Try to reset the phone by holding the sleep and home button for about 10sec, until the Apple logo comes back again. You will not lose any data by resetting, but it can cure some glitches.
If this does not help, setting it up as new device would be the next step:
How to erase your iOS device and then set it up as a new device or restore it from backups
If still no luck, set it back to factory settings, without using the latest backup and add the rest of the details manually. If the phone works after that, restore from the backup. If the problem comes back, the backup contained damaged data, causing this issue.
Use iTunes to restore your iOS device to factory settings - Apple Support