Hey everyone,
So I have seen this issue arise in quite a few customer's who have brought their phones into my companies store. (I work for a 3rd party to Apple but we repair their phones and what not) There are a few fixes I have seen that fix it a couple you guys have said already but sometimes they do fix the issues.
The first one is what we call DFUing a phone. Essentially what you are doing is wiping the phone of everything and reinstalling iOS.
First you want to make sure that you back up to iCloud. Don't back up to iTunes in this case. If it is a software issue it will replicate when you restore from iTunes backup.
1. Put the phone into recovery mode.
A. Hold lock button and home button at same time until you see the apple logo then let go of lock button while still holding home button until you see the iTunes logo and the lightning connector below it.
2. Plug in your to your computer. iTunes should open. Now at this point the best way to reinstall the os is using a file that is stored locally on your machine. There are a few good websites that have the iOS files store so you can download them. Make sure you get the correct file and have it downloaded somewhere you can remember it.
3. Once iTunes is open you should hit option(mac) or alt(windows) at the same time you click restore.
4. A dialogue box will pop up and you will select the file that you want to restore to.
5. It will go through the reinstall and iOS will boot back up for you.
If this doesn't work get a new sim card from your carrier and then once you do, do the following:
Settings -> General -> Reset -> Reset All Network settings
If that doesn't work then it is very possible that it's a hardware failure and you need to get in touch with Apple.
I hope this helps some people!