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Are you sure the iMac is really shutting down or just turning off the backlight? The tell-tale sing would be to let the unit fully boot-up (wait 3 minutes) and the press the button the apple keyboard that ejects the CD-ROM. If the CD-ROM door opens, you iMac is running with disabled or broken backlight.
The inverter circuit on the iMac G4 is known to be weak.
There could be another problem, though. The iMac G4 is also known to have weak internal power supplies. Here's how you can tell:
Hold down the CMD OPT "O" and "F" keys at the same time while pressing the power button. This will tell the iMac that you want to boot into the Open Firmware shell, bypassing the Mac OS X startup. If this works, your power supply is good, the inverter is good and you have a problem with your Open Firmware boot string. To fix the boot string, Power on the unit while holding down the keys CMD-OPT-P-R to zap the parameter RAM (PRAM) and reset everything to defaults.
If you were not able to boot into the Open Firmware shell, your problem is either a failed power supply or a failed screen backlight.
The last possible culprit is your iMac could suffer from "Bulging Capacitor Syndrome", where a batch of iMac G4s, iMac G5s another electronics were fitted with a bad batch of capacitors. Eventually, the caps will rupture and cause irreparable damage to the logic board. Only way to verify this is to pull the unit apart and visually inspect. If you see little green or blue cylinders that seem to be bulging or oozing a thick oily goo, that's your problem. Only recourse is to shop around on eBay and look for a replacement logic board.
try these articles:
also boot to cd and run disk first aid. If it can repair the you have 3 options
1: try 3rd part utility program such as disk warrior or drive genius.
2: try to back up.. you can use target disk mode and hopefully you hard drive will mount, if you dont know how to do target disk mode do a search. if not drive saver can save data but it's expensive
3: erase and install
The power supply is part of the motherboard in the iMac G4, so it's hard to change it. The repair would certainly cost more than the price of your iMac.
You can try to take off the small battery, and reboot it to see if it changes.
To open the mac lay it down, screen on the table (on a t-shirt). Unscrew the four screws (they wont come out, so just unscrew them. Pull the screws and take off the silver plate. Then you'll have access to the inside.
Some of the start up files have been corrupted or are missing. Start up from the OSX system disc and them run Disc Utilities from the installer toolbar. Check the hard drive for errors. Repair any errors if they are found. If it's not possible for Disc First Aid to repair the hard drive you will need to replace the drive and re-install OSX.