The ESP Guitar Company

There are a bunch of possible causes.  Are your strings properly stretched?  Are all strings out of tune?  Just one?

Are you sure it's the nut?    If so, take off the locking screws, and see if the caps are worn.  If they are not worn, make a mark on the string, behind the nut.  A little dot with a sharpie will work perfectly.  Play with your tremolo, and then look at the marks you made on all the strings to see if they moved.  If so, you have diagnosed your problem.

It could also be that it's not set up correctly.   It's also possible that the knife edges are shot, which is the worst case scenario.  This usually is the problem when the strings stay sharp on pulling up on the trem, and flat when diving.  If this is the case, try some lubrication, like simply rubbing a pencil around the point of contact, or use some nut sauce.

Again, there are many things that could be the cause.  The first step is to start diagnosing the problem.  Start with the marks on the strings behind the nut.  Let us know your findings.

It was the strings slipping under the caps on the locking nut.  Use a nail file and scuffed the bottom of the caps. Now works like a dream.  Thank you.