  $J/\psi$ production in p-p ultra-peripheral collisions through the elastic
and inelastic photoproduction processes, where the virtual photons emitted from
the projectile interact with the target, are studied. The comparisions between
the exact treatment results and the ones of equivalent photon approximation are
expressed as $Q^{2}$ (virtuality of photon), $z$ and $p_{T}$ distributions, and
the total cross sections are also estimated. The method developed by Martin and
Ryskin is employed to avoid double counting when the different production
mechanism are considered simultaneously. The numerical results indicate that,
the equivalent photon approximation can be only applied to the coherent or
elastic electromagnetic process, the improper choice of $Q^{2}_{\mathrm{max}}$
and $y_{\mathrm{max}}$ will cause obvious errors. And the exact treatment is
needed to deal accurately with the $J/\psi$ photoproduction.
