iPhone 5 Troubles

Hi all,

Recently I've been having this problem with my iPhone 5. I picked up my phone and it was suddenly off. When I tried to turn it back on, the Apple logo would show up for a few seconds and then the screen would turn black again. It wasn't low on battery, but I tried to plug it in anyways, thinking maybe it died. When I plugged it in, the Apple logo came on as if it were booting up, but then the logo faded within approximately five seconds. After it faded, a new apple logo reappeared, and it faded again. This kept repeating itself over and over when plugged in. It continues when unplugged, and only stops when I hold the home and lock button at the same time for around ten seconds. I searched for hours to find a solution, and the only one given was to restore it, and I tried that because I had recently made a backup for the phone. The only problem with this was that the iPhone would show the same repeating fading boot up that it did before when plugged in, and for some reason my iTunes nor my computer read that the phone was plugged in. The phone simply didn't pop up. I'm really hoping that my phone isn't broken, and that I can fix this. Any feedback is very helpful.

Thanks