On weak exponential stability of evolution operators in Banach spaces (Q988136)

According to the authors, an \textit{evolution operator} is a family of (not necessarily linear) operators \(\{ U(t, s)\mid t \geq s \geq 0 \}\) on a Banach space \(E\) satisfying \(U(t, t) = I\), \(U(t, s)U(s, r) = U(t, r)\) \((t \geq s \geq r)\) plus a continuity assumption and the exponential growth condition \[ \|U(t, t_0)u\| \leq Me^{\omega(t - t_0)} \|u\| \quad (t \geq t_0 \geq 0 , \;u \in E) \, . \tag{1} \] If (1) is reinforced to \[ \|U(t, t_0)u\| \leq Ne^{-\nu(t - s)} \|U(s, t_0) u\| \quad (t \geq s \geq t_0 \geq 0, \;u \in E) , \tag{2} \] then \(\{U(t, s)\}\) is uniformly exponentially stable. Finally, if (2) holds for \(t_0\) depending on \(u\), the evolution operator is weakly exponentially stable. The results are on various properties of \(\{U(t, s)\}\) that are equivalent to weak exponential stability.