Let g Є C[a,b] such that g(x) Є [a,b], for all x Є [a,b]. Suppose, in addition, that g' exist on (a,b) and that a constant 0 < k < 1 exists with |g'(x)| <= k, for all x Є (a,b)
Then, for any number Po in [a,b], the sequence defined by
Pn = g(Pn-1), n >= 1
converges to the unique fixed point p in [a,b].
Of course this is for just a fixed point for a function of one variable. Just work with your interval so that those conditions are satisfied.