Maybe you should have a look at the book "Computational Methods in Engineering" of S. P. Venkateshan

If I understood your question correctly, my answer is the following: for a fixed-point equation x=g(x), whenever the iterative sequence x_(n+1)=g(x_n) is convergent and the function g is continuous, then the limit is necessarily a fixed point of. Simple like that.

There are classical fixed point theorems, among those the most celebrated Banach Contraction Mapping Principle, which asserts that for any (strict) contraction on a complete metric space there exists a unique fixed point (which can be retrieved by the above fixed-point scheme). The rate of convergence is in general linear, but in specific cases (like Newton's method) the rate may be quadratic or higher.

I am not sure whether it is the answer you expected. 

One of the reason (probably the main reason) for the name 'fixed point Iteration(FPI)' for the method is as follows.

Consider the map(of a country e.g.) having two versions, one is the distorted (possibly bigger) of the other. Place one(smaller) map on the other. In the maps, each town in the country will be at different point(on the table), EXCEPT only one town which will be on the same point(called fixed point) on the table.

To find the fixed point town, follow the algorithm below, which is (in one  dimensional version ) the fixed point algorithm.

Assume that the distortion is represented by a function g(x,y). In 1D version, it is g(x).

1. Choose any town in the smaller map and get its point on table. ( get an initial x_0)

2. Locate the town in bigger map for that point ( x_1 = x_0 )

3. Find that town in smaller map ( x_0 = g(x_1) )

4. Go to step 2 ( x_1 = x_0 ... )

After some iterations, you will be nearing the fixed point town on both maps.

I use this example to explain the origin of FPI using(drawing) two maps of Sri Lanka in  numerical methods classes.

Hope this might help.

I recommend you Petko D. Proinov book "Approximations of fixed points and applications foe numerical solution of nonlinear equations" to appear.

Thank you.