Adapting arguments and plagiarism I'm currently working on my PhD thesis. I have several suggested problems to work on, some of them are very similar to some problems that my advisor have worked before and published already, either in his thesis or papers. Basically, the main difference is in the dimension of some singular sets (his works are mainly on the isolated case, but I'm working on a case with a far more hairier, non-isolated singular set), which we were unsure if the argument would hold but it seems that the adaptations I've made were fine.
Not that the nature of the problem matters, but the approach I'm making worries me. It seem to me that if there would be a 'railroad' to prove the results I'm working on, it would be the same path that he followed to write his own results, with different objects. That's the way I've been advised to work, and it's been producing results.
Is this a reproachable approach? Of course, there is the problem of using similar introductions (and in that subject I've read this previous question Does this qualify as "self plagiarism" or something? , only one that got close to my problem) sinde the objects being studied by me and that has been studied by him were so similar. 
 A: What you describe seems to me to be a normal mode of mathematical
progress, and I would urge you simply to carry on! Ride that train
as far as you can.
It often happens that someone's mathematical results can be
improved or generalized in various ways, and when this is
possible, it is mathematically desirable that the generalization
be undertaken well.
You may be worried that the value of this work is less than some
other totally original work. If the generalizations are routine,
then indeed that may be true. But from what you say, this doesn't
seem to be your case. Many generalizations are not routine and
such work is definitely worth doing.
Finally, let me caution you to guard yourself against a certain
mistake that sometimes undermines motivation for a young
researcher. Namely, it often happens in mathematical research that
we begin in a state of terrible confusion about a topic; as
research progresses, things only gradually become clarified. After
hard work, we finally begin to understand what is the actual
question we should be asking; and then, after fitful starts and
retreats, we gain some hard-won insight; until finally, after
laborious investigation, we have the answer.
But alas — it is at this point that the crippling illness
strikes. Namely, because the researcher now understands the
problem and its solution so well, he or she begins to lose sight
of the value of the very solution that was made. The mathematical advance begins to
seem trivial or obvious, perhaps without value. Having solved the problem so well,
the mathematician becomes a victim of his or her own success.
Because all is now so clear, it is harder to appreciate the value
of the achievement that was made.
Please guard against this disease! Do not denigrate your
achievement simply because it seems easy after you have made it.
In many mathematical realms, the actual achievement in research is
that certain issues and ideas become easy to understand. Please look
upon the ease of the answer at the end as part of the achievement itself, and
think back to the initial state of confusion at the beginning of
the work to realize the value of what you have done.
So please carry on and ride that railroad as far as it will take
you.
A: One major aim of a PhD is to study a problem so intently that suddenly, after much hard work and perseverance, the solution becomes obvious - or at least that it becomes obvious what the approaches to the problem would be and why one of these approaches is likely to be superior to the other approaches. Be aware that everyone has good ideas - and groundbreaking approaches tend to build on earlier approaches. My PhD was "simply" the combination of three existing approaches in a novel way. But to get to the point where it was obvious that these three things together were what was necessary or would produce a solution took 2 1/2 years' work.
A: Just do what your adviser tells you to do and don't worry. Adviser has to approve your thesis before you defend or publish it. With his/her approval who can blame you in plagiarism from your adviser? 
EDIT. Concerning all other sources (other than your adviser) you should do all you can do to make sure where the ideas come from and give proper references.
