An iBook G4 says "iBook G4" on the bezel below the display.
For a more detailed descriptions and more definitive identification of iBook models, check out this list:
Let us know exactly which model it is so we are better able to assist you.
It sounds like you may be having logic board problems. Enough Dual USB G3 iBooks suffered from the logic board problem that Apple began the iBook Logic Board Repair Extension Program in January of 2004.
Do the symptoms look familiar? After leaving it sit overnight, does squeezing the case to the left of the trackpad allow you to start up and work for a bit as long as you keep the pressure on it there? If so, you may want to consider trying the shim fix suggested by John Sawyer here in the Apple Discussions iBook G3 (Dual USB) Forum:
[Y]ou can often fix this yourself by removing the iBook's bottom housing, and placing a shim of any sort, about 1mm to 1.5mm thick, onto the raised square on the bottom shield. I use a Scotch mounting square--you can get them in hardware stores and many grocery stores. When you reinstall the bottom case, it will press against this shim, which will press against the graphics chip, and may allow the chip to come into better contact with the logic board.
CJS Macintosh Repair
Instructions for removing the lower case:
If the shim fix doesn't work (or you're leery of trying it), DT&T Services in Fremont, California offers the cheapest repair I've found. They have a six-month warranty on the repair.
Once you get it fixed, try never to pick it up by the left front corner alone. The theory is that picking it up by the left front corner alone can contribute to logic board failure when the case flexes, causing the graphics chip to come loose from the logic board. If you're picking it up in your left hand (as I often do mine), pick it up in the center under the trackpad. Good luck.
Please note: Because a G3 iBook is quite old at this point, if you are faced with paying a repair bill on it, it may be wiser to put that money toward a new (or used) MacBook. They are quite a spectacular upgrade from the iBook G3.
Again, good luck with whatever you decide to do.