The structure of the sets of nonuniformness of weak exponential dichotomous linear differential systems

E. B. Bekriaeva

Trudy Instituta matematiki, Tome 23 (2015) no. 1, pp. 12-26

Résumé

Linear $n$-dimensional system with piecewise continuous and bounded coefficients on the time half-line is called weak exponential dichotomous system if there exist positive constants $\nu_1$ and $\nu_2$ and a decomposition $\mathbb{R}^n=L_-\oplus L_+,$ such that system's solutions satisfy two inequalities for arbitrary $t\ge s\ge0$: а) if $x(0)\in L_-,$ then $\|x(t)\|\le c_1(x)e^{-\nu_1(t-s)}\|x(s)\|;$ b) if $x(0)\in L_+,$ then $\|x(t)\|\ge c_2(x)e^{\,\nu_2(t-s)}\|x(s)\|,$ where $c_1(x)$ and $c_2(x)$ are positive constants, which, in general, depend on the choice of the solutions ($c_1(x)\ge1$ and $c_2(x)\le1$). For $\varepsilon\in(0,1]$ the set of $x(0)\in L_-,$ for which one cannot take $c_1(x)=\varepsilon^{-1}$ in the estimate а) is called the first $\varepsilon$-nonuniformness set, and the set of $x(0)\in L_+,$ for which one cannot take $c_2(x)=\varepsilon$ in the estimate b) is called the second $\varepsilon$-nonuniformness set of weak exponential dichotomous linear differential system. We obtain the necessary and sufficient condition for one-parameter family of sets depending on a parameter $\varepsilon\in(0,1]$ to be the first (second) $\varepsilon$-nonuniformness sets of a weak exponential dichotomous system.