I solved their problem. Should I involve them?
If you solved an open problem working independently, the polite thing to do is acknowledge the source(s) of the problem in your paper, and perhaps send a preprint to the source(s). 

But, if they have not contributed towards the project, it would be inappropriate to include them as co-authors. Co-authorship is properly used to recognize joint work, not as a gift.  If you are able to write a publishable paper of your work, there is no reason etiquette would require offering them a co-authorship, based on what is written in the question here.

On the other hand, it might well be reasonable, depending on your field, to ask someone else if they would like to join with you to extend your work to something that is publishable. For example, if you cannot provide enough context or application to make a publishable paper, and they can, then you could collaborate with them. 

The main drawback of the collaboration (I don't view collaboration in itself as a drawback) is that it would take time for them to work on their piece, delaying publication. 

One polite alternative would be to send them your preprint and ask them whether they feel it is publishable, or whether additional material is needed. If they say that additional material is needed, that might open the door to a collaboration. 

P.S. added a few days later I do agree with others here that it would be very valuable to discreetly inquire about what the others meant by "welcome to the project". The question describes them as "not colleagues" and "their only contribution has been to make me aware of the open problem I resolved"; this answer is written under the assumption that those are correct. If the others don't think they are correct, you run a significant risk of burning bridges. The advice to write a joint paper combining your work with theirs, which was also mentioned in other answers, shouldn't be dismissed too easily.