Square integrable solutions and stability of a second-order stochastic integro-differential equation

Area of Science:

• Mathematics
• Dynamical Systems
• Stochastic Analysis

Background:

• Nonlinear stochastic integro-differential equations (SIDEs) with delays are crucial in modeling complex systems.
• Existing research often simplifies these models by addressing fewer complexities simultaneously.
• Understanding the stability and boundedness of solutions is essential for practical applications.

Purpose of the Study:

• To investigate the stochastic asymptotic stability, boundedness, and square integrability of solutions for a specific class of second-order nonlinear SIDEs with multiple variable delays.
• To develop a unified analytical framework that incorporates stochastic perturbations, time-varying delays, and integro-differential memory terms.
• To extend and generalize existing results in the field by relaxing restrictive assumptions.

Main Methods:

• Construction of a specialized Lyapunov-Krasovskii functional (L-KF).
• Application of stability theory for stochastic differential equations.
• Analysis of the impact of multiple variable delays and memory terms.
• Utilizing techniques to handle nonlinearities and stochastic perturbations.

Main Results:

• The proposed Lyapunov-Krasovskii functional (L-KF) effectively addresses the combined complexities of the equations.
• Theoretical conditions for stochastic asymptotic stability, boundedness, and square integrability of solutions are derived.
• The framework generalizes and improves upon previous findings in the literature.
• Numerical simulations confirm the theoretical predictions and demonstrate practical applicability.

Conclusions:

• The study provides a robust theoretical foundation for analyzing complex nonlinear stochastic integro-differential equations.
• The developed methods offer a more comprehensive approach to understanding the behavior of delayed stochastic systems.
• The findings have implications for modeling and control in fields utilizing such equations, validated by simulations.