  We present optical follow-up observations with the DDOTI telescope of
gravitational-wave events detected during the Advanced LIGO and Advanced Virgo
O3 observing run. DDOTI is capable of responding to an alert in a few minutes,
has an instantaneous field of about 69 deg$^{2}$, and obtains $10\sigma$ upper
limits of $w_{\rm lim}=18.5$ to 20.5 AB mag in 1000~s of exposure, depending on
the conditions. We observed 54\% (26 out of 48) of the unretracted
gravitational-wave alerts and did not find any electromagnetic counterparts. We
compare our upper limits to various possible counterparts: the kilonova
AT~2017gfo, models of radioactive- and magnetar-powered kilonovae, short
gamma-ray burst afterglows, and AGN flares. Although the large positional
uncertainties of GW sources do not allow us to place strong constraints during
O3, DDOTI observations of well-localized GW events in O4 and beyond could
meaningfully constrain models of compact binary mergers. We show that DDOTI is
able to detect kilonovae similar to AT~2017gfo up to about 200~Mpc and
magnetar-powered kilonovae up to 1~Gpc. We calculate that nearby ($\lesssim$200
Mpc) afterglows have a high chance ($\approx$70\%) to be detected by rapid
($\lesssim$3 hours) DDOTI observations if observed on-axis, whereas off-axis
afterglows are unlikely to be seen. Finally, we suggest that long-term
monitoring of massive BBH events with DDOTI could confirm or rule out late AGN
flares associated with these events.
