On uniform stability with growth rates in Banach spaces

On some growth concepts for dichotomic behaviors of evolution operators

“…We need to prove that the two relations from the Definition 2.4 are satisfied for the particular case when h(t) = e t . The first one follows in a similar manner as the necessity proved in Theorem III.1 from [9]. For the second relation, if (U, P ) has u.h.g and (t, s) ∈ ∆ then there are M > 1 and ω > 0 such that…”

“…Let (t, s) ∈ ∆. The relation (uhg 1 ) follows by using similar arguments as in the sufficiency of Theorem III.1 from [9]. For (uhg 2 ) we have…”

“…In what follows, we maintain all the notations and the concepts presented in our previous papers (see [3] and also [4][5][6][7][8][9][10]).…”

“…We have to prove that (uhg 1 ) and (uhg 2 ) are true in the particular case when h(t) = t + 1 for the pair (U h , P h ). For the first statement, see the necessity of the Theorem III.2 from [9].…”

"On Uniform Instability with Growth Rates in Banach Spaces"

“…We obtain generalizations of some instability with growth rates results as well as versions of stability with growth rates theorems for the case of instability. ( [2], [4], [5], [6], [7], [8], [9]).…”