The lens cover on my Sony CyberShot DSC-W50 won't open automatically. The two doors normally separate to open, however, they now act like they are stuck on something (as if it needs to be oiled or cleaned on the underside of the door).

See this page with 7 tips on how to try to fix this problem:

Do It Yourself Digital Camera Repair: Fixing a Lens Error on a Digital Camera

If you can't fix it following those tips, you will probably need to replace the camera. The camera is old enough that it is probably out of warranty (unless you bought an extended warranty), and the cost to repair it will probably exceed the camera's value so you should just put the money towards buying a new camera.

Actually, the link above is for a different issue. Sounds like what you're experiencing is a stuck automatic lens cover. These can also be repaired at home, relatively easily. Here's a more appropriate link for your issue from the same blog (great blog isn't it :-) )

http://camerarepair.blogspot.com/2007/11/repair-of-stuck-lens-covers.html