  In this work we construct certain general bundles $<\mathfrak{M},\rho,X>$ and
$<\mathfrak{B},\eta,X>$ of Hausdorff locally convex spaces associated with a
given Banach bundle $<\mathfrak{E},\pi,X>$. Then we present conditions ensuring
the existence of bounded sections $\mathcal{U}\in \Gamma^{x_{\infty}}(\rho)$
and $\mathcal{P}\in \Gamma^{x_{\infty}}(\eta)$ both continuous at a point
$x_{\infty}\in X$, such that $\mathcal{U}(x)$ is a $C_{0}-$semigroup of
contractions on $\mathfrak{E}_{x}$ and $\mathcal{P}(x)$ is a spectral projector
of the infinitesimal generator of the semigroup $\mathcal{U}(x)$, for every
$x\in X$.
