wide range of physical events, engineering systems, and biological processes. They depict dynamic systems involving second derivatives. This research analyzes the asymptotic stability, boundedness, uniform stability, uniform boundedness, and an ensemble of SODE with variable delay and its square integrability of solutions. We investigate these dynamics using the Lyapunov-Krasovskii direct technique and provide an alternative Lyapunov-Krasovskii functional (LKF), which streamlines and extends previous literature. Numerical examples demonstrate the validity and usefulness of our results. Furthermore, we provide implications on the uniform stability and integrity of the problem's solution under investigation. Finally, we explain the practical consequences of our approach by using additional instances from particular circumstances.