This kind of thing is called fixed point iteration. Starting with being whatever, you let and more generally for every .

There are various assumptions that ensure that the sequence converges (for instance, here the function is contracting: where , hence Banach's fixed point theorem applies). Then, if , necessarily we have (taking the limit in ).

So that your constants are just the fixed points of and : the numbers such that and (they are unique). I don't think there exits a simple explicit formula for them.

You probably can find more about this if you look for "Picard iteration", or "Banach fixed point theorem" with Google.