A new version of a theorem of Minh–Räbiger–Schnaubelt regarding nonautonomous evolution equations

Abstract

We develop a new version of a known theorem obtained by Van Minh, Räbiger, Schnaubelt in [N. Van Minh, F. Räbiger and R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line, Integral Equ. Oper. Theory 32 (1998) pp. 332–353]. We rely completely on the classical ‘test functions’ method designed by Perron in 1930. The advantage of such a version is that is more readable since the classical method of Perron have been known for decades and that we do not involve a sophisticated mathematical machinery. Our approach is in contrast with the general philosophy of ‘autonomization’ the nonautonomous system, since we do not require to attach the evolution semigroup. Also we point out a discrete-time version of our approach extending some known results given by Li and Henry.

Keywords:

• (non)autonomous evolution equations
• exponential dichotomy
• admissibility
• evolution semigroup

AMS Subject Classifications::

• 93D25
• 93D20
• 34G10
• 37D20
• 34C45