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Admissibility and nonuniform exponential dichotomies

Nonuniform exponential dichotomy represents the nonuniform hyperbolicity for evolution family. Considerable efforts have been made to the relationship between the existence of nonuniform exponential dichotomies and the admissibility of a pair of function classes. In order to reduce the effect of nonuniformity, most results on this aspect need Lyapunov norms in function classes. However, it is difficult to construct an appropriate Lyapunov norm before the existence of a nonuniform exponential dichotomy is established, which happens in the converse problem from admissibility to dichotomy. In this paper, we prove the relationship without a Lyapunov norm. Then we apply our results to giving the relationship between nonuniform exponential dichotomy and invertibility of a differential mapping on some function spaces.

evolution family ; nonuniform exponential dichotomy ; admissibility ; differential operator ; infinite-dimension.

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