randomly shutting off, repeatedly even on start up

My macbook pro is obviously a few years old now and starting to show its age in more ways than one. I recently updated to Yosemite right from snow leopard (the laptop was originally shipped with leopard). Now the battery had been declining for a while and for about the last 6 to 12 months it has only run while plugged in and if unplugged it would shut off after less than a minute. Because of this I opted to get a new battery from the site and did so about a week ago. Since putting the new battery in my laptop has run without being plugged in however while using the laptop it will randomly shut off even when there is still lots of battery % left. Often on start up it will shutdown and restart the start up process multiple time. Now that you all have a description of the problem my question is, is this a battery issue or something else. I have read quite a number of help forums and some people say its a heat problem some say its a battery, others say the logic board is going and some say it has to do with the power control unit (could it have to do with the update as i couldnt try running without magsafe) . Any help would be great and if its a battery issue still i need to know so i can get a replacement as it is brand new. Thanks all