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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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What is the algebraic structure of functions with fixed points?

So I just noticed that the set of functions with a fixed point $$f(x_0)=x_0,$$ are closed under composition $$(f\circ g)(x):=g(f(x)),$$ and with $e(x)=x$, the inverible functions even seem to form ...
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Homeomorphic to the disk implies existence of fixed point common to all isometries?

A fellow grad student was working on this seemingly simple problem which appears to have us both stuck. (The problem naturally came up in his work so isn't from the literature as far as we know). Let ...
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Does every continuous map from $\mathbb{H}P^{2n+1}$ to itself have a fixed point?

This question is motivated by Fixed point property of Cayley plane and idle curiosity. In the link, it is shown that every continuous map from the Cayley plane to itself has a fixed point and the ...
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Generalization of “easy” 1-D proof of Brouwer fixed point theorem

In topology class, before we learned the homotopy-based proof of Brouwer's fixed point theorem, the professor mentioned the easy proof of the one-dimensional case: just draw the graph $f(x) = x$ in $[...
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Prove the map has a fixed point

Assume $K$ is a compact metric space with metric $\rho$ and $A$ is a map from $K$ to $K$ such that $\rho (Ax,Ay) < \rho(x,y)$ for $x\neq y$. Prove A have a unique fixed point in $K$. The ...
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Contraction mapping in an incomplete metric space

Let us consider a contraction mapping $f$ acting on metric space $(X,~\rho)$ ($f:X\to X$ and for any $x,y\in X:\rho(f(x),f(y))\leq k~\rho(x,y),~ 0 < k < 1$). If $X$ is complete, then there ...
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Why does the fixed point theorem justify the existence of the factorial function?

I was learning about fixed point theorem in the context of programming language semantics. In the notes they have the following excerpt: Many recursive definitions in mathematics and computer ...
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Algebraic fixed point theorem

I was wondering if there are some "algebraic" fixed point theorems, in group theory. More precisely, given a group $G$ and a group morphism $f : G \to G$, what conditions on $G$ and $f$ should we ...
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Prove existence of unique fixed point

Let $f(x)$ be a strictly decreasing function on $\mathbb{R}$ with $|f(x)-f(y)|<|x-y|$ whenever $x\neq y$. Set $x_{n+1}=f(x_n)$. Show that the sequence $\{x_n\}$ converges to the root of $x=f(x)$. ...
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$f(a)-f(b)$ is rational iff $f(a-b) $ is rational

Prove that the continuous function $f:\mathbb{R} \to \mathbb{R}$ satisfying $f\left(x\right)-f\left(y\right) \in\mathbb{Q} \iff f\left(x-y\right) \in \mathbb{Q}$ is of the form $ f\left(x\right)=ax+...
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Converse of a fixed-point theorem

I'm having some trouble furnishing a proof here. Let $(E, d)$ be a metric space such that any $k$-Lipschitz function has a fixed point for $0 < k < 1$. Does it follow, then, that $E$ is ...
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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,...
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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....
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Leray-Schauder fixed point theorem

I know the proof of the Schauder fixed point theorem which states Schauder fixed point theorem : If $D$ is a non-empty , convex and compact subset of Banach space $B$ and $T:D \to D$ a continuous ...
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Fixed Point Property for a special space?

Suppose $X$ is a compact connected metric connected space and for every $\epsilon >0$ , there exists a continuous surjective function $f : X \rightarrow [0,1]$, that for all $y \in [0,1]$, the ...
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Why use the Lefschetz Zeta function?

Given a compact, triangulable space $X$ and a continuous function $f: X\rightarrow X$, then we define the Lefschetz number $\Lambda_{f}$ by $$\Lambda_{f} = \sum_{k\geq0}(-1)^{k}Tr(f_{\ast}\vert H_{k}(...
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Fixed points in computability and logic

I asked this question on CS.SE, too: https://cstheory.stackexchange.com/questions/27322/fixed-points-in-computability-and-logic I would like to understand better the relation between fixed point ...
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Completeness can be omitted from Banach Fixed Point Theorem?

In Kreyszig's Functional Analysis, page no. 303, exercise no. 3 says that completeness cannot be omitted from Banach's Fixed Point Theorem. But if we take $f(x)=x^2$ from an incomplete metric space $(-...
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In this case, does $\{x_n\}$ converge given that $\{x_{2m}\}$ and $\{x_{2m+1}\}$ converge?

I'm playing around with a sequence $\{x_n\}$ defined by $$ x_{n+1}=\frac{\alpha+x_n}{1+x_n}=x_n+\frac{\alpha-x_n^2}{1+x_n}. $$ Here $\alpha\gt 1$, and $x_1\gt\sqrt{\alpha}$. I'm trying to compute $\...
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Is the Knaster-Tarski Fixed Point Theorem constructive?

According to Tarski's Fixed Point Theorem, for a complete lattice $L$, and monotone function $f:L \rightarrow L$, the set of fixed points of $f$ forms complete lattice. Definition of $lfp(f)$ and $...
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How to determine the outcome of the recursive sequence $a_n=\frac{1}{\operatorname{abs}\left(a_{n-1}\right)-1}$

Depending on the starting value, the end result of this iterative sequence appears to be very variable, for example, if the starting value $a_0\ =\frac{b}{c}$ where $b$ and $c$ are integers, then ...
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Schwarz's Lemma, fixed points question

This is from an old qualifying examination question. If f is holomorphic in the unit disk $D$ and $|f(z)|<1$ for all $z\in D$. Suppose also that $f$ has two distinct fixed points in $D$ then $f(z)=...
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Is there a simple proof of Borsuk-Ulam, given Brouwer?

(Brouwer) Any continuous function from a convex compact subset K of a Euclidian space to itself has a fixed point. Given this lemma, is there a simple proof of: (Borsuk-Ulam) Any continuous ...
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Fixed points of the Gamma Function?

I am interested in complex values of $z$ such that $$ \Gamma (z) =z$$ Clearly, the one trivial value of $z$ is 1. Also, looking at a graph of the gamma function on the real axis, I can tell that there ...
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Continuous function on closed unit ball

Take a continuous mapping $f: \bar{B^{n}} \rightarrow \bar{B^{n}}$, where $\bar{B^{n}}$ is a closed unit ball in $\mathbb{R}^{n}$. Assume that $f(x) \neq x$ for every $x \in \bar{B^{n}}$. Define ...
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If you didn't already know that $e^x$ is a fixed point of the derivative operator, could you still show that some fixed point would have to exist?

Let's suppose you independently discovered the operator $\frac{d}{dx}$ and know only its basic properties (say, the fact it's a linear operator, how it works on polynomials, etc.) If you didn't know ...
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If $x_{n+1}=f(x_n)$ and $x_{n+1}-x_n\to 0$, then $\{x_n\}$ converges

Let $f:[0,1] \rightarrow [0,1]$be a continuous function. Choose any point $x_0 \in [0,1]$ and define a sequence recursively by $x_{n+1}=f(x_n)$. Suppose $\lim_{n \rightarrow \infty}x_{n+1}-x_n =0$, ...
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Blackwell's condition for a contraction: Why is boundedness neccessary?

I'm trying to understand the proof that certain operators $T$ are a contraction if they fulfill Blackwell's sufficient conditions. In particular, I try to understand why the operator $T$ has to map ...
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Mapping homotopic to the identity map has a fixed point

Suppose $\phi:\mathbb{S}^2\to\mathbb{S}^2$ is a mapping, homotopic to the identity map. Show that there is a fixed point $\phi(p)=p$.
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Common knowledge as a fixed point

I read on a wikipedia page that from the modal logic formalization CK can be formulated as a fixed point. If it also holds for the set theory formalization? If it does, where I can find about it? ...
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Continuous Brouwer's fixed point theorem via Stokes's theorem?

Let $B$ denote the closed unit ball in $\mathbf{R}^n$. Brouwer's fixed point theorem states that every continuous map $f:B\to B$ has a fixed point. There is a simple proof using Stokes's theorem, at ...
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eventually constant maps

Let $f:[0,1]\to [0,1]$ be a continuous function with a unique fixed point $x_{0}$ Assume that $\forall x\in [0,1], \exists n\in \mathbb{N}$ such that $f^{n}(x)=x_{0}$. Does this implies ...
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Finding a functor satisfying a recursive equation

Say I want to find a functor satisfying something like: \[FA = 1 \sqcup (A \times FA).\] Equivalently, I want to find for each $A$ a fixed point of: \[GB = 1 \sqcup (A \times B)\] (I'll worry about ...
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Minimax Theorems V.S. Fixed Point Theorems.

Is there any relationship between the minimax theorems $$ \mbox{Regularly hypothesis on } f:U\times V\to\mathbb{R} \implies \min_{u}\max_{v}f(u,v)=\max_{v}\min_{u}f(u,v) $$ and fixed point ...
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Convergence of fixed point iteration for polynomial equations

I'm looking for the solution $x$ of $$x^n+nx-n=0.$$ Thoughts: From graphing it for several $n$ it seems there is always a solution in the interval $[\tfrac{1}{2},1)$. For $n=1$, the solution is ...
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A variation of the square peg problem

This question spawned from this recent thread. The notorious square peg problem states that any continuous, simple and closed curve $\gamma$ in the plane contains the vertices of some square. It has ...
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does this Newton-like iterative root finding method based on the hyperbolic tangent function have a name?

I've recently discovered that modifying the standard Newton-Raphson iteration by "squashing" $\frac{f (t)}{\dot{f} (t)}$ with the hyperbolic tangent function so that the iteration function is $$N_f (...
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Fixed point of a polynomial mapping - what's the relation between the two views

Let $\sigma : \Bbb{C}^3 \to \Bbb{C}^3$ be a polynomial mapping. Let $P:= \Bbb{C}[x,y,z]$ denote the space of polynomial in 3 variables. Then $\sigma$ induces a (linear) mapping $\tilde{\sigma} : P\...
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Fixed point: linear operators

I ask my question in two parts: though the topic is similar, I would like to distinguish linear and general cases since methods may be too different while my questions are broad. Consider a space $X$ ...
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For a continuous function $f$ satisfying $f(f(x))=x$ has exactly one fixed point

Let $f \colon [ 0, 1] \to [0, 1]$ be a continuous map such that $$ f\big( f(x) \big) = x \ \mbox{ for each } x \in [0, 1], $$ and $$ f(x) \neq x \ \mbox{ for at least one } x \in [0, 1], $$ ...
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Why is this easy “proof” of Brouwer's Fixed Point Theorem not correct/common?

Brouwer's Fixed Point Theorem states, essentially, that any continuous function on a closed disc to itself has a fixed point. I am familiar with the proof based on the impossibility of a retraction ...
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Can the proof of fixed point theorems ever be constructive?

Overall, Brouwer fixed point theorem and Kakutani fixed theorem are non-constructive. Is there any established paper that demonstrates that there exists constructive proofs that do exactly what these ...
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Opposite of a contraction mapping

I am taking Real Analysis and we recently went over the Banach Fixed-point Theorem, also commonly known as the Contraction Mapping Theorem which states: If $(X,d)$ is a complete metric space, and $f:...
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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 ...