Abstract
This article is concerned with a well-known theorem of Barbashin which states that an evolution family {U(t, s)} t≥s≥0, or simple 𝒰, is uniformly exponentially stable if and only if 𝒰 satisfies the integral condition . In fact, the author formulated the above result for non-autonomous differential equations in the frame work of finite-dimensional spaces. The aim of this article is to give discrete and continuous versions of Barbashin-type theorem for the case linear skew-evolution semiflows. Giving up disadvantages in Barbashin's proof, we shall extend this problem, based on the recent methods. Thus we obtain necessary and sufficient conditions for uniform exponential stability, generalizing a classical stability theorem due to Barbashin.
Acknowledgements
The author is grateful to the referees for carefully reading this article and for their valuable comments. The author was partially supported by the grant TN-10-08 of College of Science, Vietnam National University, Hanoi (Dai hoc Khoa hoc TU Nhien, Dai hoc Quoc Gia Ha Noi).