engleski

Asymptotic stability of nonuniform behaviour

This paper is devoted to exponential dichotomies of nonautonomous difference equations. Under the assumptions that $(A_m)_{; ; ; ; m\in \Z}; ; ; ; $ is a sequence of bounded operators acting on an arbitrary Banach space $X$ that admits a uniform exponential dichotomy and that $(B_m)_{; ; ; ; m\in \Z}; ; ; ; $ is a sequence of compact operators such that $\lim_{; ; ; ; \lvert m\rvert \to \infty}; ; ; ; \lVert B_m\rVert=0$, D. Henry proved that either the sequence $(A_m+B_m)_{; ; ; ; m\in \Z}; ; ; ; $ admits a uniform exponential dichotomy or there exists a bounded nonzero sequence $(x_m)_{; ; ; ; m\in \Z}; ; ; ; \subset X$ such that $x_{; ; ; ; m+1}; ; ; ; = (A_m+B_m)x_m$ for each $m\in \Z$. In this paper we prove Henry's result in the setting of \emph{; ; ; ; nonuniform}; ; ; ; exponential dichotomies. Then we obtain a result on roughness of the nonuniform exponential dichotomy and give stability of Lyapunov exponents. In addition, we establish corresponding results for dynamics with continuous time.

nonuniform exponential dichotomy, roughness

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