You have a problem with the power board section of the TV then.
If it takes a while to warm up then you have capacitors that are beginning to fail on the power board section inside the tv.
If your good at soldering and any kind of basic electronics, this can be fixed by replacing all of the caps on the power section of the board just after the main internal fuse section.
The sign of a bad cap is a doming of the top side where it should be flat, or can dome at the bottom where it contacts the circuit board. and or a sticky fluid or residue is present on the cap.
To replace them all you have to do is get the voltage of each cap, and its farad rating example 2200uf.
Until you do this the tv will just get to the point where it never turns on.
Its a cheap fix, and this will often happen to most Lcd tv`s or monitors of an age of four years or more, depending on the quality of the components used.
Radio shack or any place like RS components, Tandy. should have a wide range of caps for what you need.
http
/uk.rs-online.com/web/