A proof of a sufficient condition for a strongly continuous
semigroup $\{ T(t)\}_{t\ge 0}$ on a Banach space $X$ to be uniformly
exponentially stable is given. This result is a simplification of an earlier theorem
by van Neerven, and concludes that a semigroup is uniformly exponentially stable
provided $\sup\nolimits_{||x||\le 1}J(||T(\cdot)x||)<\infty$ ; here $J$ is a
certain nonlinear functional with certain natural properties. A
non-autonomous version of this theorem for evolution families is also given.
This implies the well-known Datko-Pazy and Rolewicz Theorems. This result is connected to the
uniform asymptotic stability of the well-posed linear and non-autonomous
abstract Cauchy problem
\begin{equation*}
\left\{
\begin{array}{rcl}
\dot{u}(t)& = & A(t)u(t),\quad t\geq s\geq 0, \\
u(s) & = & x\in X.
\end{array}
\right.
\end{equation*}