POWER SUPPLY CAPACITORS

Chances are it is the Power Supply. It is a Megmeet MLT386X which is hard to find, but sometimes IS available on ebay. There are places that will repair the old one advertising on ebay as well. If you remove the covers running across the heat sinks and see that the capacitors are bulging, the repairs are within the capability of anyone who knows how to use a soldering iron.
If you are handy with a soldering iron, LCDalternatives sells a kit with ALL electrolytic capacitors prone to fail for $15.95 plus shipping. I bought the kit and fixed mine. It took a couple of hours to unsolder and replace the capacitors. Powered up and has run for last two years with no problems.

Go to EBAY and look up a capacitor repair kit for your model kit. They cost $20 to $30. This is to fix the power supply board which is the middle board. Be sure to note polarity of the capacitors you replace. There is a positive and a negative on them and that must be observed. Use a magic marker and mark the top of each cap as you replace it. Replace all caps supplied in the kit even if old ones don't look puffed out. This should fix your power supply. I did this on a friend's set over a year ago and the power supply is still ok.

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.

http://www.youtube.com/watch?v=Mm51C_RDIZE
http://www.youtube.com/watch?v=BNHHrgX_6cs&feature=related
http://www.youtube.com/watch?v=8l0AUj8QUkg&NR=1&feature=fvwp
http://www.youtube.com/watch?v=Y7b_nTaZYcU&feature=related

Backlight related video here:
http://www.youtube.com/watch?v=6XoF6QZ1dmw&feature=related

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.

There's a couple transistors on the Power Supply board that are probably bad. On mine, the solder joint was bad on the bottom side of the board that pointed me at those transistors in particular. Part number is FQPF9N50C. Also, you could get a refurb or new power supply board of ebay or send your board in for service (also on ebay).Or there's a "repair kit" you can get that contains 4 transistors and a couple caps I think. If you are an electronics junky and like soldering stuff, that would be the way to go... I just ordered the two transistors off ebay for like $6.
http://cgi.ebay.com/ws/eBayISAPI.dll?ViewItem&item=320632712822
http://cgi.ebay.com/Samsung-HP-T4254-Internal-Replacement-Circuit-Board-/230575679230?pt=LH_DefaultDomain_0&hash=item35af61a6fe
http://cgi.ebay.com/REBUILD-YOUR-SAMSUNG-BN44-00161A-Power-Supply-/200513469692?pt=LH_DefaultDomain_0&hash=item2eaf88bcfc

You can this unit from;

https://www.discount-merchant.com


MLT386X PCB - POWER SUPPLY MAIN FOR AKAI LCT3701AD - List Price: $349.99
Our Price: $179.99