  We solve the anisotropic, full-bandwidth and non-adiabatic Eliashberg
equations for phonon-mediated superconductivity by fully including the first
vertex correction in the electronic self-energy. The non-adiabatic equations
are solved numerically here without further approximations, for a one-band
model system. We compare the results to those that we obtain by adiabatic
full-bandwidth, as well as Fermi-surface restricted Eliashberg-theory
calculations. We find that non-adiabatic contributions to the superconducting
gap can be positive, negative or negligible, depending on the dimensionality of
the considered system, the degree of non-adiabaticity, and the coupling
strength. We further examine non-adiabatic effects on the transition
temperature and the electron-phonon coupling constant. Our treatment emphasizes
the importance of overcoming previously employed approximations in estimating
the impact of vertex corrections on superconductivity and opens a pathway to
systematically study vertex correction effects in systems such as high-$T_c$,
flat band and low-carrier density superconductors.
