What are some easy to understand applications of Banach Contraction Principle? I know that Banach contraction principle guarantees a unique solution to problems of the form $$f(x) = x$$ But for the life of me I cannot understand why this problem is important at all. I don't remember encountering such a question in calculus or ODE (at least not the most important problems)
Further more all the examples I can find that applies Banach contraction principle some way always involves tough integral equations (plus hardcore optimal control theory) and some proof that guarantees the uniqueness of the solution. To be honest I learned existence and uniqueness for ODE a long time ago and has never once in my life used it ever since.
What are some simple to understand or physical applications of Banach contraction principle?
 A: Many problems can be reformulated to precisely ask for a fixed point of a function. A very simple example is the following. Suppose that $A$ is an $n\times n$ matrix, thought of as a function $A\colon \mathbb R^n\to \mathbb R^n$. Solving $Ax=b$ is an extremely important problem (I hope this does not require elaboration). Now, $Ax=b$ holds if, and only if, $Bx=x$ where $Bx=x - Ax + b$. So, here the problem of solving $Ax=b$ is transformed, via defining a new function $B\colon \mathbb R^n \to \mathbb R^n$, to a fixed point problem. If you can now equip $\mathbb R^n$ with a metric structure (for instance, one of the many norms you can place on $\mathbb R^n$) such that $B$ is a contraction, then you obtain an iterative process guaranteed to converge to the solution of the system no matter which initial point you use to start the process. 
Of course, there are precise methods to solve linear equations, but they tend to be computationally hard. The Banach Fixed Point theorem in this case is thus very useful.  
