Ahoy there...Your case would appear to be the classic dying cold cathode tube(s) . After a long service the tubes will grow a carbon track down the tube and it won't take much length to cause an increase in current from high voltage unit . This current is detected and unit shuts down . They can be sourced and replaced , I have seen them from American sources for approx. $30 for 10 40 inch .
HOWEVER I have also watched an LED conversion on youtube which may be a smarter choice as the rest of the tubes will meet that fate sometime . It is a picky , tedious dangerous(to your screen not you so much) process .
If you are mechanically adept and don't use force on the flat wires(there should be ZIF sockets or a stiffened handle , be careful)
you stand as good chance . You need a good sized work space though .
Good luck
Hugh