Admissibility criteria for nonuniform dichotomic behavior of nonautonomous systems on the whole line

We give new criteria for nonuniform dichotomy of nonautonomous systems on the whole line in terms of admissibility relative to an integral equation. In our approach the input space I(R,X) is an intersection of spaces that can be successively minimized and the output space C(R,X) can be one of some well-known spaces of continuous functions. Using computational arguments, we show that the admissibility of (C(R,X),I(R,X)) leads to a nonuniform exponential dichotomy. We expose a complete analysis of the connections between admissibility and nonuniform dichotomy on the whole line and we also discuss several interesting consequences. Moreover, we obtain the explicit expression of the growth rates for dichotomy in terms of the initial exponential growth and the norm of the input-output operator. Finally, we present a direct application of the main result in the case of evolution families which admit uniform exponential growth.