My mac computer will not update new software

Hi

My IMac is very old and is not under warranty anymore just for reference. I have taken my computer to an Apple Retailer to get major problems fixed before and the issue's subsided. But one problem is still here.

My IMac will not update new software, everytime I have attempted to update my software before and it does nothing, if the software happens to load and finish then when it says "to finish the update you will need to restart your computer" that is the problem, my computer cannot update new software because it does not restart properly and therefore cannot do any initail updating and it is becomming very fustrating. I have no idea what to anymore. I have pretty much just given up the fact that my computer is completely ancient.