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.