Asymptotic behavior of solutions to fractional stochastic multi-term differential equation systems involving non-permutable matrices

TL;DR

This paper investigates the long-term behavior of solutions to Caputo stochastic multi-term differential equations with non-permutable matrices, establishing conditions for existence, uniqueness, and asymptotic separation rates.

Contribution

It provides new results on the asymptotic separation rate and properties of solutions for Caputo SMTDEs with non-permutable matrices, extending previous understanding.

Findings

Established global existence and uniqueness of solutions.

Proved asymptotic separation property between solutions.

Derived the asymptotic separation rate for solutions.

Abstract

In this paper, we study the exact asymptotic separation rate of two distinct solutions of Caputo stochastic multi-term differential equations (Caputo SMTDEs for short). Our goal in this paper is to establish results on the global existence and uniqueness and continuity dependence of the initial values of the solutions to Caputo SMTDEs with non-permutable matrices of order $\alpha \in (\frac{1}{2},1)$ and $\beta \in (0,1)$ whose coefficients satisfy a standard Lipschitz condition. For this class of systems, we then show the asymptotic separation property between two different solutions of Caputo SMTDEs with a more general condition based on $\lambda$. Also, the asymptotic separation rate for the two distinct mild solutions reveals that our asymptotic results are general.