# 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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### 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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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 $\tilde{\sigma} : P\... 3answers 199 views ### For a continuous function$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 $$f(x) \neq x \ \mbox{ for at least one } x \in [0, 1],$$ ... 2answers 511 views ### 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 ... 1answer 530 views ### 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 ... 2answers 2k views ### 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:...
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 ...
### 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 ...