  Recent intensive experimental studies revealed that (Y,Lu)Ni$_2$B$_2$C is an
anisotropic s-wave superconductor. In addition, its gap function possesses deep
point minima, whose ratio of the gap anisotropy is more than 10. On the
theoretical side, however, it is nontrivial to understand the origin of such a
peculiar superconductivity. In the present paper, we propose a mechanism of the
s-wave superconductivity with deep gap minima, based on the theoretical model
where strong electron-phonon coupling as well as the moderate magnetic
fluctuations coexist. By analyzing the strong coupling Eliashberg equation, we
find that s-wave superconducting gap function owing to the electron-phonon
coupling becomes highly anisotropic as the magnetic fluctuations increases. The
set of model parameters for realizing the strong gap anisotropy in the present
model will be appropriate for (Y,Lu)Ni$_2$B$_2$C. According to the present
mechanism, (groups of) pair of gap minima appear at points on the Fermi surface
which are connected by the nesting vector ${\bf Q}$, in both cases of s-wave
superconductors and non s-wave ones. We briefly discuss other superconductors
with highly anisotropic gap function, e.g., PrOs$_{4}$Sb$_{12}$ and
Na$_{0.33}$CoO$_2$.
