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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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 ...
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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 ...
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System of equations and the Brouwer's Fixed-Point Theorem.

Let's consider the following system of equations: \begin{eqnarray}{ e^{xyz} = \frac{x}{\sqrt{e^{2xyz}+1}}\\ \cos(x+y+z) = \frac{y}{\sqrt{e^{2xyz}+1}}\\ \sin(x+y+z) = \frac{z}{\sqrt{e^{2xyz}+1}} }...
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Existence of Solution for $Ax+\alpha g(x)=0$

Let $A$ be an $n \times n$ non-singular matrix and $g:\mathbb{R}^n \to \mathbb{R}^n$. I want to prove that for sufficiently small $\alpha$ the equation $\alpha g(x) + Ax = 0$ has a solution. Any ...
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Fixed point method for non-homogeneous ode and pde

During my course in mathematical methods for physics, my professor introduced us to the "method of fixed point" for studying non-homogeneous systems of odes, odes and pdes. Although I think I ...
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Intersection of sets in Banach convex space

Let $X$ be uniformly convex Banach space. $f:K\rightarrow K$, such that $\parallel fx-fy\parallel \leq\parallel x-y\parallel\,\,\forall x,y\in K $, with $K$ a nonempty, closed, convex, bounded subset ...
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How is the triangle inequality being used here?

I'm reading the Banach fixed point theorem (https://en.wikipedia.org/wiki/Banach_fixed-point_theorem) and at one point, this inequality is used: https://puu.sh/AS4uY/78caef5bc6.png However, I'm not ...
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is it always possible to choose a small enough positive $\varepsilon$ such that $0 < c_n (\varepsilon) < 1$?

Let $ Y_{n, m} (t) = \left\{ \begin{array}{ll} t & m = 0\\ t + h_{n, m} \cos (\pi n) \tanh \left( \frac{Z (Y_{n, m - 1} (t))}{| \Omega (t) | \prod_{k = 1}^{n - 1} \tanh (Y_{n,...
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2answers
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Confusion about proving $f$ is not a contraction

Let $f:[0,2]\to\mathbb R, f(x)=\frac{1}{3}(4-x^{2})$ I know that $f$ is a contraction $\iff$ $\exists L \in ]0,1[$ such that for all $x, y \in [0,2]: |f(x)-f(y)|\leq L|x-y|$ I am having trouble ...
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Basic function theory notation: $f_n$, $f^n$ and $f(n)$

Im trying to learn some concepts by reading notation, but I need to know if I understand it correctly. I read that one could use the index notation $f_n$ instead of $f(n)$, is that right? Wikipage ...
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How can the distance metric of two different metric spaces be comparable?

I come from a machine learning background and recently was working on understanding self normalizing networks. It requires me to know the proof of banach fixed-point theorem which needs me to know ...
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1answer
161 views

How Gelfond find his limit for $\exp(\pi) $? [duplicate]

$$ a_0 = \frac{1}{\sqrt 2} $$ $$ a_{n+1} = \frac{( \sqrt {1 - a_n^2} -1)^2}{a_n^2} $$ $$ \lim_{n \to \infty} \frac{4^{\frac{1}{2^n}}}{a_{n+1}^{\frac{1}{2^n}}} = \exp(\pi) $$ How did Gelfond find ...
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Prove that a specific fixed point iteration is locally convergent [duplicate]

Let $g : I \rightarrow \mathbb{R}$ be a $C^1$ map such that $g'(x) \ne 0$ for any $x$ in $I$. Assume that there exists $r \in I$ such that $g(r)=0$. Prove that for $\eta \in I$ sufficiently close to $...
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1answer
66 views

Example of stable fixed point equation

Schauder's fixed point theorem says that any continuous function $f:K\to{K}$, where $K$ is a nonempty convex and compact subset of a normed linear space $Y$ admits a fixed point. I came across this ...
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Knaster-Kuratowski-Mazurkiewicz theorem implies the Brouwer fixed point theorem

I have this version of the theorem: let $x_1,\ldots,x_n$ $n$ points from $\mathbb{R}^N$ and $F_1,\ldots,F_n$ closed set such that $$ \operatorname{cov}\{x_{1_i},\ldots, x_{i_k}\}\subset F_{i_1}\cup\...
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25 views

Fixed points of cubic transformation

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a Lipschitz continuous operator and let fix$(f)$ denote the set of fixed points of $f$. Define the operator $g = (1-a) f + a(1-b) f^2 + a b f^3$, for ...
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39 views

Fixed points of quadratic transformation

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a Lipschitz continuous operator and let fix$(f)$ denote the set of fixed points of $f$. Define the operator $g = a f + (1-a) f^2$, for some $0 < a ...
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114 views

fixed point theory for $\ln x=x^2-1$

if we choose $$ X_{n+1} = \sqrt {1+\ln(x_n)} \quad \text{or} \quad X_{n+1} = \sqrt {1-\ln(x_n)} $$ for the fixed point theory, you draw the graphs of $\ln(x)$ and $x^2-1$ you'll see two intersection ...
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2answers
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Connected, oriented and compact smooth manifold can't retract on the boundary

i would like to proof that a compact , connected and oriented $n$-manifold $M$ can't retract onto $\partial M$. My tool is the De Rham cohomology. En effet, if there exists $f:M\to \partial M$ such ...
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1answer
225 views

Y combinator as an application of Lawvere's fixed point theorem

Lawvere's fixed point theorem states that that in a cartesian closed category, if there is a morphism $ϕ: A \to B^A$ which is point-surjective (i.e., for every point $q : 1 \to B^A$ there exists a ...
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1answer
24 views

Uniqueness of interception between functions

Suppose I have three functions $f(x,y)$, $g(z)$ and $h(z)$. These three functions are differentiable and continuous in the reals. I am interested in the intersection: $$ z^*=f(x^*,y^*)\\ x^*=g(z^*)\\...
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1answer
71 views

every nonsurjective continuous function from $S^2$ to $S^2$ there exist a fixed point?

can someone please help me to show for every nonsurjective continuous function from $S^2$ to $S^2$ there exist a fixed point? i think since the fuction is not surjective it doesn't contain at least ...
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2answers
38 views

Start value for which an iterative function converges

How do i find a start value for which the function: $x = - \exp(x)$ converges. I know how to solve it by making a graph and picking a start value but i don't know how to find the start value for ...
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Doubt in Brouwer fixed point theorem.

Consider $S^1 =\{(x,y)\in \mathbb{R}^2 : x^2 + y^2 =1\}$ , $D=\{(x,y)\in \mathbb{R}^2: x^2 + y^2 \leq 1\}$ How to show that If $f : D \to S^1$ is a continuous mapping, then there exists $x \...
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1answer
37 views

Let $f: \mathbb R\to \mathbb R$ be a continuous function. Which of the following are sufficient conditions for $f$ to have a fixed point in $[0, 1]$?

(a) $f(0)=f(1)$ (b)$f(1)<0<f(0)$ (c) $0<f(1)<f(0)$ (d) $f(0)<0<1<f(1)$ To obtain a fixed point, we should find $x=f(x)$ but how do I obtain the necessary conditions? What ...
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1answer
48 views

fixed point of a multivalued map

Suppose I have a map $S:X \to Y$ between Banach spaces which is multivalued, so that $S(x)$ is a set. I have shown that $S$ takes a closed ball of radius $R$ to itself, and it also is such that if $...
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76 views

how to show an iterative equation converges to a fixed point

let's consider the equation $y'(t) = 16-y^2 = f(t,y)$ for all $t\in [0,1]$ and $y(0) = 0$. How small should the step size $h$ be to ensure the following equation converge for all $i \geq 0 $. ...
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1answer
109 views

Why application of Sharkovskii's Theorem to Collatz is wrong?

I thought about this years ago but never had it resolved. Sharkovskii's Theorem has a corollary according to Wikipedia that if there exists a 3-cycle on a map on $\mathbb{R}$, then the map has $n$-...
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22 views

Solving multivariable modulo equations?

I am being asked to find fixed points of further iterations of the Cat map, which is defined as: $$C(x,y) = (2x+y , x+y) \pmod 1 $$ for $(x,y)$ an element of $[0,1)$ Hence the equation to solve the ...
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1answer
60 views

$a_0=0$, $a_{n+1}=a_n-\frac{1}{2}(a_n ^2-a)\rightarrow \sqrt{a}$ using banach's theorem

Let's have a sequence $a_0=0$, $a_{n+1}=a_n-\frac{a_n ^2-a}{2}$. This sequence converges for $0<a<1$ to $\sqrt{a}$. One way of proving this is to prove that $T(x)=x-\frac{1}{2}(x ^2-a)$ is a ...
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Analysis of function using fixed point

Let $f$ be function definded by $$f(x)=\frac{x^3+1}{3}$$ has 3 fixed point say $\alpha ,\beta ,\gamma$ where $-2 < \alpha < -1$,$0 < \beta < 1$,$1 <\gamma <2$I wanted to show ...
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183 views

If fixed-point iteration has linear convergence, how can Newton's Method have quadratic convergence?

Newton's Method for finding the roots of a function can be considered a type of fixed point iteration of $g(x) = x - \frac{f(x)}{f'(x)}$, since $f(k) = 0 \rightarrow g(k) = k$. But it is well-known ...
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Prove that there is no fixed point

Let $f: \mathbb R \to \mathbb R$, whereby $f(x):=(x+\sqrt{x^{2}+1})/2$ with $x \in \mathbb R$. Show that $f$ fulfils: $\forall x, y \in \mathbb R$ with $x \neq y$ $d(f(x),f(y))<d(x,y)$, whereby ...
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112 views

Existence of unique fixed point in compact Metric space [duplicate]

Let $(X,d)$ be compact. Show: for a map $f$ that when $\forall x, y \in X$ with $x\neq y$ $d(f(x),f(y))<d(x,y)$ is fulfilled. Then $f$ has a unique fixed point.
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324 views

Banach fixed point theorem ode

I am attempting to do this problem here for studying purposes for an exam I have in a couple months. I was hoping to get some help... For part a) I was trying the following: Given a contraction ...
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1answer
62 views

Finding fixed point of a function

I wanted to show that in general, T has a unique fixed point and to determine that fixed point of T such that $Tf = f$ $T : (C[0,1], \left\lVert.\right\rVert_{inf} \rightarrow C[0,1], \left\lVert.\...
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Edelstein's fixed point theorem

I just read that Edelstein fixed point theorem did not assume compactness in his original proof of his fixed point theorem for contractive mappings, but rather the existence of a point $a \in X$ whose ...
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3answers
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Why does iterating in different ways produce different solutions?

Recently, I came across the following equation: $$2^x=4x$$ To solve it, I decided to iterate. Firstly I stated: $$x_{n+1}=\frac{2^{x_n}}{4},x_0=1$$ and found a solution of $x\approx 0.3099069324$. ...
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1answer
111 views

Edelstein Theorem

Let (M,d) be a compact metric space and $d(f(x),f(y)) < d(x,y) $ for all $ x\neq y$ Prove that if $f$ is a continuous fuction then there is a unique $x_0 \in M $ such that $f(x_0)=x_0$ I know that ...
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1answer
68 views

Shrinking map theorem

Let $D$ be a compact subset of $\mathbb{R}$ and let $f$ be a shrinking map defined on $D$, show that $f$ has a unique fixed point. This is a different version of the contracting map theorem in ...
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22 views

Errors reduce at each step of Fixed-point iteration

We have known that if $g: [a,b] \rightarrow [a,b]$, $g$ is differential in $(a,b)$, $k \in (0,1)$ and $|g'(x)| \leq k, \forall x \in (a,b)$ then $g$ has a unique fixed point $x^*$. Moreover, for all $...
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Understanding Selection Theorems

Can someone explain me what is the role of Selection Theorems? Especially; what is the point of the Continuous Selection Theorems for the set valued mappings? Is it just to guarantee a fixed point?
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28 views

How to check for fixed points [closed]

How to check for fixed points in such type of questions
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1answer
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Fixed point theorem (At least one fixed point) [closed]

In fixed point theorem, If g is a continuous function g(x) in [a,b] for all x in [a,b], then g has a fixed point in [a,b] i.e. c belongs to[a,b] such that g(c)=c According to this theorem, We have ...
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2answers
111 views

Why is Brouwer's fixed point theorem considered a theorem in topology rather than $n$-d real analysis?

Brouwer's fixed point theorem states that if a continuouos function $f$ maps a compact, convex set to itself, then $f$ has a fixed point in that set. All these concepts are topological concepts, ...
3
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1answer
105 views

Geometric Proof of Perron-Frobenius II

The following is proved in these lecture notes. Let $A$ be an $n\times n$ real matrix with all entries positive. Then $A$ has a unique positive eigenvector (up to positve scaling), and the ...
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54 views

General conditions for the Fixed point property on the Quotient topology

Today in my topology class (undergraduate) we introduced the basics for Fixed Point theory for topological spaces (i.e.Fixed point property, retracts, contractability, etc.) and we were discussing the ...
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1answer
42 views

Common fixed points of two polynomials over $\mathbb{C}$

According to this question (Ritt's classification), given two commuting polynomials $f,g \in \mathbb{C}[z]$, there exists a common fixed point of them. By commuting we mean: $f(g(z))=g(f(z))$. Of ...
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58 views

Reason for using homology groups in high dimensional Brouwer fixed point theorem.

I'm new here and to the differential geometry. I find a problem when I'm trying to prove Brouwer's fixed point theorem using topology. Suppose $f: D^{n}\to D^{n}$ has no fixed point, define $F: D^{n}...
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57 views

Is this rough intuition for Brouwer's fixed point theorem in $\mathbb R^2$ correct?

To prove Brouwer's fixed point theorem on a closed disk: For any continuous function $f$ that maps a closed disk in $\mathbb R^2$ to itself, there exists a point $x_0$ in that disk, such that $f(...