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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...

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  • $\begingroup$ Have you seen Vsauce's video on fixed points youtube.com/watch?v=csInNn6pfT4 - might give you some ideas. $\endgroup$ – James Oct 18 '16 at 11:45
  • $\begingroup$ It's also known as the Contraction Mapping Theorem or Contraction Mapping Principle - in case that helps to extend your search. $\endgroup$ – Max Freiburghaus Oct 18 '16 at 11:45
  • $\begingroup$ I have a hard time imagining what a mathematical theorem could be used for other than "applied mathematics/other theorems which can be proved from it". $\endgroup$ – mrp Oct 18 '16 at 12:08
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    $\begingroup$ Proving the existence of solutions of ODEs can be done using the Banach's fixed point theorem. The inverse function theorem can be proved this way. General geometry requires it in order to escape finite-dimensional spaces, and that is useful for working with spaces of curves, etc. $\endgroup$ – DisintegratingByParts Oct 18 '16 at 14:40
  • $\begingroup$ @James: Funny, but he talks rather about Brouwer's fixed point theorem. $\endgroup$ – Jochen Oct 18 '16 at 15:00
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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.

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  • $\begingroup$ I'd only seen this logic for a crumpled map above an uncrumpled one before! Much better set-up here. $\endgroup$ – Pete Caradonna Mar 8 '17 at 19:48
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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 $G$.

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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.

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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!

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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.

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