I have an Olympus stylus 720 SW, when I turn it on the lens cover either won't open or opens only partly.

SOURCE: olympus 770 sw lens cover not closing

Your problem may be due to weak/worn out batteries or corrosion on the battery contacts inside the camera which can prevent the full power of the batteries from flowing into the camera. Try this free fix before you do anything else: remove the batteries and wipe the camera contacts firmly with a dry cloth (heavy corrosion may require cleaning with a wire brush, steel wool, or sandpaper). Remove any residue that may have fallen into the battery compartment during cleaning, then wipe both ends of the batteries and place them back in the camera. This cleaning clears the problem about 90% of the time. If it doesn't work for you, chances are that your batteries need to be replaced because they're just too worn to properly power the camera. And then, of course, there's the possibility that your camera may have a problem that requires professional repair.

SOURCE: Olympus 720 sw LCD Screen blank & white

Just in Time for Christmas then!
So decisions - live with out the screen- it appears to be a connexion failure or a main board failure.
Or replace the camera - a repair will cost almost as much.

SOURCE: Olympus Stylus 1030 SW

Try soaking in a bowl of clean warm water.  I had the same problem and this resolved it.  gotta love water proof camera!!!

SOURCE: the lens cover on my olympus stylus 770sw is stuck

I have a u770 with a broken lens. I would like to either buy yours to repair mine or I am willing to sell mine so you can repair yours. Email at (g4)(kpp)@(hotmail).(com) - remove the brackets - trying to avoid spam