Why won't my phone restore?

I plugged in my phone to charge after sending a text and 15 minutes later it turned off. I haven't dropped or spilled anything on it. It will not turn on unless its plugged into a wall plug or ihome. It says connect to iTunes. I connected it to my computer and have the latest iTunes and iPhone software. I have to force start my iphone by pressing the power button and home button at the same time and once it starts turning on I keep holding the home button until iTunes pops up on my computer. I click restore on iTunes and it will start to restore for about ten seconds and then my iphone will turn off. (It sort of flashes like when a screenshot is taken and then is completely black) then on the computer it says error 2001. I've tried using different power chords and unplugging multiple times. I've also restarted my computer many times and nothing seems to work. I've also tried DFU mode. I've had this phone for about two months now so it's still new but it also means I can't return it to apple. (Note: it's an iPhone 4.)