I'd just like to say that ahs70's post on Dec 1, 2013 at 1:17am on Pg. 9 worked for me as well.
If your iPhone's battery is acting up you need to determine if it's a software issue or a hardware/battery issue. To do that, after syncing your iPhone, erase/hard reset all data and settings from your iPhone to what it was from the factory as new. This can be found in Settings/General/Reset/Erase All Content ansd Settings. Then test it out for a day or two. If it's still acting up then it's likely a hardware issue and you will need to get your battery replaced. If your warranty is done than check out this link to get a battery kit to replace your iPhone battery yourself: http://www.ifixit.com/Store/iPhone/iPhone-5-Replacement-Battery/IF118-001#produc tDescription
If your IPhone is now working properly after hard resetting it than it's most likely a software issue. This battery issue only started after upgrading to the new iOs7, so it is obviously related to the update not properly installing itself. What has worked for me and many others is to manually update to iOS 7.0.4 using the full downloaded version of the update which is about 1.32 Gigs.
But first you need to find out what model of iPhone you have, click here to determine if it's GSM or CDMA: http://support.apple.com/kb/HT3939?viewlocale=en_US&locale=en_US
Then go here and scroll down to download the iOS 7.0.4 complete update for your device: http://www.downloadios7.org/download-ios-7-0-4-ipsw-file.html
Then scroll up on the download page and follow the instructions of updating your iPhone manually following the instructions below: "IPSW with iTunes"
And your done.
Problem solved.
I only posted this because it was a very frustrating problem that noone really seemed to figure out a solution for.
I hope this helps.
Emile Beaulieu