Discrete and continuous versions of Barbashin-type theorem of linear skew-evolution semiflows

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.

Keywords:

• exponential stability
• Barbashin theorem
• linear skew-evolution semiflows

AMS Subject Classifications::

• 34D05
• 34D20

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).