This exact same thing is happening to me....my phone was low on battery today whilst at work. The next time I checked it I just thought the phone was out of battery but when I got home and put it on charge it turns on for about 3 seconds to the Apple logo and then flicks off. This process is on repeat. There is nothing you can do because putting it on charge doesnt work longer than 3 seconds, and cant restore it on the computer because when you take it off the charger it has no battery to respond to the computer. This was a relatively new phone for me, had it for less than a year and have taken good care of it..........