Imac G5 Applications quit

Try repairing permissions and making sure you have the latest updates.

Which apps? And what apps do you have running when it happens. I have that issue when I have a lot of apps running with large files attached to those apps (PS, InDesign, and such) - with the system up time being a week or better. Maybe it has something to do with pageouts.

Firefox bums out reguarly on my iMac, so I use Safari, which bums out slightly less reguarly. The Firefox developers admitted the OS X version of Firefox wasn't developed to the same standard as other versions, since they don't use it much.

Repair Permissions. Anytime when updating or installing something that asks for the administrator password, repair permissions afterwards. Here is where you can find out how to do it. https://forums.macrumors.com/threads/89736/

Try deleting the Safari plist file.