First of all, do note I don't have a floyd guitar to be sure of this, but it sounds either that the strings are new and weren't stretched properly, or the setup isn't well done. For the first one, it's very simple to try. Unlock the nut and stretch the strings. Tune to pitch, stretch them again...do this until it stops going out of tune. Then tune to pitch and lock the nut. Do your crazy stuff and see if it holds up.
If not, then it might be a setup related problem which you can send to a guitar tech, or if you are up to an adventure, you can set it up yourself. There's a great guide here:
http://www.ibanezrules.com/tech/grunge/grunge_1.htm
If you are unsure though, best take it to a tech.