# 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.

133 questions
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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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### To prove : If $f^n$ has a unique fixed point $b$ then $f(b)=b$

If $f: \mathbb R \to \mathbb R$ be a function such that for some $n_o \in \mathbb N$ , the $n_o$th iterate of $f$ has a unique fixed point $b$ , then how to prove that $f(b)=b$ ? I cant think of ...
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### Analogue to Fixed Point Theorem for Compact metric spaces

If $\omega:H \rightarrow H$ (into) is continuous and $H$ is compact, does $\omega$ fix any of its subsets other than the empty set?
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### Confused about a version of Schauder's fixed point theorem

I have read this: We have a map $S:W_0 \to W_0$. Moreover $W_0$ is not empty, convex, and weakly compact in $W$. Thus we can apply Schauder's fixed point theorem: Schauder's fixed point ...
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### Brouwer's fixed point theorem implies Sperner's lemma

I'm looking for a proof that Brouwer's fixed point theorem implies Sperner's lemma. The proof should be understandable by an undergraduate. Thanks in advance!
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### Edelstein Theorem

Let (M,d) be a compact metric space and $d(f(x),f(y)) < d(x,y)$ for all $x\neq y$ Prove that if $f$ is a continuous fuction then there is a unique $x_0 \in M$ such that $f(x_0)=x_0$ I know that ...
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### The composition of non-linear entire functions with no fixed points has infinitely many fixed points.

The entire [non-linear] functions $f$ and $g$ do not have fixed points. Show that $f \circ g$ has infinitely many fixed points. How do you prove this statement? Or if it is not true as stated, what ...
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### Fix-Point Theorem Proof.

Firstly, the assignment: Let $a,b \in\mathbb{R}$ and $a < b$. Furthermore let $f: [a,b] \rightarrow [a,b]$ be monotone increasing. Show that if $x:= \mathbf {sup}\{y \in [a,b] \| \ y ≤ f(y)\}$ ...
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### Why does the logistic map $x_{n+1}=r x_n(1-x_n)$, with $x_0\in (0,1]$, diverge for $r>3$?

I'm studying some basic behaviour of the logistic map $x_{n+1}=r x_n(1-x_n)$, with $x_0\in (0,1]$ and $r\in (0,4]$, for a project. I can't seem to figure out why the map does not converge (i.e. has no ...
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### Fixed points of contractions in metric spaces

How do I prove that all contractions on a complete, non-empty metric space has exactly one fixed point? What I know: I know that all contractions are continuous and that completeness of $A$ means ...
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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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### 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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### 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 ...