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Matrix measure and asymptotic behaviors of linear advanced systems of differential equations. (English) Zbl 1502.34073

The authors study the convergence of solutions for the linear system of advanced differential equations \[x'(t)+\sum_{k=1}^NA_k(t)x(t+h_k(t))=0,\ \ \ t\ge t_0, \tag{1}\] where \(x\in\mathbb{R}^n\), the nonnegative functions \(h_k(t)\) and the \({(n\times n)}\)-matrices \(A_k(t)\) are continuous on \([t_0,+\infty)\).
Using the matrix measure \(\mu(A)=\lim_{\varepsilon\to0}\frac{1}{\varepsilon}(\|I+\varepsilon A\|-I)\) and the Banach fixed point theorem, the authors present sufficient conditions for the convergence and exponential convergence of the considered system.
Reviewer’s remark: The paper contains inaccuracies, and the proof of the main result is incomplete.

MSC:

34K06 Linear functional-differential equations
34K25 Asymptotic theory of functional-differential equations
47N20 Applications of operator theory to differential and integral equations
34K20 Stability theory of functional-differential equations
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