On the Uniqueness and Existence of Solutions for Linear and Nonlinear Impulsive Second-Order Differential Equations with Applications to Neural Oscillators

Abstract

This paper investigates the uniqueness and existence of solutions for second-order differential equations from both classical and modern perspectives. In the first part, classical techniques such as energy functions and the Grönwall–Bellman inequality are employed to re-establish the uniqueness of solutions for second-order linear homogeneous equations with continuous coefficients. In the second part, the analysis is extended to nonlinear impulsive systems inspired by biological neural oscillators and Central Pattern Generators (CPGs), which include time-dependent damping and instantaneous state changes. Existence and uniqueness of periodic or almost-periodic solutions are established using topological fixed-point theory and Lyapunov-like functions under biologically realistic assumptions. Numerical simulations using MATLAB are presented to validate the theoretical findings. The results have direct applications in bioengineering, neural dynamics, and control systems where hybrid continuous-discrete behavior is present.