Thanks Vincent, the following works really well. When I suspected that we had a problem I created a seach log. With the mods you have suggested a large number of the "not founds" are getting hits.
this is how the code now looks:
$sql = "SELECT authors_id as id, CONCAT( authors_first,' ',authors_last) as name FROM authors
WHERE '%$query%' LIKE concat('%', authors_first,'%') AND '%$query%' LIKE concat('%', authors_last,'%')
OR authors_first LIKE '%$query%' OR authors_last LIKE '%$query%'
ORDER BY authors_first";