I have dedicated time and effort to solve a theoretical problem posed by two other people. Now I think I have resolved it, and I have material for a publication.
Should I involve them as co-authors?
Here are more details:
those other two people (hereafter, "they") are academics (as I am), but not colleagues: I have met one of them once, the rest of contact was by e-mail.
while I resolved their problem, I have no idea of the relevance or applicability of it (they are in a less theoretical field than I am). I assume they do;
their only contribution has been to make me aware of the open problem I resolved;
my work has developed completely independently, with methods, techniques and formalism utterly different from theirs; indeed, I think they'd never have been able to reach the result with their approach;
at one point, they informally made me their collaborator, by just saying "welcome to the project" by e-mail, and sending me a draft of their attempts to resolve the problem; we had very little contact on the project, besides that;
one of them is "strategically relevant" for me: despite the mere feeble connection established by the fact that we were working on the same problem, he provided recommendation letters and proactively promoted me when I was in need;
I already have a paper illustrating my solution; however, since I don't have the competence to understand how this result is useful in the application domain, my paper at the moment is missing proper introduction, conclusions, and, generally, context;
One strategy could be to publish two disjoint papers, one with their problem with due context, and the other with my solution.
However, I would like to hear any piece of advice or consideration on how to move: I would like to do what is globally most convenient for me (e.g., balancing the good of having a publication with that of keeping my allies in the academia), while not infringing any ethics.