My sony kds 55a200 picture
There's a good chance you have failing electrolytic capacitors either in the power section or the inverter section or both.
Any caps in these sections that look bulged at the top, or bulged/leaking at the bottom (look for discoloration of the circuit board around the base of the capacitor) need to be replaced.
If you repeatedly turn it off and on, eventually it'll probably stay on, but every time you turn it off, the unit will get harder and harder to start up until one day it just won't.
Sometimes you have to do the opposite to start it up and unplug it for some time and then try again.
If you aren't tech savvy, don't worry, read the rest of this solution and watch the videos.
If you are handy with a soldering iron and can identify the power supply and/or inverter / FM section for the backlights, an inexpensive handful of capacitors will likely fix you right up.
Match the capacitance on the capacitors. Go over voltage if you can, and still have them fit.
IE - it's not a bad idea to replace a 10V cap with a 16V or 25V or even a 50V, but don't replace a 680uF cap with a 500uF or a 1000uF (unless you are SURE it's only doing supply output ripple filtering,
and even then, you should go OVER, not under the uF rating).
Most of the caps that go are 10V 1000uF or 3300uF.
I found some great videos of the procedure (for many Samsungs with the same issue) on youtube.
Backlight related video here:
As you can see, this issue spans plasma TVs, as well as large and small LCD TVs and monitors, amplifiers, computer power supplies and motherboards, and other electronic equipment.
The parts are cheap, and skill required is minimal.
A great parts source is Digi-Key, and you can order the parts online at www.digikey.com
They typically cost under a dollar a piece plus a flat shipping rate.
The parts usually arrive one business day later.
If you watch the third video, you will see that even someone with no soldering experience can perform these repairs as demonstrated by the woman in the video.