Your model is an LCD, according to what I looked up. Therefore, subcomponents holding a lethal shock is not a significant issue, as they are with CRT monitors. CRT monitors, on the other hand, are extremely dangerous if opened.
Viewsonics are generally a good brand, and by no means the only one to fall victim to this.
As the other posters have replied, since the problem appears after roughly 5 min., then the culprit is usually heat.
While there are other components that might be involved, the most likely part to be failing is the lamp assembly / assemblies. These are thin tubular -- bulbs, for lack of a better term at the moment. After prolonged exposure to excessive heat they begin to fail, and this is common for LCD monitors. Some will display black horizontal lines across the screen and eventually fail to the point that the monitor screen goes black with a dim suggestion of a picture, more, less, or none. The monitor still recieves power, as evidenced by a green light next to the power button, but it shows lines, or no discernible picture, unless it is a ghost barely seen in dim light.
Some companies still sell these lamp assemblies online, if you specify the length you need and for which make and model. Installing them can be very tedious and time consuming, and there is no guarantee it will work when you finish. This is why most computer repair companies will decline to work on them, as the cost in labor is usually more than the cost of a newer (and better) LCD monitor.
However, to give you hope, it is possible that one or more capacitors on the video board (if it is seperate from the main board or control board) or caps on the main board might be only beginning to fail and might cause this. Reduced or inconsistent power to the lamp assemblies might also cause the lines, much as a failing flourescent bulb begins to display faint "banding." You said the caps looked good, but check again for even the slightest suggestion of swelling at the tops or bottoms of the caps. This is tough because some of the smaller caps might fail without any swelling. If you have an electronics store nearby that is willing to order you the caps you need, and you are confident in your soldering skills, and if you have the time, order and replace them all. Caps are usually less than a buck or two. If time is an issue, then it's time for a new LCD monitor.
Sorry I don't have better news.