Two Generalized Classes of Exponentially Dichotomous Linear Differential Systems on the Time Half-Line without Uniform Estimates for the Solution Norms. I

Abstract

We consider two generalizations $A{\cal E}_n$ and $W{\cal E}_n$ of the class ${\cal E}_n$ of linear $n$-dimensional exponentially dichotomous systems on the half-line. The definition of these classes differs from the classical definition of exponential dichotomy in terms of estimates for the norms of solutions in that we allow these estimates to hold starting from some time depending on the solution (the class $A{\cal E}_n)$ or the constant factors in the estimates to depend on the solution (the class $W{\cal E}_n)$. We give an equivalent definition of the class $W{\cal E}_n$ in the language of Bohl exponents and prove that the proper inclusions ${\cal E}_n\subset A{\cal E}_n \subset W{\cal E}_n$ hold for $n\geq 2$.