Formula omitted]-admissibility and exponential dichotomies of cocycles

Abstract

We analyze the existence of (eventually no past) exponential and ordinary dichotomies of an exponentially bounded, strongly continuous cocycle { Φ ( θ , t ) } θ ∈ Θ , t ∈ R + over a continuous semiflow σ. Our main tool is the ( L p ( R + , X ) , L q ( R + , X ) ) -admissibility condition (i.e. there exist p , q ∈ [ 1 , ∞ ] , ( p , q ) ≠ ( 1 , ∞ ) , such that for each input f ∈ L p ( R + , X ) and θ ∈ Θ , there exists x ∈ X such that the output u ( ⋅ ; θ , x , f ) : R + → X , u ( t ; θ , x , f ) = Φ ( θ , t ) x + ∫ 0 t Φ ( σ ( θ , s ) , t − s ) f ( s ) d s belongs to L q ( R + , X ) ). We prove that the above admissibility condition implies that the output is bounded above by the input but nonuniformly with respect to θ. Requiring that the boundedness to be uniform with respect to θ, we prove that the above admissibility condition assures the existence of a no past exponential dichotomy for { Φ ( θ , t ) } θ ∈ Θ , t ∈ R + . Variants for ordinary dichotomy and also complete characterizations for the exponential dichotomy of cocycles are obtained. It is worth to note that we involve a concept of a “no past” exponential dichotomy for cocycles weaker than the well-known concept defined by Sacker and Sell (1994) [33]. Our definition of exponential dichotomy follows partially the definition given in Chow and Leiva (1996) [2] in the sense that we allow the unstable subspace to have infinite dimension. The main difference is that we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable space is invariant under the cocycle). Thus we generalize some known results due to Perron (1930) [23], Daleckij and Krein (1974) [6], Massera and Schäffer (1966) [14], Nguyen van Minh, Räbiger and R. Schnaubelt (1998) [18].