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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Fixed point, bounded derivative

Let $p\in\mathbb{N}$. Let $f:I\to\mathbb{R}$ differentiable in the closed interval $I$ (bounded or not), with $f(I) \subset I$, and let $g = f\circ f\circ \cdots \circ f = f^p$, where $\circ$ means ...
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145 views

Prove that intervals of the form $(a,b]$, $[a,b)$, $(-\infty,a]$, $[a,\infty)$ do not have the fixed point property.

Prove that intervals of the form $(a,b]$, $[a,b)$, $(-\infty,a]$, $[a,\infty)$ do not have the fixed point property. In the case of open intervals, I can derive that they do not have the fixed point ...
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Show that the only intervals having the fixed point property are the closed intervals.

By Fixed Point Theorem, I know that it deals with closed interval, for eg, [0,1]. And this theorem will be false if [0,1] is replaced by (0,1). The counter example will be $f:(0,1)\rightarrow (0,1)$ ...
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42 views

Can there be a metric space where no contraction has a fixed point?

We know that: If $X$ is a metric space, then every contraction has at most one fixed point. (Note: if metric space is complete, then we have existence and uniqueness) I wonder if there can be a ...
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103 views

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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52 views

How many fixed points does this function have?

The function is $f :\overline{\Bbb R}\to \overline{\Bbb R}, x \mapsto x^5$. So does it have $3$ or $5$ fixed points ? Thanks in advance !
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Why is this easy “proof” of Brouwer's Fixed Point Theorem not correct/common?

Brouwer's Fixed Point Theorem states, essentially, that any continuous function on a closed disc to itself has a fixed point. I am familiar with the proof based on the impossibility of a retraction ...
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26 views

Prove that $(Y,\mathcal T_1)$ also has the fixed point property

Let $(X\mathcal T)$ have the fixed point property and let $(Y,\mathcal T_1)$ be a space homeomorphic to $(X, \mathcal T)$. Prove that $(Y,\mathcal T_1)$ also has the fixed point property. I know ...
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78 views

Prove that $f(x) = \cos(x)$ has a unique fixed point.

The following question comes from Chapter 6.4, Exercise 4 on page 156 in the set of notes Topology Without Tears. Using Exercise 2 and 3, show that while $f: \mathbb R \to \mathbb R$ given by $f(x)...
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32 views

Where should I begin the study of fixed point theory, especially of multi-valued maps?

How should one begin one's study of fixed point theory, especially of multi-valued maps? What background --- in topology, analysis, functional analysis, algebra, and set theory --- should one have? ...
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16 views

Fixed points and banach spaces [closed]

Let $B$ be a closed ball centered at $0$ in a Banach space $E$ and $F:B\to E$ be a contractive map such that $F(x)=-F(-x)$ for every $x\in\partial B$. Show that $F$ has a fixed point. Any ideas?
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29 views

Function with unique fixed point of all orders

Let $f:[a,b]\to [a,b]$ be a continuous and monotone function. For each $n\ge 1$, there exists $x_n\in [a,b]$ such that $$f^n(x_n) = x_n,$$ where $f^n(x) = f(f(\cdots(f(x))\cdots))$ for $n$ times. The ...
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15 views

Using Fixed point iterations for solving system of linear equations

Given a system of $n$ linear equations $$ x_i=\sum_{k=1}^{n}a_{ik}x_k+b_i \quad i=1,2,...,n$$ I'd like to employ the fixed point iteration method to find $x_i$. The fixed point iteration define $$ x_i^...
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1answer
42 views

A Bending Buzz Wire Game

There is a wire connecting an exit and an entry point. At the entry, the wire has height $0$, at the exit, it has height $1$. Since the wire is connected, the wire has height $1/2$ somewhere, whatever ...
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42 views

Complete a proof that $F(x,y)$ is contracting.

Can anyone fill in the dots in this proof? Let $D := [0,\frac{1}{2}]^2$. Show there is exactly one $(x,y)=(x^*,y^*)\in D$ such that \begin{align*} x &= \frac{x^3}{2} + y^4 + \frac{1}{4} \,, \...
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2answers
57 views

Prove that a function is contractive

I'm stuck with the following. I need to prove that in $D:=[0,1]\times[0,1]$ the function $F$ is contractive, where $F:\mathbb{R}^2\rightarrow\mathbb{R}^2$ is defined as: \begin{align} F(x,y):=(\frac{...
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15 views

PDEs - prove continuity of operator

Consider the following nonlinear problem $$ \begin{cases} -div(a(u)\nabla u ) =0 & \text{in $\Omega$} \\ u=0, & \text{on $\partial \Omega$ } \end{cases} $$ We can assume $\Omega$ to be a ...
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1answer
35 views

Show there exists $C\in\Bbb{R}^n$ such that $|C-A_i|=|B-A_i|+u_i$, with $A_i,B\in \Bbb{R}^n$ and $u_i$ close enough to $0$

Let $A_1,...,A_n,B$ be vectors in the $n$-dimensional Euclidean Space, such that they are never on the same affine $(n-1)$-dimensional subspace. (What? Is that a way to say they span $\Bbb{R}^n$?). ...
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1answer
101 views

Example of contraction mapping theorem failing for strict metric map

Is there an example of $f: [0,1] \to [0,1]$ s.t. $|f(x)-f(y)|<|x-y|$ but a sequence $x_0,f(x_0),f^2(x_0)...$ doesn't converge to its fixed point? where $f^n$ denotes repeated application. Also, ...
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60 views

Proving a function isn't homotopic to a map to the boundary

Let $X$ be a compact Hausdorff space and $S$ a finite dimensional, convex, Hausdorff space. Moreover, let $f:X \to S$ be a closed, surjective map with $\tilde{H}^q(f^{-1}(s))=0$ for all $q\ge 0$ and $...
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1answer
49 views

Can there be a limit cycle without a fixed point in 3D space?

I am working with a population dynamics model. Basically, I have a nonlinear ODE in $R^3$ space, (X,Y,Z), and I know that if I start in the an open region ($0<X<1,0<Y<1,0<Z<1$, ...
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23 views

Fixed Point in the Space of Rational Functions

Let $\mathcal R$ be the space of rational functions and $F: \mathcal R \to \mathcal R $ be a function that transforms a rational function into another rational function. Is there a fixed point ...
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73 views

Open ball does not have fixed point

How we can prove that the open ball in $R^n$ does not have fixed point property (by algebraic topology concepts)? I know $D^n$ -closed ball in $R^n$- has fixed point property by Brouwer's theorem, but ...
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30 views

Establish the sufficient condition $|g'(x)| < 1$ for convergence of an iteration using the Banach fixed point theorem?

If $x_n = g(x_{n-1})$ is an iteration, it converges if $g$ is continuously differentiable and $|g'(x)| < 1$. The Banach FPT says that if $T$ is a contraction on a complete metric space $X$ then it ...
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50 views

Fixed Point and Contraction Mapping

Consider $Tf(x) = \int_0^x e^{-f(s)^2} \; ds$ for $x \in [0,\infty)$. I want to use the contraction mapping theorem to show that $T:C^1([0,\infty)) \to C^1([0,\infty))$ has a unique fixed point. From ...
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1answer
45 views

Banach fixed-point theorem for a recursive functional equation

I was asked to prove that the functional equation $$ x(t) = \begin{cases} \frac{1}{2} x(3t) + \frac{1}{2},& 0 \leq t \leq \frac{1}{3}\\\ f(t), & \frac{1}{3} < t\leq \frac{2}{3}\\\ \frac{1}{...
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1answer
28 views

Is this an isolated equilibrium point?

I've just been learning the definition of an isolated equilibrium point. From my understanding of this definition, I would expect (as an example) the point $x=1$ to be an isolated fixed point for the ...
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1answer
32 views

How can I see that we have $\Lambda_a=s\Lambda_a\mathbin{\dot\cup} \big(s\Lambda_a+(1+s)\big)\mathbin{\dot\cup}s\Lambda_b$ (Silver mean substitution)

Consider the (Silver mean) substitution $$\varrho:\begin{aligned}&a\mapsto aba\\&b\mapsto a\end{aligned}.$$ If we take $w^{(1)}=a|a$ and $w^{i+1}=\varrho(w^{(i)})$, then we get: $$a|a\...
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Topology: every continuous function on $\mathbb{R}^2$ scales a point

The question is simple: Suppose $f : \mathbb{R}^ 2 \to \mathbb{R}^ 2$ is continuous. Show that there exist $\lambda > 0$ and $x \in \mathbb{R}^2$ such that $f(x) = \lambda x$. So basically, we ...
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1answer
77 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
51 views

What's the meaning this DOT notation?

I'm reading a chapter in a Model Checking book. I came across this chapter "Symbolic Model Checking", in which the author mentions Fixed Point representation. I don't know how to explain the context, ...
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1answer
36 views

What can we say about the convergence of these fixed-point iterations for $\phi:\mathbb{R}\to \mathbb{R}$

Let $\phi: \mathbb{R}\to \mathbb{R} \in C^2(\mathbb{R})$ and let $x^{*}$ be a fixed-point of this function. Further assume that $|\phi'(x^{*})| \neq 1$. We define two sequences $\begin{align} &...
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3answers
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Holomorphic map from closed convex domain in hilbert ball into itself has fixed point

Let $B=\{x \in \ell_2, ||x|| \leqslant 1\}$ - Hilbert ball $X \subset B$ - open convex connected set in Hilbert ball, $\bar{X}$ - closure of $X$. $F: \bar{X} \to \bar{X}$ - continuous map that ...
4
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1answer
65 views

Continuous map in $\mathbb{R}^2$ has a (scaled) fixed point

Let $\phi:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be a continuous map. How do I prove that there exist $a>0$ and $x\in\mathbb{R}^2$ such that $\phi(x)=ax$? What I know: I thought maybe this can ...
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33 views

Transform system to polar and sketch phase portrait. Show that $(0,0)$ is an unstable focus.

Transform the system $$x' = y - x(x^2+y^2-1)$$ $$y' = -x - y(x^2+y^2-1)$$ to polar coordinates, and sketch the phase portrait. Show that it has a unique limit cycle and that all ...
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1answer
30 views

Problem with proof of the upper hemicontinuity of correspondence

I have a problem with a proof I found here of the upper hemicontinuity of the best-reply correspondence in the Nash Theorem. Below there is the proof, and here my problems: Problems: Is here ...
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1answer
75 views

Find the fixed points of the system, and sketch the trajectories of the system

I am given the following system: $$x' = [(x-1)^2 + y^2]y$$ $$y' = -[(x-1)^2 + y^2]x \tag{*}$$ where $x = x(t), y = y(t)$. I am supposed to Find the fixed points of the system, and ...
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Show Existence of Fixed Point in $\mathbb{R}^n$ with Euclidean Metric

Consider the closed unit ball in $\mathbb{R}^n$, $B = \{ x \in \mathbb{R}^n : \|x\| \leq 1 \}$ with the Euclidean metric. Then let $g : B \rightarrow B$ be a function such that $$\|g(x) - g(y)\| \leq \...
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If $X$ has the FPP then does $X\times I$ have the FPP.

If $X$ is compact subset of $\mathbb{R}^2$ and all continuous maps $f:X\rightarrow X$ have a fixed point, do all continuous maps $f:X\times I\rightarrow X\times I$ have a fixed point?
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45 views

Uniqueness of solution to integral equation with “endogenous” kernel

Let $x,y\in C$ and consider the functional equation $T:y\mapsto x$ implicitly defined by the following integral equation: $$ x(t) = \int_{-\infty}^\infty K(x(t),t,s) y(s) \, ds, $$ with $K$ given. ...
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22 views

Does this functional equation have a (unique) solution?

Does the following equation have a (unique) solution for $b_1$? \begin{equation} b_1 = \dfrac{1}{2b_1} \left\lbrace -Var(p) + Cov \left(\left[ (p + b_1 + b_0 + \alpha)^2 -4b_1(p + b_0) \right]^{0.5},...
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78 views

closed unit ball without a fixed point.

What I've done so far: Let $B=\{x\in\mathbb{R}^n : \|x\| \leq 1\}$ be the closed unit ball in $\mathbb{R}^n$ equipped with the standard Euclidean metric. Let $f \colon B \to B$ be a function such ...
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1answer
14 views

Find 3 fixed points of function with 2 arguments

I'm looking for a way to determine the fixed points of a function with 2 arguments. The specific function is the following: $F:\bigl( \begin{smallmatrix} x \\ y \end{smallmatrix} \bigr) \mapsto \...
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36 views

Proof that fixed points of a null field are zero

In this context, a null field means a field constructed of planar waves $e^{I k_{\mu} x^{\mu}}$ that satisfy the null condition $k_{\mu} k^{\mu} = 0 \implies c^2 k^2 = \omega^2 $ Suppose we have a ...
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2answers
26 views

Understanding fully the proof of the uniqueness of a fixed-point

I was reading the proof of the uniqueness of the fixed-point. The uniqueness is stated as follows: ... If, in addition (to the fact that we know that $\exists$ a fixed-point in a range $[a, ...
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1answer
33 views

How many iterations are required to reduce the convergence error by a factor of 10?

I've the following function $$g(x) = x^2 + \frac{3}{16}$$ for which I found the two fixed points $x_1 = \frac{1}{4}$ and $x_2 = \frac{3}{4}$. I noticed that the fixed-point iteration $$x_{k+1} = g(...
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2answers
31 views

Find if a fixed-point iteration converges for a certain root

I'm asked to find if the fixed-point iteration $$x_{k+1} = g(x_k)$$ converges for the fixed points of the function $$g(x) = x^2 + \frac{3}{16}$$ which I found to be $\frac{1}{4}$ and $\frac{3}{4}$. ...
3
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0answers
41 views

Extension of Kakutani's fixed point theorem.

Can the Kakutani's fixed point theorem's be extended to say that there exists a fixed point inside the set (not on boundary)(I am not sure how to formally state this). For a $n$-dimensional compact, ...
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1answer
29 views

Provinf Orbits are eventually fixed or eventually prime-2-periodic

Please I need help with this question: Let $S = \{a,b,c,d\}$ be a finite set and suppose that $f \colon S \to S$ has the property that: $$f(a) = f(b) = f(c) = d$$ Prove that each of the orbits of ...
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1answer
50 views

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 ...