Issues such as this one are usually caused by bad electrolytic capacitors used inside the monitor. With time they lose their capacitance and their equivalent series resistance increases. Such problems are common on computer power supplies, but can happen on anything that used electrolytic capacitors.
The several restarts are required because as the capacitors age, more time is required for them to charge and when the screen is restarted several times in succession, they don't have enough time to discharge while the screen is off, so the stored amount of charge increases with each restart.
The electrolytic capacitors are often used with voltage regulators an that could explain the restarts. Monitor will usually have pre-programmed amount of time in which the regulator output must stabilize. If it doesn't safeties will kick in and turn it off. Once the amount of charge in the capacitors is enough for the regulators to work correctly, the screen will stop turning off and will work more or less normally.
The solution to this problem isn't very easy and probably not very cheap. You (or a repairman) will have to locate the failing capacitors and replace them, hopefully with capacitors of higher quality.
Another option is to trash the monitor and get a new one. Depending on the costs of repairs, this could turn out to be a cheaper option than fixing the monitor, if you don't know how to replace the capacitors yourself.
