I would agree that this is an SQL issue. A colleage showed me this trick the other day:

... AND START_TIME < $datetime_end AND END_TIME > $datetime_start
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This includes all records that start or end within the period or that span it entirely without the need for OR conditions. Of course, I should make the obligatory comment that you should consider using placeholders - see The fine art of database programming - rather than including the values directly in your SQL string.