Initially I'd suspect you've got a problem with the main power board within the television.
Take the back off the tv and locate the printed circuit board that the mains power cable is connected to - this is the main power board and can often fail.
Hopefully you can identify what a capacitor looks like, as you will need to inspect all the capacitors to see if any have blown which would cause a problem that you describe.
The difference between a good capacitor and blown capacitor is quite subtle, but you need to look at the end of the capacitor and you will see a cross stamped into the silver metal end. The metal end will be flat on a good capacitor, but when blown, will be bulging outwards slightly and sometimes split along grooves of the cross.
If you find any blown capacitors, carefully read the data off the side of it and order replacements off ebay. They are only a few pennies each so a cheap fix if you can do basic soldering. You will need to know the voltage and how many microfarads (uf) the capacitor is, and order exactly the same. Simply de-solder the bad capacitor and replace with the new one, but it's vital to make sure that it is soldered onto the board with the correct polarity - the white line down the side of the capacitor must line up with the white line on the bad one you removed from the board (this is the negative terminal which will also be marked on the board)
If you don't find any blown capacitors, you will need to perform some specialist tests which I feel are too involved or require specialist skills beyond the scope of this forum