Right English wording for "counterexamples to a theorem" This question is about the right English wording.
I give here what I call "counterexamples to Banach fixed-point theorem". What I do, is that I look to what happen if some hypothesis of the theorem are not fulfilled.
I call this "counterexamples to Banach fixed-point theorem". However, I was told that this is not a proper English wording as a theorem cannot have a counterexample! I'm a French native speaker and it seems that a straight translation of "contre-exemples au théorème du point fixe de Banach" is not good.
What would be the appropriate wording to this situation?
 A: A possible English wording is "what happens if we drop the hypothesis of the Banach fixed-point theorem?"
A: I have looked at your website before, and I remember being a bit dissatisfied with the terminology "counterexamples"- from a technical point of view. For me, a counter-example is some constructible object that demonstrates the falsehood of some statement. Equivalently, a counter example demonstrates the truth of the negation of the original statement. So for (counter-)example, take the statement "All functions are invertible". That is, for every $f$ there is some $f^{-1}$ such that for all $x$ in the domain of $f$, it is true that $f^{-1}(f(x)) = x$. A counter example to this is the function $f(x) = x^2$. Because, assuming an inverse function $f^{-1}$ we have
$$1 = f^{-1}(f(1)) = f^{-1}(1) = f^{-1}(f(-1)) = -1$$ which is clearly a contradiction. So this is a counter example to the claim.
On the other hand, as one commenter points out, from a "marketing" point of view, "counter-example" sure sounds catchy. But ultimately I find it misleading.

Edit: I realise I didn't really answer your question! In your case, I would say "Counter-examples to variations on the theorem statement" or something along those lines. Or go with the answer above. There are really a lot of ways to say it, but I would certainly avoid "counter examples to the ____ theorem".
A: How about simply "counterexamples related to the Banach fixed point theorem"?
Or if you want to be more precise, "counterexamples to possible strengthenings of the Banach fixed point theorem".
A: I am also not a native English speaker, so not sure it this is a good suggestion, but I would use something like Necessity proofs or Necessity demonstrations or Necessity examples (meaning: examples that prove that every assumption of the theorem is necessary).
A: This is much more a mathematical question than an English language question. (Note that I didn't even look at the maths)
On your website you say "We look here at counterexamples to the Banach fixed-point theorem when some hypothesis are not fulfilled." 
Basically this theorem, like most theorems, takes the form "if conditions A, B and C are fulfilled, then we get the result X". And you are giving results that demonstrate that if A is not fulfilled, or B is not fulfilled, then we don't get that result X. You call A, B, and C "hypothesis". That's wrong. They are not hypotheses. They are preconditions of the theorem. 
You then demonstrate that you don't get the result X. That is not a counterexample to the theorem, since the theorem never claimed you would get the result X without all the preconditions. 
I'd suggest "Examples for the necessity of the preconditions of the Banach fixed-point theorem". There are probably many possible ways to put this. 
All that said, when you were told "a theorem cannot have counterexamples", that is not quite right. "Theorem" is not used for a mathematical statement that is true, but usually for one that is widely believed to be proven. Sometimes not even that is necessary; Fermat's Last Theorem was called Fermat's Last Theorem for many, many years when no proof was known, and when it was not even know if a proof existed. So it is entirely possible that something called a "Theorem" is actually wrong and that there are counterexamples. If you managed to give an actual counterexample and that counterexample has been widely accepted, the name would probably get changed. 
A: I would go for non-inclusive examples or exclusive examples, as in essence any violation to the hypotheses/preconditions of the theorem are supposed to produce results that are excluded from when every condition is fulfilled. 
For instance, on the level of definitions, when defining the term 'rectangle', you can state that a square and a triangle are inclusive and exclusive examples of a rectangle respectively.
A: Great question, great discussion.
Perhaps "special case" or "classic example" is appropriate. For instance when discussing the implications between continuity and differentiability the special case or classic example is the Weierstrass function continuous everywhere, differentiable nowhere. This demonstrates that continuity does not imply differentiability.
