  We apply the Migdal-Eliashberg theory of superconductivity to heavy-fermion
and mixed valence materials. Specifically, we extend the Anderson lattice model
to a case when there exists a strong coupling between itinerant electrons and
lattice vibrations. Using the saddle-point approximation, we derive a set of
coupled nonlinear equations which describe competition between the crossover to
a heavy-fermion or mixed-valence regimes and conventional superconductivity. We
find that superconductivity at strong coupling emerges on par with the
development of the many-body coherence in a Kondo lattice. Superconductivity is
gradually suppressed with the onset of the Kondo screening and for strong
electron-phonon coupling the Kondo screening exhibits a characteristic
re-entrant behavior. Even though for both weak and strong coupling limits the
suppression of superconductivity is weaker in the mixed-valence regime compared
to the local moment one, superconducting critical temperature still remains
nonzero. In the weak coupling limit the onset of the many body coherence
develops gradually, in the strong coupling limit it emerges abruptly in the
mixed valence regime while in the local moment regime the $f$-electrons remain
effectively decoupled from the conduction electrons. Possibility of
experimental realization of these effects in Ce-based compounds is also
discussed.
