Barbashin type characterizations for the uniform polynomial stability and instability of evolution families

The aim of this paper is to give some discrete and continuous versions of Barbashin type theorems for the uniform polynomial stability and instability of evolution families in Banach spaces, by using Banach sequence spaces in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">ℋ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>ℕ</m:mi> <m:mrow> <m:mi/> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {H(N≥ 1)} and Banach function spaces in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">ℋ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>ℝ</m:mi> <m:mrow> <m:mi/> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {H(R≥ 1)} , respectively. Variants for uniform polynomial stability and instability of some well-known results in the exponential stability theory and polynomial stability theory are obtained.