iPhone random shut down

My iPhone 5 shut down twice within the last three days.

The first time happened when I was opened safari, and the screen froze. Usually, if I wait it out, the scree will un-freeze and then everything goes back to normal. This time, pink and green vertical stripes appeared and then the screen went black.

I tried to turn it on by pressing the power and home button at the same time, and it showed a connect to iTunes symbol. When I connected it to my laptop, iTunes popped up and said that I needed to update or reformat my phone. I started the download, and I left it alone.

After two hours, I checked the download, and saw that it had failed. At this time, I tried to turn on my iPhone using the method of pressing both the home and power button, but this time, it did not turn on. After more than an hour of trying, it powered up, with more than 80% battery.

I did not witness the second time my phone shut off. I was going to check and see if I had any updates, so I pressed the power button. The phone did not light up to show the lock screen (as normally it would). I tried to turn it on using the method above, but it still won't.

Does anyone have any ideas to fix it? Or is it finally the time to get a new phone?