Repeated power cycles after a couple of hours running.

I have a Dell 15" Flat Panel with the same symptoms. I found that if I tapped on the monitor, it would work for awhile which suggested a loose connection. I dismantled the monitor and removed the main board so that I could see the soldered connections. With the board positioned with the power connection facing away from me, I inspected all solder joints on the main board using a magnifying glass. On the left side, I saw two capacitors and found insufficient solder around the leg of one of the capacitors. I touched a soldering iron with a small dab of solder to the connection. After reassembling, I have experienced no further cycling. Going on two weeks now, it seems to have cured the problem.

before it goes out, does the picture compress on one of the sides?
it's possible the horizontal output transistor is overheating. this is usually caused by a dried out electrolytic capacitor in the driver circuit. if you're not sure what i'm talking about, have it repaired. if you understand what i just said, i'd be glad to help you find the problem.