Iphone 5c keeps shutting itself off. And other issues?

I have a 32GB iphone 5c and about a week ago it started shutting itself off for no reason. It doesn't matter if I'm in the middle of something or if I've just unlocked it, it will shut itself off, display the black screen with the apple logo for about 10-15 seconds and then start working again by itself. I'm running the latest software updates and everything so I'm not sure what's going on. Also, when texting for about the past week as well it seems to be going in slow motion. I can have a whole sentence typed and the letters slowly appear onscreen. Kind of delays when I hit send on that particular text, and then it goes back to normal. I have over 9GB free of my memory so I have no clue why its all of a sudden having issues. Any advice would be greatly appreciated.

Hi connect phone to pc open i tunes add a song and disconeect, you prob dnt need to add song just connect to i tunes and some faults go bk to normal

Sometimes it's a hardware problem with the battery or electronics, but often enough it's a software issue that a few simple steps can fix. Try this: restoring your iPhone or iPad in iTunes. ITunes is a safer bet than iCloud because it actually offloads, re-installs, and reloads your data rather than doing everything in-place and on your device. That can sometimes shake loose bad bits that even a hard reset or iCloud restore can't.
1. Connect your iPhone, iPod touch, or iPad to iTunes on your Mac or Windows PC via Lightning or 30-pin Dock connector.
2. Click on the Device tab at the top left.
3. Click on Backup.
4. Wait for the backup to finish.
5. Click on Restore
6. Wait for the restore to finish.

Data loss due to restore can't be found back iPhone Data Recovery (http://www.imyfone.com/iphone-data-recovery/ When you're all done, see if your iPhone or iPad stays on.
If not, Apple for help.

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