Matrix measure and asymptotic behaviors of linear advanced systems of differential equations

In this article, we study the convergence and exponential convergence of solutions for the linear system of advanced differential equations x ′ t + ∑ k = 1 N A k t x t + h k t = 0 , t ≥ t 0 ≥ 0 . The idea used here is to construct appropriate mappings by the fundamental matrix solution of x ′ t = A t x t . Then, we apply the matrix measure and Banach fixed point theorem to obtain sufficient conditions satisfying convergence and exponential convergence of the considered system. The obtained theorems generalize and improve previous results of Dung (Acta Math Sci 35(3):610–618, 2015).