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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.

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1answer
806 views

Does a compact connected complete linear order have the fixed point property?

Would the same arguments used for showing $[0,1]$ has the fixed point property hold in this general case? What could go wrong? EDIT: The fixed point property can be interpreted in two ways that I ...
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How can I prove $x_{n+1} = e^{-x_n}$ is convergent?

I'm doing a practice problem which asks to prove that the sequence defined by $x_{n+1} = e^{-x_n}$ is convergent (or rather "study the convergence of $(x_n)$"). So I'd like to try and find some ...
7
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Does the sequence $(x_n)$ given by $x_{n+1} = -16+6x_n+\frac{12}{x_n}$ converge?

Question. If $x_0$ is sufficiently close to $2$, then will the sequence obtained as $$x_{n+1} = -16+6x_n+\frac{12}{x_n}$$ converge to 2 ? My attempt : I have shown that if $x_0$ is close to ...
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Every increasing function from a certain set to itself has at least one fixed point

I need a hint for the following question: Let $S$ be a nonempty ordered set such that every nonempty subset $E\subseteq S$ has both a least upper bound and a greatest lower bound. Suppose $f:S \...
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1answer
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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 $T(J)\supseteq J$,...
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112 views

Free $\mathbb{Z}_{2}$ action on the plane

Motivated by the following question we ask: Is there a free action of $\mathbb{Z}_{2}$ by homeomorphism on $\mathbb{R}^{2}$? Lie groups with no free $\mathbb{Z}/2\mathbb{Z}$ action
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If two continuous maps of an interval commute, then they agree at some point

Let $f,g:[0,1] \rightarrow [0,1]$ be continuous functions such that $f\circ g =g\circ f$. Prove that there exists $x \in [0,1]$ such that $f(x)=g(x)$
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There exists x on closed interval such that f(x)=x

If $f$ is a continuous function on a closed interval, how can I show that there exists some $x$ on $f$ that $f(x)=x$? I know it will require the Intermediate Value Theorem. Initially I thought of ...
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If $f^N$ is contraction function, show that $f$ has precisely one fixed point. [duplicate]

If $f$ is a mapping of a complete metric space $(X, d)$ into itself and $f^N$(composite $f$ for $N$ times) is a contraction mapping for some positive integer $N$, then $f$ has precisely one fixed ...
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Fun with Newton's Method - Infinitely many cycles

I'd like to preface this problem by saying that I have absolutely no clue if it is solvable or not. This is just the result of some musings, and I'm looking for either some guidance, or to be pointed ...
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521 views

Is there a proof for the Central Limit Theorem via some fixed point theorem?

This question arose in my mind when I learned that the Gaussian is a fixed point for the Fourier transform. On the other hand, in e.g. the Banach fixed point theorem we have convergence to a fixed ...
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Proof $\lim\limits_{n \rightarrow \infty} {\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}=2$ using Banach's Fixed Point

I'd like to prove $\lim\limits_{n \rightarrow \infty} \underbrace{\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}_{n\textrm{ square roots}}=2$ using Banach's Fixed Point theorem. I think I should use the ...
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If $f: \mathbb R^n \to \mathbb R^n$ is a contraction, then $x-f(x)$ is a homeomorphism

I am stuck in following homework question. Let $f : \mathbb R^n \to \mathbb R^n$ be a uniform contraction and $g(x) = x - f(x)$. Investigate whether $g : \mathbb R^n \to \mathbb R^n$ is a ...
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Do there exist sets $A\subseteq X$ and $B\subseteq Y$ such that $f(A)=B$ and $g(Y-B)=X-A$?

This is a little exercise I've been fiddling with for a while now. Let $f\colon X\to Y$ and $g\colon Y\to X$ be functions. I want to show that there are subsets $A\subseteq X$ and $B\subseteq Y$ ...
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Fixed point: sets and measures

Let $X$ be a Borel space with a Borel measure $\mu$. Suppose $\xi: X\times X\to\mathbb R_{\geq 0}$ is a continuous function and put $s(x) = \{y\in X:\xi(x,y) = 0\}$. For any set $b\in\mathcal B(X)$ we ...
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How to figure out the starting point for this Mandelbrot?

My answer for another math stackexchange question, asked by Gottfried, involved observing Mandelbrot bifurcation for the iterated function in question, $f(z)\mapsto z-\log_b(z)$. In particular, for ...
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Fixed Point Theorems

Theorem 1. Let $B=\{x\in \mathbb R^n :∥x∥≤1\}$ be the closed unit ball in $\mathbb R^n$ . Any continuous function $f:B\rightarrow B$ has a fixed point. Theorem 2. Let $X$ be a finite dimensional ...
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1answer
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Least common fixed-point

I have been reading a book, "Introduction to Lattices and Order", and I'm trying to solve exercise 8.29 as the following in it: Suppose that $P$ is a complete lattice and let $F$ and $G$ be order-...
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Fixed points in category theory

Is there a usual way to express the concept of fixed points in category theory? What would I say to express that a morphism has a fixed point? Thank you!
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Banach theorem example

By Banach fixed point theorem, if a metric on a metric space $X$ is such that $d(f(x),f(y))\leq K d(x,y)$ for $K\in (0,1)$ then $f$ has one unique fixed point. Is there an example where $d(f(x),f(y))\...
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What can Lefschetz fixed point theorem tell us about the group of automorphisms of a compact Riemann surface?

Let $X$ be a compact Riemann surface of genus $g>1$, $f\in Aut(X)$, a biholomorphism of $X$ onto itself, $x\in X$ a fixed point of $f$. Since tangent map of a holomorphic map (on the real tangent ...
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Continuous function on unit circle has fixed point

The question I have is: Let $f: S^1 \rightarrow S^1$ be a continuous function, where $S^1$ is the unit circle. Prove that if $f$ is not onto, then $f$ must have a fixed point.
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Must a holomorphic function from $D(0,1)$ to $D(0,1)$ have a fixed point?

Must every holomorphic function $f:D(0,1)\longrightarrow D(0,1)$ have a fixed point? I know that any holomorphic function with two fixed points is the identity: $f=Id$, but I can't find out an ...
6
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Fixed point property of spaces having same homotopy type

Suppose X and Y have same homotopy type.X as a topological space has fixed point property.Can we conclude anything about fixed point property of Y?
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Does the Brouwer fixed point theorem hold for the torus minus an open disk?

Let $S:=T^2\setminus{\rm int}(D^2)$ be the torus minus an open disk. Must every continuous function $f:S\to S$ have a fixed point? This is essentially the conclusion of the Brouwer fixed point ...
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Is there a simple way to prove the Brouwer fixed Point theorem?

The quest may be for references but I want to know if there is a simple way to prove the Brouwer fixed point theorem! That is if a function $f:\bar{B}\to\bar{B}$ is continuous then $f$ admits one ...
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Contraction mapping does not hold in metric space

Let $X=\mathbb{Q}\cap [1,2]$, i.e $X$ is the set of rational number between 1 and 2 inclusive. We can consider $X$ to be a metric space by endowing it with the usual distance function, i.e for $x,y \...
6
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197 views

Variable in upper bound of sum

I need to find a solution $x$ of the following equation: $$\sum_{n=0}^{\left[\frac{0.9}{x}\right]} (1-nx) = 45$$ where $[.]$ denotes the nearest integer function. I am an engineer and I'm currently ...
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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 $x\approx 0.3099069324$. ...
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Firm non-expansiveness in the context of proximal operators

$\newcommand{\prox}{\operatorname{prox}}$ Probably the most remarkable property of the proximal operator is the fixed point property: The point $x^*$ minimizes $f$ if and only if $x^* = \prox_f(x^*) $...
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1answer
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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,...
6
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1answer
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Fixed point combinator (Y) and fixed point equation

In Hindley (Lambda-Calculus and Combinators, an Introduction), Corollary 3.3.1 to the fixed-point theorem states: In $\lambda$ and CL: for every $Z$ and $n \ge 0$ the equation $$xy_1..y_n = Z$$ can ...
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Showing a function is Frechet Differentiable?

I just started learning the Frechet Derivatives. So I have a function $H:\mathbb{R}^{N\times n}\to\mathbb{R}^{N\times n}$, i.e. $U^T\in\mathbb{R}^{N\times n}$ and $$H(U^T)=GW\times (F(U))^T+S\times U^...
6
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1answer
551 views

Complete Condition in Banach Fixed Point Theorem

Can someone provide an example to show that for the Banach fixed point theorem, that is if $T : X → X$ is a contraction in a complete metric space $(X, d)$ then $T$ has a unique fixed point that $X$ ...
6
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1answer
265 views

A property of continuous bijective maps in the plane.

Take a non-empty open connected bounded set $D$ of $\mathbb{R}^2$ and let $f:D\to D$ be a continuous bijective map. It is well known that $f$ could have no fixed points i. e. $f(p)\not=p$ for all $p\...
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1answer
485 views

If $f:S^2\to S^2$ is homotopic to the identity does it have a fixed point? [duplicate]

Question: Let $f:S^2\to S^2$ be a continuous map that is homotopic to the identity. Does $f$ necessarily have a fixed point? I thought about this question after learning the proof of Brouwer Fixed ...
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1answer
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Composition of projections has a fixed point in a Hilbert space

Let Let H be a Hilbert space with an inner product ⟨⋅,⋅⟩ : H×H→R, and induced norm $∥⋅∥ : H→R_+$ Let $C_1$ and $C_2$ be closed, convex, nonempty, disjoint subsets of $H$ with at least one of ...
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1answer
169 views

A homeomorphism of the plane has a periodic point if it leaves a compact set invariant

Let $h$ be a homeomorphism of $\mathbb{R}^2$ onto itself such that $h(K)=K$ for some compact subset $K$ of $\mathbb{R}^2$. Show that $h$ has a periodic point in $\mathbb{R}^2$. My idea is consider a ...
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1answer
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How to show the optimization/ODE fixed point iteration steps converge?

I have $\vec{C} = G(\vec{\beta})$ by solving a system of ODE numerically. Thanks for the help of Robert the ODE can be found in this link: Solving a system of ODE Also $\vec{\beta}$ should satisfy $$...
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1answer
256 views

Corollary of Banach fixed-point theorem

Let $(X, \left\lVert\cdot\right\rVert)$ be a Banach space. Let $A:X\to X$ be a linear map and $\nu\in \mathbb{N}$ such that $A^k:X\to X$ is a contraction for every $k>\nu$. Is it true that for ...
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1answer
263 views

Proof for the existence of a second fixed point in Poincaré's last geometric theorem

In "Geometry and Billiards" by S. Tabachnikov the author proves Poincaré's last geometric theorem: "An area-preserving transformation of an annulus that moves the boundary circles in opposite ...
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1answer
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Is Poincare-Hopf index theorem connected with Leftschetz fixed point theorem?

Lefschetz Fixed Point Theorem: For a compact triangulable space $X$, and a continuous map $f:X\rightarrow X$, we have $$\sum_i(-1)^i\mathrm{Tr}(f_*|H_k(X,\mathbb{Q})=\sum_{x\in\mathrm{Fix}(f)}\...
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219 views

A function over the integers and its fixed points

Define $f:\mathbb{N}\rightarrow\mathbb{N}$ as follows, $f(n)$ is the number of times the digit "1" is needed if we were to write all integers between 1 and $n$ (inclusive) in base 10. So for example $...
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0answers
176 views

Every $f : [a, b] → [a, b]$ has a fixed point where $f$ is continuous. Deduce the intermediate value theorem.

Every $f : [a, b] → [a, b]$ has a fixed point and $f$ is continuous (on $[a,b]$). Deduce the intermediate value theorem. I managed to show the other way, now I'm here. I know that $f(c)=c$ for some $...
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Limit of a Discrete Dynamical System, Part 2

In my previous post (i.e., Limit of a Discrete Dynamical System) the following system was considered: $$\left[\begin{array}{c}x_{t+1}\\y_{t+1}\end{array}\right]=\left[\begin{array}{c}y_{t}/b\\x_{t}-1+...
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473 views

Closest fixed point to a convex set

Consider the compact convex sets $Y \subset X \subset \mathbb{R}^n$, and a Lipschitz continuous function $f : X \rightarrow X$. Assume that $f$ has multiple fixed points. (From Brouwer's theorem, $f$ ...
6
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1answer
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Converse to Lefschetz fixed point theorem (counterexample)

The celebrated Lefschetz fixed point theorem, in its simplest form, says (following Wikipedia) that if $f\colon X \to X$ is a continuous map of a compact triangulable space $X$ to itself, then $f$ has ...
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Fixed point of a shrinking map. Proof.

Could you please verify my proof? Let $f:X \to X$ be a shrinking map on a compact metric space $(X,d)$. In other words: $d(f(x),f(y)) < d(x,y)$ for all $x,y \in X$. Prove: $f$ has a unique ...
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161 views

A fixed point theorem [duplicate]

Let $\emptyset \not = X\subseteq \Bbb{R}^n$ be convex and compact and let $\cal{A}$ be a family of affine maps from $\Bbb{R}^n$ into $\Bbb{R}^n$ such that $X$ is invariant under each element of $\cal ...
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369 views

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 ...