What are some easy Banach fixed point theorem applications?

Question

I have to write an essay on applications of Banach's fixed point theorem and need some ideas of topics to write. Thing is, I don't want it JUST filled with iterative methods for zeros of functions and operators, I would like some more varied applications.

So what are some applications which don't need too much context to understand/state and which aren't iterative methods for zeros of functions? Like in areas other than applied mathematics/theorems which can be proved using it, etc...

Answer 1

One can prove a theorem commonly referred to as "the elementary domain invariance" theorem. It asserts the following.

Let $A$ be a Banach space and $U$ an open set inside $A$ and suppose $T\colon U \rightarrow X$ denotes a linear contraction. Then the map $x \mapsto x - Tx$ defines a homeomorphism onto its image.

To be more specific, it exploits a corollary to Banach's contraction principle, namely that for a complete metric space $X$, any contraction $\varphi \colon B(x_0,r) \rightarrow X$ has a fixed point whenever $$ d(\varphi(x_0),x_0)<(1-\alpha)r $$

with $\alpha$ being the Lipschitzan constant and $x_0\in X$.

The elementary domain invariance theorem may be further applied to deduce the rather basic fact in operator algebras:

Suppose $A$ denotes a Banach space and $T\colon A \rightarrow A$ denotes a bounded operator fulfilling $|| I-T || < 1$. Then $T$ is invertible and $$ ||T^{-1}|| \leq \frac{1}{1-||I-T||} $$

Other than mathematical theorems, I know fixed-point theorems are crucial in determining solutions to differential equations.

Answer 2

Here is a "fun" application of the Banach fixed point theorem:

Pick a map of your city. Drop it on the floor of your room/office/classroom.

Prove that there exists unique point on the map which sits exactly above the corresponding point in the room.

Answer 3

The Banach Fixed Point Theorem plays an important role in guaranteeing the existence of solutions to common problems in economics. Consider some $\beta \in (0,1)$ as a discount factor. One then often wishes to choose an infinite sequence of 'controls' $\{u_t\}_{t=0}^\infty$ to maximize:

$$ \sum_{t=0}^\infty \beta^t r(x_t, u_t) $$

subject to some state variables $\{x_t\}_{t=0}^\infty$ which obey some law of motion $x_{t+1} = g(x_t, u_t)$, and with $x_0$ a fixed boundary condition. We usually assume that $r(\cdot, \cdot)$ is as smooth as needed, and strictly concave and increasing, bounded, and the set:

$$ \{(x_{t+1}, x_t) : x_{t+1} \le g(x_t, u_t), u_t \in \mathbb{R}^k\} $$ to be convex and compact. The objective is to find some time-invariant solution function $h$ that maps the state $x_t$ into a choice of control $u_t$ such that this policy function solves the above problem.

To this end, one speaks of the value function: $$ V(x_0) = \max_{\{u_t\}_{t=0}^\infty}\sum_{t=0}^\infty \beta^t r(x_t, u_t) $$ i.e. the function that returns the optimal value of the problem, given an arbitrary initial condition. We of course do not know $V(x_0)$ ex ante. But if we did know it, the optimal policy could be computed by solving, for all $x$:

$$ \max_{u}\big[r(x,u) + \beta V(g(x,u))\big] $$ where we have imposed the law of motion on $x$. If we can solve this above equation for all $x$, we can find the functions $V,h$ that would constitute a solution. The relation between them may be expressed as a Bellman Equation:

$$ V(x) = \max_{u}\big[r(x,u) + \beta V(g(x,h(x)))\big] $$

This is a functional equation: the right-hand side may be viewed as some mapping $T_V(x)$, and our problem has a solution if there exists some $V = T_V$, i.e. a fixed point of $T$. In fact, given nice assumptions on primitives, it is a mapping from $B(X)$, the space of all bounded functions of states (endowed with the sup norm to render it a complete metric space) into itself. In light of Blackwell's sufficiency conditions, $T$ is a contraction map, and hence there exists some unique value function $V$ which solves the above! It also gives an algorithm for solving numerically for this value function: pick a random $V_0$ and iterate $T$; your convergence will be geometric!

Answer 4

Here is a completely different application, which is an overkill but I think is nice.

Consider a triangle $ABC$ in the plane. Let $A_1, B_1, C_2$ be the midpoints of $BC, AC, AB$.

There is an obvious affine (i.e linear + translation) $F: ABC \to A_1B_1C_2$ which takes $A \to A_1, B\to B_1, C \to C_1$ and is a contraction. Geometrically, the mapping takes $ABC$, halves it, reflects it, and then moves it to fit $A_1,B_1,C_1$.

The function $F : ABC \to A_1B_1C_1 \subset ABC$ is a contraction, thus has an unique fixed point, lets call it $G$.

Let now $A_2$ be the midpoint of $B_1C_1$. Since $AB_1A_1C_1$ is a parallelogram, it is easy to show that $F(AA_1)=A_1A_2$.

Therefore $$F|_{AA_1}: AA_1 \to A_1A_2 \subset AA_1$$ is a contraction. This shows that this function also has a fixed point inside $AA_1$. This is a fixed point of $F$ inside the triangle, and hence must be the unique fixed point of $F$.

This shows, that the fixed point $G$ must be on the median $AA_1$.

Same way, $G$ is on $BB_1, CC_1$.

Therefore, we proved that the medians in a triangle are concurent at some point $G$.

Answer 5

I expanded the idea of the user N. S.

Imagine a picture containing a picture of itself and the picture it contains contains a picture of itself, etc.. This is called mise en abyme. Here's an example.

Then the Banach fixed-point theorem states that there is a unique fixed-point in that picture. This fixed-point is normally called vanishing point in photography.

This differs from the idea given by N. S., because when throwing a map down with the new map containing the first map as well, the co-domain must be restricted in order to get a convergence of the points being both in the real and mapped world for the second map.