This question has an obvious answer: No. As well as a more complicated answer. (Which doesnt exactly answer what your question literally asks but what it could ask to have a more interesting answer :P ).
Trying to stay awake can only be best way to fall asleep if you completely fail at it. Either because you are ignorant or stupid. This means that for most people that know anything about the subject will try to do activities that stimulate their bodies or brains in a way that will not allow them to fall asleep.
If they are familiar and willing to do any of the fallowing to stay awake they will not fall asleep:
- putting legs in iced water
- standing up from time to time
- electrocuting themselves
- engaging in an activity which requires mental concentration.
- Drinking coffee etc etc..
If on the other hand person is completely ignorant then trying to stay awake might have strange results.
If a person tries his hardest to stay awake then he is bound to concentrate on that task which means he will not fall asleep as fast. Just like trying to fall asleep makes one concentrate on the task at hand making it harder (mostly impossible if not exhausted) to fall asleep.
So no matter what it is trying at any activity is not the best way to fall asleep. (i know there is research behind it because i was reading about it but i cant find it :/ )
Now if your question means best way to fall asleep is to try to stay awake without moving much in a comfortable environment then it could possibly be the best way to fall asleep for some people.
Some people might try to fall asleep really hard but because they have problems with it usually, but because staying awake comes naturally to them they will not try so hard. In fact some people might assume they have to do nothing to stay awake and just let go and relax which is exactly what you need to fall asleep. For such people it might be the best way to fall asleep.
Anybody that can find some research on both ways to stay awake and the best way to fall asleep (actual scientific research) please post below in comments so i can have actual scientific examples in my answer (I really couldnt find anything but its late here and im tired :P )
Also IMHO i think best way to fall asleep is to exhaust yourself like hell during the day both physically and mentally. If you try to solve really hard math problems after you have ran for 3-8 hours straight when time comes to hit the hay you will be asleep before you have time to think about it. (my humble opinion is just an opinion and is not really part of the answer)