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Questions tagged [fixed-point-theorems]

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) etc.

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Betti sum, cup-length, Lusternik-Schnirelmann category, and critical points

I am trying to make some order in the notions of cup-length, sum of Betti numbers, LS category, critical points of functions. Let $M$ be smooth closed compact manifold. We denote by Crit($M$) the ...
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Bannach Fixed Point Theorem for rotation and scaling the punctured unit disk

I'm trying to show that a weak contraction is not sufficient for the Bannach Fixed point theorem. The best example I could think of is the function defined on the punctured unit disk to itself given ...
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Show that $\sqrt{x+2}$ is a contraction

Let $f:X \rightarrow X$ where $X=[0,\infty)$ be defined as $f(x)=\sqrt{x+2}$. I have to show that this mapping is a contraction and find its unique fixed point. The second part is easy: by the CMT, it ...
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Fixed points of Banach.

Let $S,T : M \hookleftarrow$. By definition. Let $M$ a set. A fixed point of an application $A:M\hookleftarrow$ is an element $\xi \in M$ satisfying $A(\xi)=\xi$. If $TS=ST$ (commute), then what ...
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31 views

Fixed point itteration

Let $x_{n+1}=\sqrt{2+x_n}$ with $x_0=0$ and $y_{n+1}=\sqrt{2+y_n}$ with $y_0=2018$. Do those two sequences have the same limit? I think the answer is "Yes", but I am not really sure. Is my argument ...
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75 views

Convergence analysis for Solving quadratic matrix equation $\mathbf{XBX=A}+\lambda \mathbf{X}$

I seek for a symmetric positive definite (PD) solution $\mathbf{X}$ for the following quadratic matrix equation: \begin{array}{cc} \mathbf{XBX=A}+\lambda \mathbf{X} \tag{1}\label{eq1} \end{array} ...
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How does PageRank deal with nodes that do not have out-links?

I will use the notation that $A_{ij}=1$ if an arrow exists from $j$ to $i$ and otherwise zero. Just to avoid confusion I use in brackets the standard convention $B_{ij}=1$ when $i$ has a directed ...
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1answer
41 views

Proof of fixed point theorem on $D^1$ using the technique used in Brouwer's fixed point theorem.

Let $f : [-1,1] \longrightarrow [-1,1]$ be a continuous function. Then using IVT I have proved that it has a fixed point. Now my question is "Can I prove this result by the technique used in Brouwer's ...
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188 views

Existence of infinite iteration of functions $f_\infty$?

Given a sequence of functions $\{f_n\}$ satisifying an iterated relation such as $f_n(x)=g(x+f_{n-1}(x))$ $f_n(x)=g(xf_{n-1}(x))$ $f_n(x)=g(x/f_{n-1}(x))$ Where $g:=f_1$ is ...
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61 views

Application of Banach contraction principle

Define $T:\mathbb R^3→\mathbb R^3$, $(x,y,z)\mapsto\left(\dfrac12\cos y +1,\dfrac23\sin z,\dfrac34x\right)$. I have checked that this example is a contraction and now I am trying to apply Banach ...
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74 views

Proof of convergence rate for iterative methods

Given the sequence $\{x_n\}$ generated from the iterative function $\Phi(x)$, $\{x_n\}$ converges with order $p$ to the fixed point $\alpha$ if: $$ \exists \lim_{n \to \infty} \frac{|x_{n+1}-\alpha|}{...
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How does one find all the fixed points of the operator defined on the factorial function and how it affects the definition of it?

I was reading these notes and I recently asked: Why does the fixed point theorem justify the existence of the factorial function? that outlined the need of fixed points for justifying the definition ...
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Why does the fixed point theorem justify the existence of the factorial function?

I was learning about fixed point theorem in the context of programming language semantics. In the notes they have the following excerpt: Many recursive definitions in mathematics and computer ...
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Infinite composition of function that has a fixed point

I have faced an interesting question working with functions that have a fixed point(i.e such $f$ that $\exists$ $x: f(x)=x$ ) So I asked myself quite a gerenal question that I did't find easy ...
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Why does apply F to a fixed point show that its a fixed point?

I was learning fixed points in the context of programming languages and the text wants on page 89 wants to show that $fix(\mathcal F)$ is a fix point by applying $\mathcal F$ to the fix point itself. ...
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55 views

Fixed point of a function defined in terms of a metric on a compact metric space

Let $X$ be a compact metric space, and assume $f : X → X$ satisfies $$d(f(x), f(y))< d(x, y), \forall x \neq y ∈ X.$$ Define a function $g : X → \mathbb{R}$ by $g(x) = d(x,f(x)), \forall x ∈ X$. ...
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A lemma for the idea in the modern proof of Banach's Contraction Mapping Theorem

Let $(X,d)$ be a metric space and $T: X \rightarrow X$ a contraction mapping with a fixed point $w$. Suppose that $x_0 \in X$ and we define $x_n$ inductively by $x_{n+1}= Tx_n$. Show that $d(x_n,w) \...
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105 views

Continuous on the unit ball – odd on the unit sphere – does it have a fixed point?

For $n\in\mathbb N$, let \begin{align*} B^n\equiv&\;\{\mathbf x\in\mathbb R^n\,|\,\lVert \mathbf x\rVert\leq 1\}\text{ and}\\ S^{n-1}\equiv&\;\{\mathbf x\in\mathbb R^n\,|\,\lVert \mathbf x\...
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Verify that $x$ is a fixed point

Given the function $f(x) = {-x^4 \over 4} + x^3 -4x + 4$ I have graphically localized two roots $\alpha$ and $\beta$ (with $\alpha < \beta$). After analyzing them with Newton's algorithm I'm given ...
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Properties of Alternating Least Squares: fixed points, monotone convergence

Let $y_i$ a scalar and $W_i \in \mathbb{R}^{n \times n}$ for all $i = 1..N$, and $a,b \in \mathbb{R}^n$ . The following optimization problem is bilinear: $min_{a,b} \sum_{i=1}^N \| y_i -a^TW_ib \|_2^...
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70 views

Is there a fixed-point theorem that can be applied to a ring endomorphism of $k[x,y]$?

Let $k$ be a field of characteristic zero (I do not mind to assume that $k \in \{\mathbb{R},\mathbb{C}\}$), and let $R=k[x,y]$ be the $k$-algebra of polynomials in two variables $x,y$. Let $f$ be a $...
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1answer
161 views

Numerical Analysis - Proving that the fixed point iteration method converges.

I am having some trouble with a numerical analysis proof related to the fixed point iteration method. The problem is as follows: Suppose that $f$ in $C^2[a,b]$ and for some $x$ in $(a, b)$ we have $...
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107 views

Fixed points of an endomorphism of a ring

Let $R$ be a commutative $k$-algebra, where $k$ is a field of characteristic zero. Let $f$ be a $k$-algebra endomorphism of $R$. ($f$ is not assumed to be either injective nor surjective). Are ...
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1answer
59 views

Minimum of repeated iteration of $n^{-x}$

Consider the repeated iteration of the function $f(x) = n^{-x}$, (meaning the result from the first calculation is the argument for the next and so on) The first value of $x$ can be any positive ...
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1answer
131 views

Prove that $f$ has a unique fixed point.

This is a practice problem for an exam I am taking. Let $(X,\rho )$ be a complete metric space and $f: X \rightarrow X$ a function. Writing $f^{n}$ for the $n$-th iterate of $f$, denote $$c_{n} := \...
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initial valued problem

Let $F : [0,a] \times [b-r,b+r] \rightarrow \mathbb{R}$, $a, b, r$ fixed real numbers, satisfying (1) for each $k \in [b-r,b+r]$, $g_k : [0,a] \rightarrow \mathbb{R}$ defined by $$g_k(t) = F(t,k)$$ ...
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On convergence of a fixed point iteration

I'm stating the problem that I'm stuck with, along with the progress that I've made. Problem : The iteration defined by $x_{k+1}=\frac{1}{2}\left(x_k^2+c\right)$, where $0<c<1$, has two fixed ...
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Convergence of a fixed point iteration with nonlinearity

If the following is too specific, my question can be simplified as fixed point iteration $${\mathbf X}_{k+1}={\mathbf G}f({\mathbf X}_k)+h({\mathbf X}_k)$$ and given that the spectrum radius of $\bf ...
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Problem in proving fixed point theorem

Let $ f(x)=x^r, 1<r<\infty, x\in \mathbb{R^+}=[0,\infty) ~and ~n\in \mathbb{N}.$ Define $$\pi(x)= 1 ~~if~~ x\leq n ~and =0 ~if~ x< (n+1).$$ Then for any $x,y\in\mathbb{R},~~ $ I want to ...
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For a continuous function $f$ satisfying $f(f(x))=x$ has exactly one fixed point

Let $f \colon [ 0, 1] \to [0, 1]$ be a continuous map such that $$ f\big( f(x) \big) = x \ \mbox{ for each } x \in [0, 1], $$ and $$ f(x) \neq x \ \mbox{ for at least one } x \in [0, 1], $$ ...
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38 views

Proving the stability of fixed point

How do I prove the following statement : $$ \forall k \geqslant 1, \ |f^k(x)-p| \leqslant a^k|x-p|$$ inductively? $p$ is a stable fixed point of $f$ and $f'(p) \leqslant a < 1$. $f$ is a ...
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Why exists a periodic point?

In Brin's book "Introduction to dynamical systems", page 10, he defines a cuadratic function $q_\mu$ for $\mu>4$. He observes that $[\frac{1}{\mu}, \frac{1}{2}] \subset q_{\mu}^2([\frac{1}{\mu}, \...
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Motivation for Banach Space in Global Picard-Lindelöf

I'm looking for the motivation of the core idea to a proof of the global existence version of Picard-Lindelöf. Global Picard-Lindelöf. Let $f:[a,b] \times \mathbb{R}^n \to \mathbb{R}^n$ be ...
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1answer
76 views

Showing that a function is a contraction

I have a function $H$ whose domain is the set $C$ of the continuous functions $[0,1] \to \mathbb{R}$. $H$ is defined by $$H(f)(t) := \int_0^t \phi(f(s))ds $$ for $t \in [0,1], f \in C$ and with $\phi: ...
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1answer
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Why a map which is close to the identity on the boundary of a disk has a zero?

Let $\mathbb{D}^2$ be the closed unit disk in $2D$. Let $f:\mathbb{D}^2 \to \mathbb{R}^2$ be a smooth map. Suppose that $$\|f|_{\partial \mathbb{D}^2}-\operatorname{Id}_{\partial \mathbb{D}^2}\|_{\...
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1answer
31 views

How does showing $\varphi \circ \iota_{\mathbb S^1}$ homotopic to $\mathrm{Id}_{\mathbb S^1}$ give us a contradiction to $f$ has no fixed points?

I am trying to prove the $2$-dimensional version Brouwer's fixed point theorem. If $f: \overline{\mathbb B^2} \to \overline{\mathbb B^2}$ is continuous and has no fixed point, and we define $\varphi: ...
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Must continuous surjection $f:\Bbb Z[\frac16]\to\Bbb Z[\frac16]$, fixed in $\lvert\cdot\rvert_3$, stabilise to approach $0$?

Must a uniformly continuous (under the 2adic metric) surjection $f:\Bbb Z[\frac16]\to\Bbb Z[\frac16]$ which is fixed in $\lvert\cdot\rvert_3$, stabilise in order to approach $0$? Let us say that some ...
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160 views

Variable in upper bound of sum

I need to find a solution $x$ of the following equation: $$\sum_{n=0}^{\left[\frac{0.9}{x}\right]} (1-nx) = 45$$ where $[.]$ denotes the nearest integer function. I am an engineer and I'm currently ...
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Rearranging to get an iterative function (fixed point)

I don't quite get why things are rearranged the way they are when trying to get an equation to be used in fixed point iteration. For example, $x^3+2x+5=0$ could be rearranged to give $\:x=-\frac{5}{...
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1answer
62 views

What sequences in $\Bbb Z[\frac16]$ converge to the fixed point $0$ in $\lvert\cdot\rvert_2$ while being fixed in $\lvert \cdot\rvert_3$?

Suppose some uniformly continuous (under the 2-adic topology) function $f:\Bbb Z[\frac16]\to\Bbb Z[\frac16]$ converges under composition with itself to $0$ (in the 2-adic topology) for all inputs, and ...
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55 views

On applications of the Lefschetz fixed point theorem

I would like to know a little bit about practical applications of the Lefschetz fixed point theorem. I am specially interested in knowing about possibple applications for the study of formal and ...
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1answer
221 views

Proving the Bourbaki–Witt Theorem using Recursion.

I am trying to prove the Bourbaki–Witt theorem, which states: Let $\left(X,\leq\right)$ be a partially ordered set and let $f:X\rightarrow X$ be a function. Suppose that $f\left(x\right)\geq x$ ...
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1answer
41 views

How to prove the existence of these numbers?

I have to prove that there are two numbers $x_1,x_2 \in (-1,1)$ that are the unique solution of the system $\{3x=\sin(x+y+13), 3y=\cos(x+y)\}$. Considering $x_1,x_2 \in (\frac{-1}{3},\frac{1}{3})$ we ...
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67 views

Coloring the vertices of a decagon

In how many ways can you color, up to symmetry, the vertices of a regular decagon using $q$ colors? (We are talking here about the dihedral group of order $20$). So I was thinking about maybe group ...
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1answer
42 views

For what c is the Banach fixed-point theorem true?

We have $g(x_1, x_2) = \frac{1}{6} \begin{pmatrix} x_1x_2+cx_2-1\\ x_1^2-x_2+1 \end{pmatrix}$ For what $c \in \mathbb{R}^+$ is the condition of the Banach fixed-point theorem in set $M = [-1,1]\times ...
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56 views

Fix points of integral transfroms

After I have worked a little bit with different integral transform, especially with the Laplace transform, I was confronted with the fact that there are some functions which remain of the same type ...
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2answers
126 views

Banach fixed-point theorem

The well known fixed-point theorem by Banach reads as follows: Let $(X,d)$ be a complete metric space, and $A\subseteq X$ closed. Let $f: A\to A$ be a function, and $\gamma$ a constant with $0\leq\...
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1answer
82 views

Continuously chosen points on lines in $\mathbb{R}^{n}$

On each line $l \subset \mathbb{R}^n, n \geqslant 2$ passing through $0$ we choose a point $a(l)$ such that $a(l)$ depends continuously on $l$. I have to prove that there exists a line with $a(l)=0$. ...
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1answer
95 views

Stability of a fixed point of a discrete dynamical system

I'm stuck with studying the stability of one fixed point of a discrete dynamical system given in exercise (3) page 44 of Petr Kůrka's Topological and Symbolic Dynamics. Could you please help me? I ...
4
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2answers
219 views

How to explain powers of $(x+1)^{2^n}$ appearing in the Babylonian approximation of $\sqrt x$?

I'm working with this iteration used for approximating square roots and trying to see what I can draw out from it, and in doing so I found something very strange that I can't logically explain. I'm ...