Had the same "playing up issues" when the 5s came out last year and the battery on mine has been running down to under 20% during the day when i am not using it , this happened last year and also emails and messages not working right and other apps slow to work or even open .
Now that was a few days ago and my phone has been on for at least 3 hours , i have turned it off and back on again , i have had emails (100 +) i have already had come again and also messages . I have also had messages sent 2 days ago coming to me for the first time . But the battery is still showing 100% which was how it was prior to 9th Sept !
Whats been going on and anyone else had similar in the last 5 days or so ? Mine is an iphone4 which has worked perfectly apart from last years around this time just before i go on holiday (same last year)