Instability with growth rates of discrete systems in Banach spaces

In this paper, we investigate new forms of instability for linear discrete-time dynamical systems in Banach spaces, focusing on characterizations described by non-exponential growth rates. We introduce and analyze the concept of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>h</a:mi> </a:math> -instability, where <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>h</b:mi> <b:mo>=</b:mo> <b:mo stretchy="false">(</b:mo> <b:msub> <b:mi>h</b:mi> <b:mi>n</b:mi> </b:msub> <b:msub> <b:mo stretchy="false">)</b:mo> <b:mrow class="MJX-TeXAtom-ORD"> <b:mi>n</b:mi> <b:mo> ∈ </b:mo> <b:mrow class="MJX-TeXAtom-ORD"> <b:mi mathvariant="double-struck">N</b:mi> </b:mrow> </b:mrow> </b:msub> </b:math> is a non-decreasing sequence that diverges to infinity. This formulation generalizes classical and nonuniform notions of exponential and polynomial instability by allowing for a variety of growth profiles. Our approach studies a linear system. We establish sufficient conditions for <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mi>h</h:mi> </h:math> -instability, investigate its connections with known instability concepts, and provide examples illustrating the sharpness of the results.