# 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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### 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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### Contractive Operators on Compact Spaces

Suppose that $T: M \to M$ is a compact contractive Operator on a nonempty compact subset $M$ of a complete metric space $X$. Show that $T$ has a unique fixed point. Further show that the sequence ...
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### Fixed points of polynomial ring homomorphism

$S=\mathbb R[x+y+z, xy+yz+zx, xyz]$ is the ring of the symmetric polynomials in $\mathbb R[x,y,z]$. Let $\psi\colon S \to R[x,y,z]$ be a ring homomorphism such that \begin{align} x &\mapsto -x,\\...
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### Every 1-Lipschitz function in the closed unit ball has a fixed point

I'm currently trying to solve the following exercise: Let B be the closed unit ball in $\mathbb R^n$ together with the euclidean metric. Show that every 1-Lipschitz function $f:B\to B$ has a fixed ...
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### What are the conditions for a compact, convex set to be homeomorphic to the closed unit ball in the plane?

A generalization of Brower's fixed point theorem says that any continuous map from compact, convex set in the plane $K$ to itself, $f:K \to K$, must have a fixed point. It is easy to see that any set ...
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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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### 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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### 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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### 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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### 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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### Showing that a function $f$ has a unique fixed point in a metric space.

Let $(X, d)$ be a compact metric space, and suppose $f : X → X$ satisfies $$d(f(x), f(y)) < d(x, y)$$ for all $x \neq y \in X$. Show that f has a unique fixed point. All I've gotten it so far ...
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### Continuous bijections from the open unit disc to itself - existence of fixed points

I'm wondering about the following: Let $f:D \mapsto D$ be a continuous real-valued bijection from the open unit disc $\{(x,y): x^2 + y^2 <1\}$ to itself. Does f necessarily have a fixed point? I ...
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### Brouwer's fixed point theorem (for unit balls) and retractions

Let $X$ be a generic Banach space over $\mathbb R$ (also infinite dimensional), and let $B$ be the unit ball in $X$. I want to prove that the following proposition $B$ is a fixed-point space if ...
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### Use the Contraction Mapping Principle to show that $x=\frac19\sin\left(3x\right) + \sqrt{x}$ has exactly one solution $x\geqslant\frac{8}{9}$

Use the Contraction Mapping Principle to show that $x=\frac19\sin\left(3x\right) + \sqrt{x}$ has exactly one solution $x\geqslant\frac{8}{9}$. I have literally no idea if this is right, please could ...
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### Proving $\displaystyle\lim_{n\to\infty}a_n=\alpha$ $a_1=\frac\pi4, a_n = \cos(a_{n-1})$

Let $a_1=\frac\pi4, a_n = \cos(a_{n-1})$ Prove $\displaystyle\lim_{n\to\infty}a_n=\alpha$. Where $\alpha$ is the solution for $\cos x=x$. Hint: check that $(a_n)$ is a cauchy sequence ...
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### Brouwer's fixed-point theorem and the intermediate value theorem?

In one dimension, Brouwer's fixed-point theorem (BPFT) can be proved easily based on the Intermediate Value Theorem (IVT). Is the inverse also true? I.e, is it possible to prove the IVT directly from ...
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### Fixed point iteration convergence of $\sin(x)$ in Java [duplicate]

Is it mathematically correct to say that $\sin(x)$ converges to zero as $x$ approaches $0$? If the $\sin(x)$ iteration is done starting at $\dfrac{\pi}{2}$ in Java, for $10^9$ iterations, the result ...
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### Question about fixpoints and zero's on the complex plane.

Define property $A$ for an entire function $f(z)$ as $1)$ $f(z)=0$ has exactly one solution being $z=0$ $2)$ $f(z)=z$ has exactly one solution $=>z=0$ (follows from $1)$ ) $3)$ $f(z)$ is not a ...
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### Suppose $f(z)$ is analytic in the closed unit disc…

Suppose $f(z)$ is analytic in the closed unit disc and $$|f(z)|<1 \quad \text{for} \quad |z|=1$$ Show that $f(z)$ has one and only one fixed point; that is, there exist a unique point $z_0$ in the ...
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### A question about fixed points and non-expansive map

Let $$K=\{x=(x(n))_n\in l_2(\mathbb{N}):\|x\|_2\le 1\ \text{ and } x(n)\ge 0 \text{ for all } n\in \mathbb{N} \}$$ and define $T:K\to c_0$ by $T(x)=(1-\|x\|_2,x(1),x(2),\ldots)$. Prove : (1) $T$ is ...
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### Fix point of $L:S^2\rightarrow S^2$

Let $L:S^2\rightarrow S^2$ be a bijective continue map. Is there exists $L$ such that $\forall x\in S^2 \Rightarrow Lx\neq x$ and $Lx\ne -x$? I mean that whether $L$ must have fix point? In fact ...
For the following functions (i) find the fixed points and (ii) use the graphical analysis to determine the dynamics for all points on R: $f(x)=2sin(x)$. I have no problem finding the fixed points, ...