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 h -instability, where h = ( h n ) n ∈ N 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 -instability, investigate its connections with known instability concepts, and provide examples illustrating the sharpness of the results.