I understood that a NTM can say only yes but it doesn't know if the input is a NO instance with a single execution. Furthermore, the NTM can diverges and, at the same time, can decide a language (source: Papadimitrou - Computational Complexity, proposition 7.1). While a TM must converges in a "yes" or "no" answer to decide the same language. Now, we know that for a NTM that decides a language L we can create a TM that decides L (theorem 2.6). But suppose that the NTM diverges for some computation. How can the TM treat this case?
[EDIT] Definition (source Papadimitrou, pg 45): A non-deterministc Turing Machine decides a language L if for any input x, the following is true: x belongs to L iff there is an acceptance path.
This definition admits machines that, instead of rejecting an input, diverge and at the same time decide a language.
Theorem 2.6 (pg 47): Suppose that language L is decided by a nondeterministic Turing machine N. Then it is decided by a deterministic Turing machine M.
Proposition 7.1 (pg 141): Suppose that a (deterministic or nondeterministic) Turing machine M decides a language L within time (or space) f(n), where f is a proper function. Then there is a precise Tring machine M', which decides the same language in time (or space, respectively). O(f(n)).
This last proposition confirms that there is a Turing machine that decides a language and that diverges for some path.