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, ...

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Fixed points of complex rational functions dilemma.

There's a theorem about fixed points of complex rational functions which states that those with degree $d$ have $d+1$ fixed points. the case where denominator has greater degree makes sense and it's ...
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Brouwer Fixed Point Theorem via the Jordan Curve Theorem

There is a proof of the Brouwer Fixed Point Theorem via the Jordan Curve Theorem ? The Brouwer Fixed Point Theorem. Let $B=\{x\in \mathbb R^2 :∥x∥≤1\}$ be the closed unit ball in $\mathbb R^2$ . ...
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How many distinct fixpoints does $\sum_{i=1}^n 2 \sinh(a_i z)$ have?

Let $n$ be a positive integer. Let $z$ be a complex number. Let $a_1,a_2,...,a_n$ be a given sequence of distinct reals larger than $\ln(2)$. How many distinct solutions are there to $\displaystyle ...
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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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Regarding fixed point

Please help! I am stuck at this question: Let X be compact and $f : X \rightarrow X$ be continuous. If $f$ has no fixed point then there exist $\delta > 0$ such that $d(a,f(a)) \geq \delta$ for ...
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Does Particular Point Topology has Fixed Point Property?

A Particular Point Topology is not compact, is path-connected but what about Fixed Point Property? Does it have fixed point property? If so how? I have been told that it should have Fixed Point ...
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Brouwer Fixed Point Theorem $f(S^1)\subset B$

I have a question about the Brouwer Fixed Point Theorem: Theorem 1.(Brouwer Fixed Point Theorem) Let $B=\{x\in \mathbb R^2 :∥x∥≤1\}$ be the closed unit ball in $\mathbb R^2$ . Any continuous ...
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Prove that a real variable function that satisfies certain conditions has a fixed point.

Let $f: \mathbb R \to \mathbb R$ be a differentiable function such that $f(0)=1$ and $|f'(x)|<\dfrac{1}{2}$. $i)$ Prove that there exists $x_0 \in [0,2] :f(x_0)=x_0$. $ii)$ Let ...
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77 views

Application of fixed point theorem in $R^n$

Let $A=(a_{ij}) \in \mathbb R^{n \times n}$ a matrix such that $|a_{ij}|<\frac{1}{n}$ for every $i,j$. Prove that $I-A$ is invertible. My attempt at a solution: $I-A$ is invertible $\iff$ ...
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191 views

Function iteration and intervals of attraction for fixed points

I am currently studying iteration sequences and I am a bit hung up on one specific bit which involves determining intervals of attraction of fixed points. I've been given a graphical method to ...
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99 views

Proving a particular function is surjective on a Banach space.

Let $(E,\|x\|)$ and let $f: E \to E$ such that $f+Id$ is a contraction ($Id$ is the identity map). Prove that $f$ is surjective and prove that, if $f$ is linear, then $f$ is a homeomorphism. The ...
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432 views

Must a holomorphic function from $D(0,1)$ to $D(0,1)$ have a fixed point?

Must every holomorphic function $f:D(0,1)\longrightarrow D(0,1)$ have a fixed point? I know that any holomorphic function with two fixed points is the identity: $f=Id$, but I can't find out an ...
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40 views

Two indexes $c$ and $b$, such that $Dom(\varphi_a)=\{b\}$ and $Dom(\varphi_b)=\{c\}$

Problem: Assume $\{\varphi_i\}_{i=1}^\infty$ is an effective enumeration of the computable functions. Find two indexes $c$ and $b$, such that $Dom(\varphi_c)=\{b\}$ and $Dom(\varphi_b)=\{c\}$. ...
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116 views

Fixed point in plane transformation.

Some one give me a idea to solve this one. It's a problem from Vladimir Zorich mathematical analysis I. Problem 9.c from 1.3.5: A point $p \in X$ is a fixed point of a mapping $f:X \to X$ if ...
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63 views

Fixed point iteration for $\sqrt[3]{a}$

So I'm given the scheme for computing $\sqrt[3]{a}$ $$x_{k+1}=px_k + \frac{qa}{x_k^2} + \frac{ra^2}{x_k^5}$$ and I have to find the p, q, and r so that this scheme is as fast as possible. Any hints ...
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70 views

Number of Fixed points of an odd degree polynomial

Let $p(x)$ be a polynomial of degree $2n+1$ with real coefficients. then $p(x)$ has (I) exactly $2n+1$ fixed points (II) at least one fixed point (III) at most one fixed point (Iv) $n$ fixed ...
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Getting a root of a continuously differentiable function by Banach's Fixed Point Theorem.

Banach Fixed Point Theorem: Consider a metric space $X = (X, d)$, where $X\neq \varnothing$. Suppose that $X$ is complete and let $T: X \to X$ be a contraction on $X$. Then $T$ has precisely one ...
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Reference request: generalized Lefschetz-Hopf fixed point theorem

Let $X$ be a smooth projective variety over $\mathbb{F}_p$. A basic (and very important) theorem is that we have equality $$ \# X(\mathbb{F}_{p^n}) = \sum_{i=0}^{2\dim X} (1)^i ...
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59 views

This system is contractive?

I have a system which has a form of find point problem, described as following $$p_i=h_i(\mathbf{p})$$ where $$p_i\in[0,1]$$ is the $i$-th components of the $n$-dimensional column vector ...
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On the Proof of the Perron-Frobenius Theorem.

The Perron-Frobenius theorem states that a square matrix with nonnegative entries has a real nonnegative eigenvalue. One possible proof uses the Brouwer fixed point theorem, and every proof I've seen ...
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Are the solutions to $\zeta \left(\frac{1}{2}+i n a\right)=a$ the same as fixed points? $n=1,2,3,…$

Are the solutions to $\zeta \left(\frac{1}{2}+i n a\right)=a$ called fixed points? $n=1,2,3,...$ $\zeta$ is Riemann zeta function. Mathematica program for finding $\zeta \left(\frac{1}{2}+i 3 ...
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Showing a sequence $x_{n+1} = Tx_n$ forms a contraction mapping

I want to show that the sequence given by $$x_{n+1} = Tx_n = x_n-\frac{(x_n^2-2)}{x_n+x_{n-1}}$$ forms a contraction mapping. That is $$|Tx_1-Tx_2|\leq c|x_1-x_2|.$$ Where $c$ is to be determined. I ...
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Continuous mapping on unit ball such that $T(\Bbb x_0)=0$

got a question from a course in functional analysis. " Let $T:\{\Bbb x\in\Bbb R: ||\Bbb x||\leq 1\}\to\Bbb R^n$ be a continuous mapping. Moreover assume that $\langle T(\Bbb x),\Bbb x\rangle>0$ ...
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Is $f : [0,\infty]\rightarrow [0,\infty]$ which is continuous and bounded has a fixed point

Question is to check if : $f : [0,\infty]\rightarrow [0,\infty]$ which is continuous and bounded has a fixed point. I have first of all considered boundedness. So, $f(x)$ should not have $x$ as ...
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37 views

Is it possible to show that the Gaussian is a fixed point of the Fourier transform using a fixed point theorem?

We know that $\text{exp}(-\alpha |x|^2)$ is a fixed point for the unitary Fourier transform if $\text{Re } \alpha > 0$. Is it possible to show this using a fixed point theorem?
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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 my fixed point proof hold?

I have am looking for existence of a fixed point for an operator that I have. I already looked at some related fixed point theorems such as Schrauder's and Rothe's. But most of them seem to require ...
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Proving $\frac{k-1}{k}$ is an attractor of the logistic map $kx(1-x)$.

Consider the logistic map $f(x) = kx(1-x)$ defined on $\mathbb{R}$. We already know $\frac{k-1}{k}$ is a fixed point of $f$, but my issue is showing it's an attractor when $k \in [1,3]$. There is an ...
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Questions about $\ln(z)$ recurrence and fixed points.

Define property $A_R$ for an analytic function $f(z)$ as $1)$ $f(z)=0$ has exactly one solution being $z=0$ for $|z|<R$ where $R$ is a radius. And $f(z)$ is analytic within the radius $R$ ...
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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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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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Importance of Banach fixed point in mathematics

I know the proof of Banach fixed point theorem in complete metric space and two example of it. It is useful to prove Picard's theorem on existence and uniqueness of ordinary differential equation. In ...
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Question about unique fixed point.

Suppose $f(x,y)\in C^{2}(R^n)$, $g=(g'_1,g'_2)'$, $g_1(y)=\arg\min_xf(x,y)$ and $g_2(x)=\arg\min_yf(x,y)$. $g$ has a fixed point at $(x_0,y_0)$. $h=(h'_1,h'_2)'$, $h_1(y)=x_0+\dot{g}_1(y_0)(y-y_0)$ ...
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Fixed point of matrix

Suppose that $a$ is a fixed point of matrix $A$, what that means? What is a fixed point of matrix? Thank you!
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How to decide if these two maps are proper?

We define a mapping $f$ of a topological space $X$ into a topological space $Y$ to be proper if the subspace $f^{-1}(C)$ is compact in $X$ whenever $C$ is a compact subspace of $Y$. Now how to ...
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Is Poincare-Hopf index theorem connected with Leftschetz fixed point theorem?

Lefschetz Fixed Point Theorem: For a compact triangulable space $X$, and a continuous map $f:X\rightarrow X$, we have ...
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Calculating the fixed points of a model

Does anyone know how I would go about calculating the fixed points of this model?
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62 views

Simulating Mixed Nash Equilibria

I have a $N$ person game where each person has a set of $M$ discrete strategies. I know from the theory that at least one mixed strategy Nash Equilibrium exists. Can someone please tell me how do I ...
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A function that is not contractive with respect to any metric

I am struggling with this homework question with is related to iterated function system and fixed point theory. The question is: Let $\Delta \in R^2$ be a filled non-degenerate triangle with ...
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No fixed points imply no periodic points

Let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a smooth injective function with $\operatorname{det}[f'(x)]\not=0 $ for all $x\in\mathbb{R}^n$. Moreover assume that $f$ has no fixed points. Can $f$ have a ...
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Contradict the Contraction Mapping Theorem

I am trying to show that the function $f(x) = 2\pi+x-\tan^{-1}x$ is contractive but has no fixed points. Finally I wish to conclude that it does not contradict the contraction mapping theorem. $f$ is ...
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An application of fixed point theorem

I want to use the fixed point method to solve the equation to find $y$: $$ y = c_1 y^3 - c_2 y$$where $c_1, c_2$ are real valued constants. So I designed $$ y_{k+1} = c_1 y_k^3 - c_2 y_k$$ to ...
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Condition for A,B s.t $x_{n+1}=\frac{112233+Ax_n^2}{Bx_n}$ converges

Given the following fixed point iteration $x_{n+1}=\frac{112233+Ax_n^2}{Bx_n}$ find values for A,B s.t the iteration converges to $\alpha$ (a root) with maximal order of convergence. Find the ...
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Does a compact connected complete linear order have the fixed point property?

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 ...
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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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Compactness of a subset of a specific bounded $L^2$ space

For my research, I am working with the set $$S = [0,1] \times [0,\delta] \times[0,\delta^2] \times \cdots $$ where $S\subset \mathbb{R}^\infty$. I am using the $\|\cdot\|_2$ norm. I was hoping to ...
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121 views

upper semi-continuity of a multi-valued function $T$ and lower semi-continuity of $d(x,T(x))$

Let $(X,d)$ be a complete metric space, $CB(X)$ the set of closed and bounded subsets of $X$, and $T:X\rightarrow C(X)$ be a multi-valued function. How can you prove this: If $T$ is upper ...
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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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57 views

Fix point of squaring numbers mod p

Take the set of integers $\{0, 1, .., p-1\}$, square each element, you get the (smaller) set of quadratic residues. Repeat until you get a fix point set. The size of this set is a function of $p$. ...
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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 ...