Iphone Will not turn on or stay on unless plugged in?

I have an iphone 3g with supposed new battery, bought it used. Anyway, worked fine the past two weeks. Left it in car overnight, wake up and battery is dead. Phone won't turn on. I go to plug it into car charger and the apple logo comes up and then fades away slowly after 20 seconds. Now tonight, I connect to my itunes via usb and nothing happens, can't restore it. Went back to my car to charge and the apple logo comes on and goes away after 20 seconds, this time doesn't fade, black screen. I get hold power and home together and get it to the itunes connect screen. However, as soon as i unplug it from car charger it goes off. What is happaning and how to fix. The guy at store said just resotore it via itunes but I can't get it to turn on or recognize it Help! Ty

Apple has finally got what they wanted. They suggest buying a new one…

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