Frodo survived in the physical sense, however he was never fully whole again (apart from missing a finger
, the mental scars would take a lifetime to heal, and the poison of Shelob I believe would return from time to time in much the same way as a malaria attack (my opinion, a possibility). The wound he recieved from the Morgul blade troubled him so he could never forget the horror of Weathertop. Frodo pre-War of the Ring did not survive, he was changed into Frodo the Hurt.