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 point of a shrinking map. Proof.

Could you please verify my proof? Let $f:X \to X$ be a shrinking map on a compact metric space $(X,d)$. In other words: $d(f(x),f(y)) < d(x,y)$ for all $x,y \in X$. Prove: $f$ has a unique ...
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310 views

Fixed point: linear operators

I ask my question in two parts: though the topic is similar, I would like to distinguish linear and general cases since methods may be too different while my questions are broad. Consider a space $X$ ...
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196 views

Closest fixed point to a convex set

Consider the compact convex sets $Y \subset X \subset \mathbb{R}^n$, and a Lipschitz continuous function $f : X \rightarrow X$. Assume that $f$ has multiple fixed points. (From Brouwer's theorem, $f$ ...
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78 views

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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34 views

Limit of a Discrete Dynamical System, Part 2

In my previous post (i.e., Limit of a Discrete Dynamical System) the following system was considered: ...
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79 views

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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103 views

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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128 views

algebraic or homotopical proof for Kakutani fixed point theorem

As Kakutani fixed point theorem is a genral case of Brouwer fixed point theorem, and one can read the proof from homotopy theory books. I wonder if there is any proof for the Kakutani using homotopy ...
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303 views

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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180 views

Vector valued contraction

I am familiar with the Banach Fixed Point Theorem and I have used it to prove existence and uniqueness of functional equations in Banach spaces like $C(X)$, the space of bounded continuous function ...
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65 views

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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107 views

Connection between codata and greatest fixed points

It's easy to see how inductively-defined data types correspond to least fixed points. Let's take the natural numbers as an example, with constructors are $0 : \mathbb N$ and $s : \mathbb N \to \mathbb ...
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132 views

Fixed point: general case

This is the second part of the question Fixed point: linear operators. Here I would like to ask you about the general case. A lot of concepts can be described or even defined as a solution of a ...
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43 views

Reference request for the proof of the Brodskii–Milman fixed point theroem for isometries

Can any one help me to access the paper M.S Brodskii and D.P Milman, On the center of a convex set, Dokl. Akad. Nauk SSSR 59 (1948) 837–840 in Russian? or to prove the theorem If $K$ is a ...
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35 views

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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35 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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44 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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29 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 ...
2
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47 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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77 views

What is the method for solving this class of problem?

$\frac{dx}{dt} = y$ $\frac{dy}{dt} = -\partial y - \mu x - x^2$ Find the fixed points and discuss stability. I'd at least like to know what I should be googling, any help is appreciated.
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74 views

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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14 views

Proof of equality of least fixed point of two continuous function

The question is simple: given a set U, a continuous function (Scott continuity) $f \colon \mathcal{P}(U) \to \mathcal{P}(U)$ and function $g(X) := f(f(X))$, prove that $g$ is continuous and its least ...
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29 views

Diffeomorphism and hyperbolic points

Suppose $f$ is a diffeomorphism.Prove that all hyperbolic periodic points are isolated. I tried using the mean value theorem using two diferent periodic points (assuming the periodic points arent ...
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63 views

Banach Fixed Point Theorem. Measurable version.

The Banach fixed point theorem has the following statement THEOREM ( Banach contraction principle). Let $(Y,d)$ be a complete metric space and $F:Y\to Y$ be contractive . Then $F$ has a uniqe ...
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33 views

Use a fixed point argument to show there exists a unique solution to the following BVP

Show using a fixed point argument that there exists a unique solution $f\in C[0,1]$ to $$ -f''(x)+\sin(f(x))=\sin(x) , x\in (0,1), y(0)=y'(1)=0 $$ This is what I have so far: We can show ...
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42 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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16 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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21 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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26 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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53 views

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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31 views

How to find fixed point for Hamilton equations

I am learning the dynamical system. I start with Taylor-Greene-Chirikov map as follow \begin{eqnarray*} && I_{n+1} = I_n + K\sin(\theta_n), \\ && \theta_{n+1} = \theta_n + I_{n+1} ...
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27 views

How to linearize and solve ODE $\dot{z}_n = \sum_{m} i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}$ for $z_n\approx 0$?

I came across a physical system which obeys the following ODE $$\frac{d z_n}{dt} = \sum_{m=1}^N i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}, \qquad n\in\{1,2,\dots,N\}$$ where $z_n \equiv z_n(t)$ are ...
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32 views

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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77 views

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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56 views

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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77 views

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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68 views

Fixed Point Iteration Scheme

I have been asked to "Find a fixed point iteration scheme for minimising $f(x) = e^{cos (x)}$". Does anybody know what a fixed point iteration scheme actually is? I know it's not Fixed Point ...
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125 views

Properties of the derivative $\frac{\delta F(\phi(x))}{\delta\phi(y)}$ , where $ \phi(x) = F(\phi(x)) $ is a fixed point problem.

Dear experts I have a fixed point problem of the type: $ \phi(x) = F(\phi(x)) $, where $x \in \mathbb R^3 $. $\phi(x)$ is differentiable non-negative function on a given domain. I am trying to find ...
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90 views

Conjecture on continuous selection of fixed points of a correspondence

I have the following conjecture to show some sort of a continuous selection of fixed points in a correspondence: Let $S$ and $\Theta $ be non-empty, compact and convex subsets of some Euclidean space ...
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134 views

Fixpoint of monotone operators

Let $X$ be some set and let $F$ be the set of all functions with a domain $X$ and a range $[0,1]$. We consider $F$ to be a partially ordered set with $f\leq g$ if and only if $f(x)\leq g(x)$ for all ...
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65 views

Fixed Point Theorem for Set-to-Set Mappings

Is there a fixed-point theorem regarding mappings of the form $T:2^S\to 2^S$, i.e. mappings that map subsets of $S$ to other subsets in $S$: $$ T:S\supseteq A \mapsto T(A)\subseteq S $$ Where $S$ ...
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112 views

length of orbits and fixed point

Given a dynamical system $ \frac{dx}{dt}= F(x(t))$ Then is there a relationship between the Cardinal of the fixed point of the classical system $ |\operatorname{Fix}(f^{m})| $ with $ f^{m}(x)= ...
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29 views

Finding the proper g(x) for fixed point iteration on $2\sin{\pi x} + x = 0$

After spending over an hour trying to get this problem I realize my trig is weak. I found: Fixed point iteration .Numerical method. The selected solution is informative, but lacking detail to really ...
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29 views

How do I verify the stability of this fixed point?

The difference equation I am given is: $ x_{n+1} = rx_n(1-x_n)$ or the map $f(x) = rx(1-x)$ I am asked to verify that "the nontrivial 2-cycle is stable for $3 < r < 1 + \sqrt(6)$ and unstable ...
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25 views

Fixed points and periodic orbits of $F(x)=x^2-1.1$

A question asked me to find the fixed points of $F(x)=x^2-1.1$, then use the fact that these points were also solutions of $F^2(x)=x$ to find the cycle of the prime period 2 for F. How do I go about ...
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18 views

About the compactness condition in Schauder fixed point theorem

The theorem is Let $X$ be a locally convex topological vector space, and let $K ⊂ X$ be a non-empty, compact, and convex set. Then given any continuous mapping $f: K → K$ there exists $x ∈ K$ ...
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17 views

Fixed element $u \in U$ and subsets difference only at the $u$ point.

I have a task with a tip to it. The task: Let U - is not empty ultimate multitude. Prove, that number of subsets of the multitude U, with even power, same as how many subsets with odd power. ...
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20 views

Firm non-expansiveness in the context of proximal operators

$\newcommand{\prox}{\operatorname{prox}}$ Probably the most remarkable property of the proximal operator is the fixed point property: The point $x^*$ minimizes $f$ if and only if $x^* = \prox_f(x^*) ...
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24 views

Fixed point of projected operator

Let $X \subset \mathbb{R}^n$ be a compact convex set and let $f: X \rightarrow X$ be Lipschitz continuous and such that $$ \left( f(x) - f(y) \right)^\top \left( x-y\right) \leq 0 $$ for all $x, y ...
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19 views

Baillon theorem in fixed point theory

Baillon proved that if $X$ is a Banach space with $\rho'_X(0)<\frac{1}{2}$, then $X$ has the fixed point property (by $\rho_X(t)$ we denote the modulus of smoothness). My questions are as ...