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)...

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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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If $T^n$ is $q$-contractive, $T$ exactly has one fixed point

Consider a complete metric space $(X,d)$ and $T\colon X\to X$. Suppose there exists $n\in\mathbb{N}$ such that the n-th power of $T$ is $q$-contractive. Show that then $T$ has exactly one fixed point $...
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Check Points are line, triangle, circle or rectangle

How to determine geometric properties of four distinct points in a plane (x1,y1), (x2,y2), (x3,y3), (x4,y4) represented in the 2-D Cartesian coordinate system, whether these four points are on a ...
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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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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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Contraction and Fixed Point [duplicate]

How do I show that for $T: X \rightarrow X$ where X is complete and $T^m$ is a contraction that T has a unique fixed point $x_0 \in X$. I know there exists $\lambda_1 \in (0,1)$ for $x, y \in X$ ...
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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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On Tarski-Knaster theorem

Let $P(X)$ denote the power set of $X$ (the set of all subsets of $X$) with a partial order given by inclusion. If $F: P(X) \to P(X)$ is monotone (order preserving), then $F$ has a fixed point. How ...
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Periodic orbits

Let $x\in\mathbb R$ be a periodic point with lenght 2 of the recursion $x_{n+1}=f(x_n)$ My book about Dynamic systems says that this recursion has a fixed point now, because a periodic point with ...
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229 views

If $f^N$ is contraction function, show that $f$ has precisely one fixed point. [duplicate]

If $f$ is a mapping of a complete metric space $(X, d)$ into itself and $f^N$(composite $f$ for $N$ times) is a contraction mapping for some positive integer $N$, then $f$ has precisely one fixed ...
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Bourbaki-Witt fixed point theorem: two questions

Consider the following theorem: Let $f\colon E \to E$ have the propery that $f(x)\geq x$, where $(E,\leq)$ is a non-void partially ordered set with the property that every totally ordered subset of $...
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227 views

Apply Banach's fixed point theorem

Let $$T:f\mapsto (x\mapsto \frac{2}{5}\int_0^1 (x^2+t^5)f(t) dt + \sin(x))$$ for any $x\in[0,1]$, $f\in C([0,1])$. I want to show that that there is a uniqu $\tilde{f}$ that solves that equation $f(...
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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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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 ...
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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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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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Proof $\lim\limits_{n \rightarrow \infty} {\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}=2$ using Banach's Fixed Point

I'd like to prove $\lim\limits_{n \rightarrow \infty} \underbrace{\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}_{n\textrm{ square roots}}=2$ using Banach's Fixed Point theorem. I think I should use the ...
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Using the Banach Fixed Point Theorem to prove convergence of a sequence

Use the Banach fixed point theorem to show that the following sequence converges. What is the limit of this sequence? $$\left(\frac{1}{3}, \frac{1}{3+\frac{1}{3}}, \frac{1}{3+\...
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Iterates of $f_b(x) = x - \log_b(x) $ - for $\log(b) \approx 0.399$: convergence to accumulation points or chaos?

In a previous question I arrived at the family of functions (depending on a real parameter b for the base of exponentiation/logarithm): $$ f(x) = x - \log_b(x) \qquad \qquad b \gt 0$$ with the ...
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Prove that $f$ has a fixed point . [duplicate]

For $f:[a,b]\rightarrow [a,b]$ is a continuous . Prove that $f$ has a fixed point . Is that true if we change $[a,b]$ by $[a,b)$ or $(a,b)$.
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Equivalence of Brouwers fixed point theorem and Sperner's lemma

I'm looking for a proof that Brouwer's fixed point theorem implies Sperner's lemma. All proof I've found just prove that there must be at least one completely colored n-simplex, not that there must be ...
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In a complete lattice every monotone function has a fixpoint (Knaster–Tarski Theorem)

L is a complete lattice, so every subset has a supremum and infimum. In addition, there exists a function $f:L \rightarrow L$ such that $a \leq b$ implies $f(a) \leq f(b)$. Prove that there exists ...
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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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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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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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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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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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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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Generalized Fixed Point Theorem

Suppose that $T: M \to M$ is a self map of a nonempty closed set $M$ in a complete metric Space $(X,d)$. Suppose further that $$d(Tx,Ty) \le k(a,b)d(x,y)$$ for all $x,y \in M$ with $0 \lt a \le d(x,y) ...
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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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Proving and understanding the Fixed point lemma (Diagonal Lemma) in Logic - used in proof of Godel's incompleteness theorem

http://en.wikipedia.org/wiki/Diagonal_lemma I am wondering about the proof of the "Fixed-Point Lemma" $\text{Mod } \Sigma$ is the class of all models of $ \Sigma$. $\text{Th Mod } \Sigma$ is the set ...
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When does Newton-Raphson Converge/Diverge?

Is there an analytical way to know an interval where all points when used in Newton-Raphson will converge/diverge? I am aware that Newton-Raphson is a special case of fixed point iteration, where: $...
7
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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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Proof that the solution to cosx = x, is the limit of a recursive sequence.

So I've got this question. Exists a sequence $a_n$ such that: $$a_0 = \frac \pi4, a_n=\cos\left(a_{n-1}\right)$$ Prove that $\lim_{n\rightarrow\infty} a_n = \alpha$ Where $\alpha$ is the solution to $...
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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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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 ...
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Fixed point combinator (Y) and fixed point equation

In Hindley (Lambda-Calculus and Combinators, an Introduction), Corollary 3.3.1 to the fixed-point theorem states: In $\lambda$ and CL: for every $Z$ and $n \ge 0$ the equation $$xy_1..y_n = Z$$ can ...
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If $f: \mathbb R^n \to \mathbb R^n$ is a contraction, then $x-f(x)$ is a homeomorphism

I am stuck in following homework question. Let $f : \mathbb R^n \to \mathbb R^n$ be a uniform contraction and $g(x) = x - f(x)$. Investigate whether $g : \mathbb R^n \to \mathbb R^n$ is a ...
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Is this a valid way to show $\chi(SL_n(\mathbb{R}))=0$?

Why does $\chi(SL_n(R))=0$? I'm going about it like this. Let $X:=SL_n(R)$. Define a map $f:X\to X$ such by $A\mapsto BA$, where $B$ is the identity matrix, except with an extra $1$ in the upper ...
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Fixed point property of Cayley plane

I want to know whether the Cayley plane has fixed point property or not. I think it is so but I am not able to prove this. It certainly does not admit maps of period two without fixed points. By fixed ...
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1answer
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Generalization of Banach's fixed point theorem

I wanted to show that if $f:X\to X$ is a function from a complete metric space to itself and if $f^k$ is a contraction, then $f$ has a unique fixed point (say $p$) and for any $x$ in $X$ $f^n(x)\...
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1answer
61 views

Brouwer transformation plane theorem

Can somebody show that BPTT, version 2 is deduced from BPTT, version 1 [BPTT, version 1] Let $h$ be a fixed point free orientation preserving homeomorphism of $\mathbb{R}^2$ Every point $m\in\mathbb{...
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Every increasing function from a certain set to itself has at least one fixed point

I need a hint for the following question: Let $S$ be a nonempty ordered set such that every nonempty subset $E\subseteq S$ has both a least upper bound and a greatest lower bound. Suppose $f:S \...
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4answers
223 views

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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1answer
96 views

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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1answer
48 views

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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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 ...
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3answers
526 views

Prove the sequence $x_n = \frac{x_{n-1}}{2}+\frac{A}{2x_{n-1}} \rightarrow \sqrt{A}$ as $n \to \infty$

Show that if A is any positive number, then the sequence defined by: $$x_n = \frac{x_{n-1}}{2}+\frac{A}{2x_{n-1}}$$ for any $n \geq 1$ converges to $\sqrt{A}$ whenever $x_0 > 0$.
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1answer
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Fixed Points and Graphical Analysis

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, ...