On the persistence of $k$-exponential separation of linear cocycles under a small perturbation

Abstract: In this paper, the concept about a $k$-exponential separation of a linear cocycle $(\tilde{F},\mathcal{G})$ on $\tilde{K}\times X$ is extended for a general linear coclye whose base space $\tilde{K}$ and fibre-map-value map $\mathcal{G}$ become a nonempty set and a continuous map from $\tilde{K}$ to $L(X)$ respectvely, by removing the prior assumptions in the classical sense that $\tilde{K}$ is a compact set contained in the Banach space $X$, and $\mathcal{G}(x)$ is compact for any $x\in K$. We prove that $(\tilde{F},\mathcal{G})$ on $\tilde{K}\times X$ admits a $k$-exponential separation if the cocycle $(\tilde{F},\mathcal{G})$ is generated from a linear cocycle $(F,\mathcal{T})$ on $K\times X$ adimtting a $k$-exponential separation with $K$ being compact in the classical sense via a small perturbation. We also obtain some consequent results with their needed concepts spinning off from the one of a $k$-exponential separation of $(\tilde{F},\mathcal{G})$ on $\tilde{K}\times X$, as well as the unified terminology system around $k$-exponential separation is normalized. We apply our results to analyze the linearized structure near by an invariant set of a system generated from a dissipative system via a small perturbation, where the small perturbation is without the restriction of compactness.