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.

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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 a (non-commutative) group. Then if one chooses…
Nikolaj-K
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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 $M$ be a metric space homeomorphic to the closed…
Dan Rust
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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 argument (with easy modification) applies equally…
Jason DeVito
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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 uniqueness is easy. My problem is to show that there a exist…
Proton
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Show that a continuous function has a fixed point

Question: Let $a, b \in \mathbb{R}$ with $a < b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$. I suppose this has to do with the basic definition of continuity. 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 $[0,1]$ and use the intermediate value theorem. Can…
Lopsy
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"Why do I always get 1 when I keep hitting the square root button on my calculator?"

I asked myself this question when I was a young boy playing around with the calculator. Today, I think I know the answer, but I'm not sure whether I'd be able to explain it to a child or layman playing around with a calculator. Hence I'm interested…
Rasmus
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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 exists an unique fixed point. But is there an incomplete…
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Is there a square with all corner points on the spiral $r=k\theta$, $0 \leq \theta \leq \infty$?

While reading about the square peg problem, I found this paper of Jerrard, where he described that for the spiral $$r = k\theta \quad 2\pi \leq \theta \leq 4\pi $$ if we join the endpoints, you can only draw one square that all of its corner points…
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Contraction Map on Compact Normed Space has a Fixed Point

Let $K$ be a compact normed space and $f:K\rightarrow K$ such that $$\|f(x)-f(y)\|<\|x-y\|\quad\quad\forall\,\, x, y\in K, x\neq y.$$ Prove that $f$ has a fixed point.
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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 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 science are given informally, but they are more subtle…
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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 demand, so that $f$ has a non-trivial fixed point…
Watson
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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 theorems in logic and…
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Why does Fixed Point Iteration work?

I have searched online for an answer, but everyone gave the method, and no one explained why is it working. I'll first write what I do understand. Let $f(x)$ be a continuous function at $[a,b]$. suppose $f$ has a root at $[a,b]$, and we want to…
so.very.tired
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