  Suppose that $Q$ is a weak$^{\ast }$ compact convex subset of a dual Banach
space with the Radon-Nikod\'{y}m property. We show that if $(S,Q)$ is a
nonexpansive and norm-distal dynamical system, then there is a fixed point of
$S$ in $Q$ and the set of fixed points is a nonexpansive retract of $Q.$ As a
consequence we obtain a nonlinear extension of the Bader-Gelander-Monod theorem
concerning isometries in $L$-embedded Banach spaces. A similar statement is
proved for weakly compact convex subsets of a locally convex space, thus giving
the nonlinear counterpart of the Ryll-Nardzewski theorem.
