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 Theorems

Theorem 1. Let $B=\{x\in \mathbb R^n :∥x∥≤1\}$ be the closed unit ball in $\mathbb R^n$ . Any continuous function $f:B\rightarrow B$ has a fixed point. Theorem 2. Let $X$ be a finite dimensional ...
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The limit of a fixed-point equation

Consider the following fixed-point equation: \begin{equation} x = (1-x)^{1-\frac{2}{a+1}} - 1, \end{equation} where $x \in [0,1]$ and $a \in [0,1]$. If we write the solution of $x$ in terms of $a$ as ...
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68 views

Fixed point and non-fixed point function

For constructing another proof I need two functions explicitly and therefore I was wondering whether there exists a function that has nowhere a fixed point and a function that (maybe depending on the ...
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315 views

Newton iteration- estimate the error

I was wondering whether there are equations available to estimate the a priori and a posteriori error for newton's method? My idea was to use that it is a fixed point iteration and therefore one can ...
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Numerical Analysis, build a contractive function

I have a question regarding Numerical Analysis. I've never been asked these sorts of questions before and don't even know where to begin. The goal of this exercise is to find a value alpha such that: ...
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A fixed point theorem [duplicate]

Let $\emptyset \not = X\subseteq \Bbb{R}^n$ be convex and compact and let $\cal{A}$ be a family of affine maps from $\Bbb{R}^n$ into $\Bbb{R}^n$ such that $X$ is invariant under each element of $\cal ...
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correctness of functional iteration and contraction proof

I need to prove: If $F$ is contractive from $[a, b]$ to $[a, b]$ and $x_{n+1} = F(x_n)$ with $x_0\in[a,b]$ then $|x_n -s| \leq C\lambda^n$ for an appropriate $C$ where $s$ is a fixed point of $F(x)$. ...
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Quesion on a detail of the proof of Schauder-Tychonoff fixed point theorem

I'm trying to understand the proof of Schauder-Tychonoff fixed point theorem on page $96-97$, in Fixed Point Theory and Applications, Ravi P. Agarwal,Maria Meehan,Donal O'Regan, which can be found ...
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55 views

Brouwer fixed poin in 1-D

In $\mathbb R_{+}$ (non-negative) I realize that any nonnegative valued continuous function $f\colon X \rightarrow Y$ where $X, Y \subseteq \mathbb R_{+}$ with a negative finite gradient has a fixed ...
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352 views

Brouwer's fixed point theorem (for unit balls) and retractions

Let $X$ be a generic Banach space over $\mathbb R$ (also infinite dimensional), and let $B$ be the unit ball in $X$. I want to prove that the following proposition $B$ is a fixed-point space if ...
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How to prove that the following iteration process converges?

I have the following iteration process: $$ p_{n+1} = \frac{{p_{n}}^3 + 3 a p_{n} }{3 {p_{n}}^2 + a } , $$ where $a > 0$. Q1: How to prove that this iteration process converges for every number ...
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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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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 ...
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periodic point of homeomorphism of plane

Let $h$ be a homeomorphism of $\mathbb{R}^2$ onto itself such that $h(K)=K$ for some compact subset $K$ of $\mathbb{R}^2$. Show that $h$ has a periodic point in $\mathbb{R}^2$.
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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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Show that sequence approaches fixed point of a function

Problem Let $f(x)$ be a differentiable function on $\Bbb R$ with $\left|\,f ' (x)\right| \leq r < 1$, where $r$ is constant. Then consider the sequence $\{x_n\}$ such that $x_1 = 0$, $x_{n+1} = ...
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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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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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37 views

Is there any space with normal structure but not uniform normal structure?

It is clear that uniformly normal structure implies normal structure. But, is there any space that has normal structure but it doesnt have uniformly normal structure?? Would you mind to give me some ...
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82 views

A smooth or analytic function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer

In A continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer, I asked about continuous function. This time, I would like to ask the ...
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A continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer

Can we find a continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ for every integer $k$ and every prime number $n$? And other points other than these are not fixed ...
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Chaos/Fixed points. I was reading a book by Strogatz and I encountered this.

Now, I always thought a fixed point implied $f(x)=x$, so somebody tell me, what is he talking about here? Thank you.
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How to show a function taking the form as $F(x,y) = 0$ is a contraction mapping?

Let $\Phi (x)$ be the cumulative distribution function of the standard normal distribution. Given $x_0$, $x_1 = \Phi(x_0-x_1)$.If $x_n$ is given, $x_{n+1} = \Phi(x_{n}-x_{n+1})$(By drawing a graph, ...
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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}}, ...
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Question about the infimum of $|f(x)-x|$ , where $f(x)=x$, $x$ is a fixed point of the nonlinear equation.

I am trying to check if the following property holds for fixed points: Suppose: $ f(x)= x $ is given, with solution $x = \theta \gt 0 $ I would like to show : $ \forall \epsilon \in (0,1), ...
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199 views

Showing a contraction without a fixed point

Suppose $f: [1, \infty) \to [1, \infty]$ defined by $f(x) = x + \frac{1}{x}$ for all $x \geq 1$. I want to prove that: \begin{equation} |f(x)-f(y)| < |x-y| \end{equation} except when $x=y$, but ...
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251 views

How many fixed points can a differentiable function have?

Let $f$ be a differentiable function on $\mathbb{R}$. Then could any one tell me which of the following statements are necessarily true? If $f'(x)\le r<1$ for all $x$ then $f$ has at least one ...
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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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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 $X$ ...
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Unique solution to $x^4 + 7x -1 = 0 $ on $[0,1]$ (Banach's fixed point theorem)

I want to show that $x^4 + 7x -1 = 0 $ has a unique solution on $[0,1]$. The idea is to use Banach's fixed point theorem. However, I see a problem with this as the statement of the theorem says that ...
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Functions $f:X\to X$ with no fixed points, for $X$ a punctured disk or a sphere.

$X$ is the punctured closed unit disc $D^2-\{0\} = \{(x, y) \in \mathbb{R}^2: 0 \lt x^2+y^2 \le 1\}$ Is the answer that $f$ maps all $(x,y)$ to $0$ which is not included in the unit disk and so ...
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Stokes' and Green's Theorem Integral Setup

a) For the vector : $$v= 4y\hat{x}+x\hat{y}+2x\hat{z}$$ evaluate, $$\int(\nabla \times v) \cdot da$$ over the hemisphere represented by the upper half plane of $$x^2 + y^2 + z^2 = a^2$$ (this is ...
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Application of Banach fixed-point theorem

I am looking for a function $f:[0,1]\rightarrow\mathbb R$ which satifies $\int_{0}^{1}\frac{\sin(f(t)-y)}{2}\,\mathrm dy=f(t)$ for $t\in[0,1]$. The first thing I do is to define a function ...
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Is antipodal symmetry really necessary for Tucker's Lemma?

Tucker's Lemma is here. Let's stay within the 2D case for now. A standard proof is constructive: (1) Pick an arced edge on the boundary of the circle. Note its labeling (for example, (1, 2)). (2) ...
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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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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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How to show something is a contraction?

If we let $X$ be a complete metric space, and let $S:X\to X$ be a map, such that $S^m$ is a contraction. We now want to show, that $S$ has a unique fixed point This is what I've thought so far: Due ...
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Limit of a sequence of fixed points also a fixed point?

Suppose I have a continuous function $f : [0,1]^n \rightarrow [0,1]^n$ (maybe $n$ is infinite). Suppose I have a sequence $\{a_n\}_{n=1}^\infty$ of points in $[0,1]^n$ where each $a_n$ is a fixed ...
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46 views

Name a stable output of a function taking 2 arguments

$\mathbb{C}$ is a fixed finite set, a fair chaotic sequence $(c_n \in \mathbb{C})$ is defined such that $\forall c \in \mathbb{C}, \exists n_0 \in \mathbb{N}, n > n_0 \wedge c_n = c$. That means ...
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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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Fixed Point Iteration and contraction principle.

So I have this function $g(x$) which is continuously differentiable on some domain in $R$ such that there exists a value $c$ with $|g'(c)| \lt 1$ and $g(c)=c$ i.e. the fixed point. I have already ...
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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 ...
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Iteration of $x/\log x$

Consider $f(x)= \frac{x}{\log x}$ iterated n times for given x in $Z^+.$ $f^2 = f \circ f.$ Let $x_1 = x^2.$ What I would like to show (or disprove) : $\exists ~\alpha = x_n - e > 0 $ such ...
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298 views

Lipschitz condition on nonlinear ODE

Suppose we have the ODE $$x''=-a\sin{x}.$$ Then let $$x'=y$$ and $$y'=-a\sin{x}.$$ So $$\mathbf X = \begin{pmatrix} y \\ -a\sin{x} \end{pmatrix}.$$ Im confused about how to show a Lipschitz ...
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Finding a functor satisfying a recursive equation

Say I want to find a functor satisfying something like: \[FA = 1 \sqcup (A \times FA).\] Equivalently, I want to find for each $A$ a fixed point of: \[GB = 1 \sqcup (A \times B)\] (I'll worry about ...
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Brouwer's fixed point theorem for infinite dimensional real space in subsystems of second order arithmetic

$\text{WKL}_0$ proves Brouwer's fixed point theorem for continuous functions on $\lbrack 0,1 \rbrack^n$, when $n$ is finite. What subsystem of second order arithmetic is needed to prove Brouwer's ...
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65 views

Show the following using fixed point theorem

Show that if A is s pxp real matrix such that $||I_p-A|| <1$, then for any choice of b$\in\mathbb{R}^p$ and $u_0\in\mathbb{R}^p$ the vector $u_{n+1}=b+(I_p-A)u_n$ converge to a solution x of the ...
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127 views

Show that a function $f(x)$ maps to a set of points.Fixed point theorem

Show that the function $f(x)=\frac{1+x^2}{2}$ maps the set of points $0\leqslant x\leqslant 1$ into itself and has a fixed point in that interval even though there does not exists a positive ...
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276 views

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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186 views

The equation $2 \cosh(3.1786803659501505 z) = z$?

Let $a$ be a positive real number and $z$ a complex number. I was wondering about the equation $2 \cosh(a z) = z$ where we solve for $z$. Clearly if $z$ is a solution than so is its conjugate. It ...