iPhone 4 won't turn on when plugged into computer nor charger

My refurbished iPhone 4 has the latest iOS 5.1.1 update and my mom's computer has iTunes version 10.6.3.25. However, recently it has been having issues holding a charge and now as of yesterday it won't turn on period. When it does it's only for a split few seconds and then it turns off again.

When it's connected to a wall charger it immediately turns on

and then lock screen apprears with the message "activation required". *Note it says it has at least 75% battery*

Upon entering my lock code my home page is only a page that says to "connect to iTunes".

When I attempt to take the phone off the charger, it dies

and is completely unresponsive as I connect it to a computer.

The computer doesn't recognize it and the screen remains black.

If I leave the phone alone on the charger it will continously go through a process of shutting itself off and turning back on. The time always restarts at around 7p or so and the date is Wednesday, December 31st. Also as it's plugged in, I manage to turn the phone off and on and reset it but the phone just keeps displaying the "activation required" screen. Can this be fixed at a Genius bar or should I have it replaced?