Actually, I’ve had this problem and fixed it. Its the cable connectors that attach the screen to the base of the phone. One of these is not connected properly, or has a bad connection, probably from being touched by human hands and has some oils from skin blocking the connection. All you have to do it take them off, clean them with alcohol, then reconnect them—just making sure they are all firmly pressed down. Check to make sure this worked before closing phone back up by simply turning it on.
If the problem persists…
There are two other things you can try. Another connection that causes this problem is the front facing camera and the proximity sensor assembly. You can try to remove and clean connections.
Finally, you can plug phone into charging cord, turn it on, then plug into computer with itunes installed (this will force reset). The blinking apple will most likely continue for for a while, but just leave plugged in, and surprisingly it will eventually reset, as long as all the connections have been cleaned of any oils from your hands.
I have had this problem on two different models I fixed, and these solutions worked both times; unfortunately, one of them had the problem resurface later, which means the board connectors were damaged when the screen was originally taken off and then entire board will have to be replaced.
Helpful hint. When reconnecting any cables, try not to press in the center. Press from one end, then other end. This will prevent them from being damaged.