Sounds like bad capacitors, This problem plagued iMac G5's. There are about 25 capacitors on the computer's logic board that can fail. They can be replaced however. There are kits on eBay that include all of them. This is usually a degenerative problem, where it gets worse, and worse over time, it can start with freezing, image distortion, and randomly shutting down, and it can get to the point where it won't even turn on. This is a tricky repair, however it can be done with a steady hand, and some patience.