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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Dense subgroup of an extremely amenble group is extremely amenable??

Recall that a topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space $K$ admit a fixed point. i.e there is a point $\xi\in K$ such that $g.\xi=\xi$ for ...
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116 views

Derivative of the Solution to a Fixed Point Iteration

Let $\theta_s$ is the solution to a fixed point equation $$\theta=f(\theta,\lambda)$$ Let $d(\theta,\lambda)$ be another function of $\theta,\ \lambda$. I know $f$ but I have no explicit expression ...
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85 views

How to compute the Lefschetz number

Given a continuous function $f: X \to X$ how do you compute: $$ \Lambda_f = \sum_{k \geq 0} (-1)^k \mathrm{Tr}(f_*|H_k(X,\mathbb{Q})) $$ which is known as the Lefschetz number. For instance let $X: ...
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1answer
284 views

How to prove $F$ is a contraction mapping?

Given a $n\times n$ matrix $\mathbf{D}$ in which each entry $\mathbf{D}_{ij} \in [0,1]$. Let $i$-th row vector of $\mathbf{D}$ denote by $\mathbf{D}_{i\ast}$. The mapping $F:[0,1]^n \mapsto [0,1]^n$, ...
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A question on fixed point property

Assume that $0<k<n-1$, Note that $\mathbb{C}P^{k}$ can be considered as a closed subset of $\mathbb{C}P^{n}$, in a natural way. We collapse $\mathbb{C}P^{k}$ to a point. The resulting space is ...
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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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Why $e^x$ never equal $x$?

Je veux savoir pourquoi $x=e^x$ n'a aucune solution dans $\Bbb R$. Lorsque j'ai essayé de tracer le graphe de la fonction $e^x$, j'ai trouvé en fait qu'elle est une fonction strictement croissante ...
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Converse to Lefschetz fixed point theorem (counterexample)

The celebrated Lefschetz fixed point theorem, in its simplest form, says (following Wikipedia) that if $f\colon X \to X$ is a continuous map of a compact triangulable space $X$ to itself, then $f$ has ...
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1answer
21 views

completeness of $A=\{(x_1,x_2)\in\mathbb R^2| \max\{ |x_1|,|x_2|\}<1\}$

Let $A=\{(x_1,x_2)\in\mathbb R^2| \max\{ |x_1|,|x_2|\}<1\}$ and $g(x_1,x_2)=\frac14(x_1^2x_2,x_1+1)$. I want to show that $g$ has a fixed point in $A$. So ...
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89 views

Fixed point method where the derivative is one - does it converge

I'm trying to see if the iterative method $x_n=g(x_{n-1})$ where $g(x)=2\sqrt{x-1}$ will converge to $2$, if I take $x_0$ that is sufficiently close to $2$. Indeed notice that $g(2)=2$. and we have a ...
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31 views

Existence of Galerkin approximations to PDE

Given a weak PDE $$ \langle b(u),v \rangle +\langle a(\nabla u), \nabla v\rangle=\langle f, v \rangle,\qquad v\in H^1_0 $$ where $b\in C(\mathbb{R})$ and $a\in C(\mathbb{R}^n,\mathbb{R}^n)$ grow ...
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49 views

Statement of Markov-Kakutani fixed-point theorem

Markov-Kakutani fixed-point theorem is usually stated as follows: "Let $E$ be a locally convex topological vector space. Let $C$ be a compact convex subset of $E$. Let $S$ be a commuting family of ...
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1answer
33 views

Trouble understanding the steps to go from $x^x = 1000$ to the fixed point of $f(x) = \frac{log(1000)}{log(x)}$

I'm following through the Structure and Interpretation of Computer Programs, and in an example, they talk about finding fixed points of function (where $f(x) = x$) They then go on to say, if we want ...
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1answer
83 views

quotient of an amenable group

I have a question about amenable groups. The notion of amenability I am using is: The action of $G$ on $k$ ($k$ locally compact in a topological vector space) is amenable if there exists a point $x ...
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52 views

Identification of (Centers of) Cycles in a Discrete Time Dynamical System

I am studying dynamics on nonlinear Discrete Time Dynamical System of the form $$ \vec{X}_{t+1} = D(\vec{X}_t), $$ where D is some nonlinear function. I was looking for a (relatively) quick ...
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1answer
109 views

Unique fixed point of a contraction defined on a closed ball which maps the boundary back into the ball

Let $X$ be a Banach space, $r > 0$, $A: K_r(X) \rightarrow X$ a contraction (where $K_r(X)$ is the closed ball of radius $r$ and center $0$ in $X$), with contraction constant $0<q<1$, which ...
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1answer
177 views

An application of Banach fixed point theorem for initial value problem

Find a condition for $\beta>0$ which implies that the differential equation system \begin{align} x'(t)&=x(t)+y(t) ,\\ y'(t)&=t^{2}+tx(t) \end{align} with initial conditions $x(0)=0, ...
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53 views

Fixed-Points Limit Set

Let $f : \mathbb{R}^n \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ be compact and convex. For all $p \in \mathbb{R}$, consider $A_p \in \mathbb{R}^{n \times n}$, with $p \mapsto A_p$ ...
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1answer
180 views

An application of Banach fixed theorem on an integral equation

I'm learning some applications of the Banach Fixed Point Theorem and I have the following question: Consider the integral equation $\displaystyle x(t)=\int_{0}^{\frac{\pi}{2}}\arctan ...
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1answer
81 views

Is there a proof for the Central Limit Theorem via some fixed point theorem?

This question arose in my mind when I learned that the Gaussian is a fixed point for the Fourier transform. On the other hand, in e.g. the Banach fixed point theorem we have convergence to a fixed ...
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245 views

if $f_{n}(x)=f(f_{n-1}(x))$then $f_{10}(x)=x,x\in [0,1]$

Define the function $f:[0,1]\to[0,1]$ by the following. $$f(x)=\begin{cases} x+\dfrac{1}{2},&0\le x\le\dfrac{1}{2}\\ 2(1-x),&\dfrac{1}{2}<x\le 1. \end{cases}$$ Let $f_1(x)=f(x)$ and ...
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95 views

Show that $f$ has infinitely many fixed points

Let $f:ℂ→ℂ$ be an entire function. Assume that the equation $f(s)=a$ has infinitely many real solutions for all reals $a$. Show that $f$ has infinitely many fixed points, i.e., there exists infinitely ...
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164 views

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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43 views

A basic question about upper hemicontinuity

Given a correspondence $f:X\rightarrow 2^X$, suppose X is a closed simplex in $\mathbb{R}^n$, and $f$ is compact-valued. We say $f$ is upper hemicontinuous if, $\forall x\in X$ and every open subset ...
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2answers
328 views

Banach fixed point theorem and Picard-Lindelöf applied to this equation (explanation needed)

Consider the following equation which holds for all $w$ in some space, $$\langle v(t), w \rangle = \langle v(0), w \rangle - \int_0^t \langle F(s,v(s)), w \rangle$$ where $\langle F(s,v),w \rangle$ is ...
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1answer
127 views

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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77 views

fixed point iteration $x_{k+1}=f(x_k)$ with multiple fixed points

I have a fixed point iteration problem in my research, as below: $$x_{k+1}=f(x_k)$$ where $$f(x)=\frac{\lambda r}{g}\sum_{i,j}\pi_if_{ij}\left[1- \exp\left\{\displaystyle ...
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36 views

On a Variational Inequality

Let $H$ be a Hilbert space with real inner product. Consider $f: C \rightarrow C$, where $C \subset H$ is closed and convex. I am not sure about the variational inequality problem: find $x \in H$ ...
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1answer
74 views

Conditions yielding a unique fixed point of a continuous differentiable function.

Let $f$ be a function defined on $[0,1]$ which is continuous for each point in $[0,1]$ and differentiable for each point in $(0,1)$. Suppose that $f^\prime (x) \neq 1$ for every $x \in (0,1)$. ...
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Converse of a fixed-point theorem

I'm having some trouble furnishing a proof here. Let $(E, d)$ be a metric space such that any $k$-Lipschitz function has a fixed point for $0 < k < 1$. Does it follow, then, that $E$ is ...
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1answer
56 views

Constructive fixed-point theorems where finite iteration yields the fixed point

I would like to show that $p$, a fixed point of some effective map $f : S\rightarrow S$, can be constructed effectively. Ideally, I would like there to exist a finite $n$ such that $p = f^n(p_0)$, ...
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63 views

How to show this is a contraction

Let's say I want to find a fun way to write a number $l$. I can procede by doing so: (as I saw here Can we get just $3$ from $\pi$?) Let $f(x) = \sqrt{2lx - l^2}$. The only fixed point is $f(x) = x ...
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49 views

Is the composition of monotone operators monotone?

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let ...
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49 views

Fixed Point of a Function

I am trying to answer the following question: Prove the function $f(x)=1-x^2$ has a fixed point on $[0,1]$. Find the value of this fixed point explicitly. I know how to prove that it has a fixed ...
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Aitkens Extrapolation

The Aitken's extrapolation can be written as $$X^n = X_{n-2} + \dfrac{(X_{n-1}- X_{n-2})^2}{(X_{n-1}- X_{n-2})-(X_n- X_{n-1})}$$ Verify it? And $X^n$ can be viewed being defined recursively by ...
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1answer
176 views

Definition of upper hemicontinuity of a correspondence.

When using and examining Kakutani's fixed-point theorem, I've got a question about upper hemicontinuity. A correspondence $f:X\rightarrow2^Y$ is a point-to-set mapping. One way to define upper ...
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1answer
106 views

How to use Markov-Kakutani fixed point theorem to show that abelian groups are amenable?

Recall that a group $\Gamma$ is amenable if there exists a state $\mu$ on $l^{\infty}(\Gamma)$ which is invariant under the left translation action: for all $s\in \Gamma$ and $f\in ...
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36 views

Heuristics for sequence convergence

Having a finite sequence of double precision floating point numbers (obtained using the fixed point iteration of a function), is there any algorithm which can be used to determine that this sequence ...
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fixed point iteration algebra problem

I am looking at an example which finds the root of: $$ f(x) = \cos(3x) \tag 1$$ using the fixed point iteration method. It uses $$ g(x) = \frac{2x+\cos(3x)}{2} \tag 2$$ However, it was my ...
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Strictly increasing on R

Choose correct options , more than one may be correct Let f be the function defined by $$h(x)=e^x (x-1)+x^2$$ we've : $h$ is positive on $(0,\infty)$ $h$ is negative on $(0,1)$ $h$ is ...
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1answer
44 views

Proving Banach's fixed point theorem

The hint tells me how to proceed but I am stuck. I define the sequence ${z_n}$ as is stated in the hint, First off, I want to prove that $|z_{n+k} - z_n| < \epsilon$ I add $z_{n+k-1}$ and ...
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1answer
36 views

Calculate the Derivative of a univariable integral at a point $4$

Considering the function below: the objective is to calculate $F'(4)$ (the derivative of $F(x)$ in the point $4$)? we know that: and that: so if I try to replace $x$ by $4$ in $F'(x)$ I get ...
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1answer
74 views

Root of a function? (Proof with Banach theorem)

Given is a function $f$: $\left [ -1,1 \right ]\rightarrow \mathbb{R}$, which is continuous and differentiable. The function $g$: $\left [ -1,1 \right ]\rightarrow \left [ -2,2 \right ]$ is a ...
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163 views

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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1answer
84 views

Bug in Brouwer Fixed Point Theorem using Sperner?

so I am just trying to illustrate to an informal audiance how to prove the Brower Fixed Point Theorem using Sperner's lemma. I seem to have trouble with the iterative application of Sperner's lemma. ...
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131 views

generalization of Banach fixed-point theorem on short maps?

If $ \ T:X \longrightarrow X \ $ is contraction, then using Banach fixed-point theorem we know that the fixed point exists and all other points converge to that point. But what happens if $T$ is not ...
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1answer
163 views

Fixed-point iteration, Convergence of a sequence?

Given is the function $f(x)=x^{3}+x-1$ on $\mathbb{R}$. Use the Fixed-point iteration for $x\in \left [ 0.5 , 1 \right ]$ to show that the sequence $\left \{ x \right \}_{n}$ converges to the ...
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1answer
64 views

Limit of a monotone function

Let $f\colon [a,b] \to[a,b]$ be a non-decreasing function in a sense that $f(x)\leq f(y)$ whenever $x\leq y$. Although there may be several fixpoints of $f$, at least one does always exist and there ...
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2answers
80 views

Has this theorem (on existence of inverse) an analogous for unbounded operators?

Let $S,T:X\to X$ be bounded linear operators, where $X$ is a Banach space. It's a consequence of the Banach Fixed Point Theorem that if $T$ is invertible and $\|T-S\|\|T^{-1}\|<1$ than $S$ is ...
3
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1answer
150 views

Showing that $f$ has exactly one fixed point

Let $\gamma$ be the circle $\{z \in \mathbb{C}: \lvert z\rvert=1 \}$. Suppose $f$ is a function analytic on an open set containing $\gamma$ and its interior and that $\lvert\, f(z)\rvert<1$ for ...