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The aim of this paper is to obtain necessary and sufficient conditions for uniform exponential trichotomy of evolution families on the real line. We prove that if p ∈ (1,∞) and the pair (C b (R,X),C c (R,X)) is uniformly p-admissible for an evolution family [InlineMediaObject not available: see fulltext.] ={U(t,s)} t ≥ s then [InlineMediaObject not available: see fulltext.] is uniformly exponentially trichotomic. After that we analyze when the uniform p-admissibility of the pair (C b (R, X), C c (R, X)) becomes a necessary condition for uniform exponential trichotomy. As applications of these results we study the uniform exponential dichotomy of evolution families. We obtain that in certain conditions, the admissibility of the pair (C b (R,X),L p (R,X)) for an evolution family [InlineMediaObject not available: see fulltext.]={U(t,s)} t ≥ s is equivalent with its uniform exponential dichotomy.
Mathematische Zeitschrift – Springer Journals
Published: Jan 26, 2006
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