My iphone 5 will not turn on after it turned off in the middle of backing it up

My iphone 5 has been acting pretty weird for the last couple of days. It will just turn off out of no where, show the dead battery symbol even though it will have plenty of charge in it (its even done this when its at 90%) and then turn back on when i press the lock button. Sometimes i have to plug it in for it to turn back on. It dawned on me today that maybe if i restored my iphone it would stop doing this. So i was just backing up my iphone when it turned itself off and now it keeps showing the dead battery symbol then the apple logo like its turning back on then it shows the dead battery symbol, then the apple logo and repeats over and over again. Ive tried unplugging it, then plugging it back in, holding down the lock button and home button, nothing will snap it out of this little cycle its stuck in. does anyone have any ideas??