In the Jehle and Reny textbook (which I should add I have not read much of beyond a few sections of interest), a theorem stating that there is always a (mixed) Nash equilibrium in finite strategic form games is proven. The book assumes that all players have the same number of actions available, but it's not difficult to imagine how this might be extended to the case where this isn't true.
What I'm interested in, however, is whether there is some extension of this to games, particularly those where there may be infinite choices. For instance, there's clearly no equilibrium in a game where a player wins by picking the highest number, but if we have, for instance, the same game, but where the number must be within the interval [0,100] (or any interval that contains its upper bound), the best response functions "converge". Similarly, I would also suspect that there need to be "well-behaved" cost and demand functions in competition models to get "good" results.
As such, I have two questions:
Is there any sort of well-defined setting in which a game with infinite strategy choices will have a Nash equilibrium?
What would relevant reading for this be?