You can actually do this one of two ways:
MySQL update join syntax:
UPDATE tableA a
INNER JOIN tableB b ON a.name_a = b.name_b
SET validation_check = if(start_dts > end_dts, 'VALID', '')
-- where clause can go here
ANSI SQL syntax:
UPDATE tableA SET validation_check =
(SELECT if(start_DTS > end_DTS, 'VALID', '') AS validation_check
FROM tableA
INNER JOIN tableB ON name_A = name_B
WHERE id_A = tableA.id_A)
Pick whichever one seems most natural to you.