Absolute Asymptotic Stability of Discrete Linear Inclusions

The article is devoted to the study of absolute asymptotic stability of discrete linear inclusions in Banach (both finite and infinite dimensional) space. We establish the relation between absolute asymptotic stability, asymptotic stability, uniform asymptotic stability and uniform exponential stability. It is proved that for asymptotical compact (a sum of compact operator and contraction) discrete linear inclusions the notions of asymptotic stability and uniform exponential stability are equivalent. It is proved that finite-dimensional discrete linear inclusion, defined by matrices $\{A_1,A_2,\dots,A_m\}$, is absolutely asymptotically stable if it does not admit nontrivial bounded full trajectories and at least one of the matrices $\{A_1,A_2,\dots,A_m\}$ is asymptotically stable. We study this problem in the framework of non-autonomous dynamical systems (cocyles).