The important word in this question is “possible.” There are many conceivable solutions that would end the shutdown, but it is not clear that any of them are “possible” because of the toxic political situation in which we find ourselves today.
Of course, it would be possible to solve this shutdown if one side or the other would simply give in. If the Republicans in the House would pass a “clean” budget resolution, or if President Obama would agree to delay the implementation of “Obamacare,” the dispute would be over. However, it does not seem particularly “possible” that either side would simply cave and give in completely to the other.
Therefore, it seems likely that some sort of a compromise will have to be reached. Perhaps the Republicans could give in and approve the budget and the President could promise to allow a series of votes on Republican proposals to amend “Obamacare.” That might satisfy Republicans who have promised to fight against the new health care law.
If no compromise of this sort can be found, it may be necessary for the shutdown to continue until one side decides that it is hurting itself politically. At that point, it might be possible for one side to give in completely.