  We include $p p \to p \Delta^{+} \mu^{+} \mu^{-}$ and $p p \to \Delta^{+}
\Delta^{+} \mu^{+} \mu^{-}$ processes in addition to the standard $p p \to p p
\mu^{+} \mu^{-}$ process both in equivalent-photon approximation (EPA) and in
exact $2 \to 4$ calculations. For comparison we calculate also the continuum
proton dissociation in a recently developed $k_{t}$-factorization approach to
$\gamma \gamma$-involved processes with parametrizations of $F_{2}$ structure
function known from the literature. The calculated cross section including all
the processes is considerably larger than the one measured recently by the
ATLAS collaboration. We calculate absorption effects for $p p \to p p \mu^{+}
\mu^{-}$ process in the momentum space. The final cross section with absorption
effects is by 10\% larger than the one measured by the ATLAS collaboration
which is difficult to explain. Several differential distributions with the
ATLAS experimental cuts are presented. It is shown that the processes with
electromagnetic $p \to \Delta(1232)$ and $p \to N(1440)$ transitions, that have
similar characteristics as the $p p \to p p \mu^+ \mu^-$ process, increase the
cross section for $\mu^+ \mu^-$ production and thus can affect its theoretical
interpretation when the leading baryons are not detected as is the case for the
CMS and ATLAS measurements. The mechanism of dissociation into hadronic
continuum is not under full control as the corresponding absorption effects are
not fully understood. We present first predictions for future ATLAS experiment
with the ALFA sub-detectors and discuss observables relevant for this
measurement.
