GRAWITA: VLT Survey Telescope observations of the gravitational wave sources GW150914 and GW151226

We report the results of deep optical follow-up surveys of the first two gravitational-wave sources, GW150914 and GW151226, done by the GRAvitationalWave Inaf TeAm Collaboration (GRAWITA). The VLT Survey Telescope (VST) responded promptly to the gravitational wave alerts sent by the LIGO and Virgo Collaborations, monitoring a region of 90 and 72 deg2 for GW150914 and GW151226, respectively, and repeated the observations over nearly two months. Both surveys reached an average limiting magnitude of about 21 in the r band. The paper describes the VST observational strategy and two independent procedures developed to search for transient counterpart candidates in multi-epoch VST images. Several transients have been discovered but no candidates are recognized to be related to the gravitational wave events. Interestingly, among many contaminant supernovae, we find a possible correlation between the supernova VSTJ57.77559-59.13990 and GRB150827A detected by Fermi-GBM. The detection efficiency...