Admissibility and exponential trichotomy of dynamical systems described by skew-product flows

The aim of this paper is to present a new and very general method for the detection of the uniform exponential trichotomy of dynamical systems. The investigation is done in several constructive stages that correspond to three admissibility properties that are progressively introduced with respect to an associated input–output system. We prove that the uniform admissibility of the pair (Cb(R,X),L1(R,X))(Cb(R,X),L1(R,X)) for the associated system is a sufficient condition for the existence of a uniform trichotomic behavior of the initial dynamical system. If p∈(1,∞)p∈(1,∞) and the pair (Cb(R,X),L1(R,X))(Cb(R,X),L1(R,X)) is uniformly p-admissible then we obtain the uniform exponential trichotomy. Next, we study whether the admissibility conditions are also necessary for the uniform exponential trichotomy. Supposing that a dynamical system has a uniform exponential trichotomy we prove that the associated input–output system has unique bounded solutions in certain subspaces. Finally we obtain that the uniform p  -admissibility of the pair (Cb(R,X),L1(R,X))(Cb(R,X),L1(R,X)) is a necessary and sufficient condition for uniform exponential trichotomy.