I have Shimano 105 brifters for a double. The front shifts to the big ring, but once once I let go of the shifter it returns back to the small ring. How can I fix this.

It sounds like you have some slack in the cable, and after you release the lever, it drops back to the small ring. Shift to the small ring position, and check to see if there is much slack in the cable. If so, use any adjusters you see along the cable to remove the slack. If no adjusters, loosen the bolt securing the cable to the derailleur, pull out the slack, and retighten the bolt securing the cable. Now try to shift up, and see if it stays.

I'm assuming that your upper stop and lower stop are set correctly, and do not let the chain go off the chainrings, either inside, or outside.