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Questions tagged [fixed-point-theorems]

Fixed-point theorem is a result about existence of fixed points, i.e. points fulfilling $F(x)=x$, under some conditions on the function $F$. Results of this type appear in many areas of mathematics, e.g. functional analysis (Banach), algebraic topology (Brouwer), lattice theory (Knaster-Tarski, Kleene) etc.

The Fixed-point theorem is a result about existence of fixed points, i.e. points fulfilling $F(x)=x$, under some conditions on the function $F$.

Results of this type appear in many areas of mathematics, e.g. functional analysis (Banach), algebraic topology (Brouwer), lattice theory (Knaster-Tarski, Kleene) etc. They also have many important applications.

The importance of this area of mathematics is reflected in the large body of literature on the topic, we can mention famous monograph by Dugundji and Granas, which has almost 700 pages.

1933 questions
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Contradiction with Banach Fixed Point Theorem

I am trying to find the fixed point of the function $g(x) = e^{-x}$. Wolfram Alpha tells me that this fixed point is approximately $x \approx 0,567$. However, if I apply the Banach fixed point theorem, I can prove that $g(x)$ has a fixed point in…
Luna
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Why is convexity a requirement for Brouwer fixed points? Shouldn't "no holes" be good enough?

Brouwer's fixed point theorem: Every continuous function $f$ from a convex compact subset $K$ of a Euclidean space to $K$ itself has a fixed point. I am wondering why the word "convex" is in there. It seems to me that it is necessary and…
GMB
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Connection between codata and greatest fixed points

It's easy to see how inductively-defined data types correspond to least fixed points. Let's take the natural numbers as an example, with constructors are $0 : \mathbb N$ and $s : \mathbb N \to \mathbb N$. Define the operation $F(X) = \{0\} \cup \{…
Ben Millwood
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A fixed point theorem for the unit disk?

In Dynamical Systems and Ergodic Theory by Pollicott and Yuri, there is an easy, one dimensional, fixed point theorem: If $T$ is a continuous map on a closed interval $J$ so that $T(J)\supseteq J$, then $T$ has a fixed point. Here the space…
user940
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fixed-point iteration

Are there functions $f$ of one real variable $x\in X$ that are not contracting maps on the set $X$ but for which, given the starting point $x_0$, the fixed-point iteration $x_n=f(x_{n-1})$, for $n=1,2,3,\dots$ will still converge to a fixed-point?
pluton
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Why does iterating in different ways produce different solutions?

Recently, I came across the following equation: $$2^x=4x$$ To solve it, I decided to iterate. Firstly I stated: $$x_{n+1}=\frac{2^{x_n}}{4},x_0=1$$ and found a solution of $x\approx 0.3099069324$. Then I again rearranged it…
Rhys Hughes
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How to measure the convergence speed of an iterative sequence?

Background Consider the following iteration: $$ \begin{aligned} x_{n+1} = \sin(x_n)\\ y_{n+1} = \cos(y_n)\\ z_{n+1} = \tan(z_n)\\ \end{aligned} $$ For any $(x_0, y_0, z_0)\in\mathbb{R}$, $x_n$ converges to $0$, $y_n$ converges to $0.739$, $z_n$ does…
Aster
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Iteration of $x/\log x$

Consider $f(x)= \frac{x}{\log x}$ iterated n times for given x in $Z^+.$ $f^2 = f \circ f.$ Let $x_1 = x^2.$ What I would like to show (or disprove) : $\exists ~\alpha = x_n - e > 0 $ such that if n is minimal with $ f^n(x^2) - e < \alpha $…
daniel
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How to explain powers of $(x+1)^{2^n}$ appearing in the Babylonian approximation of $\sqrt x$?

I'm working with this iteration used for approximating square roots and trying to see what I can draw out from it, and in doing so I found something very strange that I can't logically explain. I'm looking for any insight into why this is the case…
Rhys Hughes
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Weissinger's Theorem. How to prove?

Theorem (Weissinger). Let $C$ be a (nonempty) closed subset of a Banach space $X$. Suppose $K : C → C$ satisfies $$\|K^nx − K^ny\| ≤ θ_n\|x − y\|, \quad x,y∈ C $$ with $\sum_n θ_n < ∞$. Then $K$ has a unique fixed point $\bar x$ such that …
Klara
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Continuous mapping and fixed points

Does a continuous mapping $f\colon \mathbb R \to \mathbb R$ which satisfies $f(f(x))=x$ for each $x \in \mathbb R$ necessarily have a fixed point?
Gian Luca
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The most important theorems in fixed point theory

What are the most important theorems in fixed point theory and why are they so important? I know some: Banach's contraction principle, Brouwers fixed point theorem, caristi fixed point theore... I also know that Banach's contraction principle has a…
Emo
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Counterexample for fixed point theorems

There are many of fixed point theorems but two of them are: Krasnoselskii theorem: Let K be a non-empty, bounded, closed and convex subset of Banach space E. Moreover let $f: K \to E$ be a contraction and let $g: K \to E$ be a compact mapping and…
patryko519
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continuous map on compact ellipse

will there be any fixed point a continuous $f$ from the ellipse $2x^2+3y^2\le 1$ to itself? Well I think yes but in a solution of a problem hint is given that NO. Just asking to assure myself if I am not missing anything of the condition of fixed…
Myshkin
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Are the practical analogies of the Brouwer fixed-point theorem meant to be trivially understood?

When reading about the Brouwer fixed-point theorem on Wikipedia there are some "real world illustrations" of what the theorem says, one of them being the following: [T]ake two sheets of graph paper of equal size with coordinate systems on them, lay…
Speldosa
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