I have had error 4004, 4013 and 4014 messages since trying to update/restore my iphone 5c. It went a bit crazy after I tried the latest OS update and won't turn on. Any ideas?

My Iphone 5c will not turn on/freezes since I tried to install the new OS on it. I've tried itunes updates and restore but get a string of 4004, 4013 and 4014 messages while trying to correct it. I'm wondering if it's even possibly turning itself off as the system tries to correct. Does anybody have any ideas. My use of the phone is fairly modest and it's not taken any knocks, been around water etc...