Evaluate mysql date range

I have a mysql table with some date ranges, for example:

start: 2015-09-10 end: 2015-09-10

Now, I have to check if another date range is inside the start-end date, even if only for some days.

1. 2015-09-02 -> 2015-09-03 is outside the range;
2. 2015-09-05 -> 2015-09-11 is inside the range;
3. 2015-09-11 -> 2015-09-12 is inside the range;
4. ...

I have these query, it works for case 1 and 2:

SELECT * FROM prenotazione WHERE id_stanza=1 AND (start BETWEEN '2015-09-11' AND '2015-09-12' and end BETWEEN '2015-09-11' AND '2015-09-12' )

For the case 3, the query returns no result, even id the given date are inside the range. Can you help me?

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Link to original article：https://stackoverflow.com/questions/32348326/evaluate-mysql-date-range