iphone 4 update to ios 7 failed - cycles in and out of restart screen
If you have a recent backup (and thus aren't afraid of doing a fresh install and restore), there are a few things to try - all of them involve putting the phone in a mode that makes iTunes prompt you to do a complete restore/update/sync afresh.


Put the phone in Recovery Mode, and restore it.


If that doesn't help:


Try putting the phone in DFU mode and restoring it.
If you find the above instructions difficult to follow, especially with the timing between actions

just download redsn0w (look for the download links at the bottom of the blog post)
run redsn0w
go to the DFU option
it will guide you step-by-step to put the phone in DFU mode
once that's done, restore it from iTunes. Although redsn0w is primarily meant for jailbreaking, it is very useful for putting a phone in DFU mode.