Does Fixed Point theorem work on an integer to integer mapping?

Question

I have an equation which I need to get its (least) fixed point for. Generally, we have:

x_{k+1} = f (x_{k})   # fixed point iteration
f: N -> N     # N is the set of natural numbers

Is it possible to use the same fixed point theorem to prove that:

1. The function admits at least one fixed point
2. Finding the fixed point is guaranteed via an iterative algorithm

if the domains are sets of integers, and if so is there a source in literature?

Note: The iterative algorithm just assumes an initial value for x and computes the next value until the values are equal.

If needed, the functions which I refer to come from worst case response time of scheduling non-preemptive fixed-priority tasks in a task set.

# ceil() is the ceiling function

# equation one
w = a + b(1 + ceil(w/c))

where w is the variable and a, b, c are constants.

The form of the equation is a dummified version from this paper of Theorem 15. But the form should suffice anyway.