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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Lefschetz fixed point theorem-type formula for a finite group acting on a Fredholm operator

Let $D:X\to Y$ be a Fredholm operator, so that its index $\dim\ker (D)-\dim \text{coker}(D)$ is defined. We can view $D$ as a complex $\mathfrak{D}:0\to X\xrightarrow{D} Y\to 0$, and for a chain map $...
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Fixed point iteration derivative derivations do not match...

Setup Suppose I have a function $g(x,s)$ with fixed parameter $s$ and seek the minimum over $x$. One can propose an iterative scheme, e.g. gradient descent, where, $x_{i+1} = h(x_i,s) = x_i - \alpha \...
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Fixed point for some continuous function.

Let $f$ be a continuous function on $[a,b]$ ( $f: [a,b]\to \mathbb R$, $a < b$) such that $\int_a^b f(x) \, dx = \frac{b^2-a^2}{2}$. How can we prove that $f$ has a fixed point in $(a,b)$ without ...
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Proof of Kakutani fixed-point theorem

I'm reading the following proof of the Kakutani fixed-point theorem, written by Prof. McMullen in his functional analysis notes. I don't understand the proof beyond the point where he appeals to ...
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Converse of Brouwer fixed point theorem

Brouwer fixed point theorem is usually stated in the following way: Let $B^n$ some closed ball of a Euclidean space, and let $f \colon B^{n} \rightarrow B^{n}$ be a continuous map. Then $f$ has a ...
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Proving $f$ is the Identity Function Given $f^n(x) = x$ for All $x \in [0,1]$ [duplicate]

Let $f : [0,1]\rightarrow[0,1]$ be a continuous function with $f(0)=0$. Let $f^n = f \circ f \circ ... \circ f$ (composition n times). Proof that if $f^n(x)=x$ for all $x \in [0,1]$ for some fixed $n \...
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How to find function $G(v_i)$, such that $\mathbb{P}\big[{\psi}(G(v_i))\big]= \frac{1}{G(v_i)}$. Is this a fixed-point problem?

The problem comes from an economic and market scenario. We have a function $$\psi(v_i)=v_i-\frac{G(v_i)-F(v_i)}{f(v_i)},$$ where the random variable $v_i$ is any real number (e.g., person $i$'s money)...
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Fixed point theorem with OSL condition

Use Banach's CMP (Fixed Point Principle) to show existence of a unique fixed point of the $d$-dimensional integral equation $$ x(t) = T[x](t) = x(0) + \int_{0}^{t} f(x(s))\,ds, \quad t \in [0,T] $$ ...
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Fixed point theorem for strict contract map

I want to prove that a function $f: S\to S$ where $S$ is a compact metric space and satisfying $d(f(x),f(y))<d(x,y)$ has a unique fixed point. My attempt : We define $g(x)=d(x,f(x))$ which is ...
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Order of convergence of linear fixed point iteration

Let $A=\begin{pmatrix}1 & 1/2 \\ -1/2 & 0 \end{pmatrix}$. Define $(x_n)$ recursively by $$ x_{n+1}=A x_n $$ for some starting vector $x_0\neq 0\in\mathbb R^2$. Let $\vert\cdot\vert$ be the ...
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Convergence of $C^2$ functions with fixed point iteration

In Ridgway Scott's Numerical Analysis, the following two problems appear: In the first exercise, there was a clear mention of $x_0$ being sufficiently close to $\alpha$. So I knew that I should be ...
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Does a periodic function on $\mathbb{R}^n$ have fixed point?

Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a continuous function. We assume that $f$ is periodic: for every $x \in \mathbb{R}^n$ and every $a \in \mathbb{Z}^n$ we find $f(x) = f(x + a)$. Is it true ...
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Exploring Convergence and Root Determination with Fixed-Point Iteration

Understanding Fixed-Point Iteration: Exploring Convergence and Root Determination The equation $$2x^2 - 7x + 6 = 0$$ has two positive roots. We aim to determine these using the following fixed-point ...
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strongly monotone lipschitz mapping fixpoint

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a lipschitz continuous function that satisfies $ \langle f(x)-f(y), x-y\rangle \geq \alpha\|x-y\|_2^2 \text { for all } x, y \in \mathbb{R}^n $ and ...
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Sublinear convergence of a fixed point iteration

A Fixed Point iteration is given by, \begin{equation} x_{k+1} = T(x_k) + q \end{equation} By Banach Contraction theorem, if $T$ is a contraction, then the above iteration convergent. In particular, if ...
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Properties of complex iterated functions. (with complex iterations)

It's been more than a year I'm in touch with iterations of functions. A lot of my posts are about it. The problem being, it's not really a mathematical topic that gets a lot discussed, in books or in ...
Pierre Carlier's user avatar
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Given $f(x)=x^3+x^2-x+2=0$ calculate $3$ iterations at$x_0=-2.4$

Given $f(x)=x^3+x^2-x+2=0$ Find an interval[a,b] such that there exists $1$ and only root Find a function $\phi(x)=x$ Find an interval where it is possible to use the simple iteration method and ...
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What are the conditions for solution of nonlinear Fredholm equations with Banach fixed point theorem?

Consider the nonlinear Fredholm integral equation of the second kind: $$ \varphi(x) = f(x) + \lambda \int_a^b F(x, t, \varphi(t)) \, dt $$ where $(f)$ and $(K)$ are given functions, $(a, b)$ are ...
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Unique local solution to ODE $\dot{u}=f(u).$

Prove that for any $u_0 \in \mathbb{R}^n,$ there exist $T=T(|u_0|)>0$ and a unique $u \in C([0,T])$ solving $$u(t)=u_0+\int_0^tf(u(s))ds,$$ for all $t \in [0,T].$ Assume that $f$ is locally ...
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Every continuous involution on R^n has a fixed point [duplicate]

I'm looking for a reference which supports the claim that every continuous involution on $\mathbb{R}^n$ has a fixed point. This fact is discussed here, but I'd like to see this as a standalone claim ...
psychicmachinist's user avatar
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Finding the contraction factor of a function

Hi I was working on an exercise where I had to determine the fixed point and the contraction factor of the function : $$F: [1,2]\rightarrow [1,2] : F(x) = \frac{x+2}{x+1}$$ I found the fixed point by ...
Jip Helsen's user avatar
2 votes
1 answer
139 views

Brouwer's fixed point theorem for quotient spaces of the unit ball

Consider the closed unit ball $\overline{\mathbb B ^n}$, and let $\sim$ be some equivalence relation on $\overline{\mathbb B ^n}$ that only identifies points on the boundary $\mathbb S ^{n-1}$ of $\...
mathplayer's user avatar
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Automorphisms of CW complexes and fixed points

Let $X$ be a CW complex, and let $F:X\to X$ be an homeomorphism that sends each cell onto some cell; notice that we could say that $F$ is an automorphism of the CW complex since it preserves its cell ...
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Predictor Corrector Scheme for Implicit Runge Kutta

I want to solve an ODE system : $$ \frac{dy}{dt} = f(y, t) $$ Since my application requires method to be symplectic, I am using an implicit runge kutta method. $$ y_{n+1} = y_n + h\sum_{i=1}^s{b_iK_i} ...
Chandan Gupta's user avatar
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An iterative method for minimizing squared errors between nonlinear functions

Suppose that $f$ and $g$ are two vector-valued functions, where $f$ has a possibly highly complex nonlinear form and $g$ is much simpler (e.g., can even be linear). Consider the nonlinear least ...
Miles N.'s user avatar
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Did R. Buckminster Fuller reference a corollary of a mathematical theorem in his book _Critical Path_?

From Critical Path (1981) , Part 2, Chapter 6 "The World Game" , Richard Buckminster Fuller & Kiyoshi Kuromiya make the Claim : It has been found that within a 100 mile radius a wind is ...
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Condition of the Brouwer fixed-point theorem

While I know it is true that: Every continuous function from a nonempty convex compact subset $K$ of an Euclidean space to $K$ itself has a fixed point. (Wikipedia) I am not sure whether a ...
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Which fixed-point theorem to use?

My first time on this forum. I have the following system of equations. A constant $x_{min} > 0$. Two functions $A(x)$ and $B(x)$ defined on $x \geq x_{min}$. $B(x)$ is strictly increasing, ...
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Prove that when a fixed-point interation function of class p converges, if all the derivatives up to p-1 are equal, they're all zero.

The function $\phi(x) \in C^p([a,b])$ and $x_0 \in [a,b]$ meet the conditions for convergence of the fixed-point iteration method, that is: $$\phi([a,b]) \subseteq [a,b] \\ \phi'(x) < 1 \mspace{5mu}...
oquiefine's user avatar
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If a continuous application $\phi:F \times T\rightarrow F$ is a contraction in F $\exists x(t) \in F$ such that$\phi(x(t),t)=0\forall t\in T$

Furthermore assume that $F \subseteq \mathbb{R}^{m}$ and is closed and $T \subseteq \mathbb{R}^{n}$ is any subset. By being a contraction on $F$ I mean $\exists \lambda$ 0 < λ < 1 such that $\...
Victor Hugo's user avatar
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1 answer
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How to measure the convergence speed of an iterative sequence?

Background Consider the following iteration: $$ \begin{aligned} x_{n+1} = \sin(x_n)\\ y_{n+1} = \cos(y_n)\\ z_{n+1} = \tan(z_n)\\ \end{aligned} $$ For any $(x_0, y_0, z_0)\in\mathbb{R}$, $x_n$ ...
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6 votes
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Counterexamples to Brouwer's fixed point theorem for the closed unit ball in $L^p$ spaces

I was curious about counter-examples for Brouwer fixed point theorem in infinite dimension spaces. I already know an easy counterexample in $l^p$ spaces, see this question. I also know one in $l^\...
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Order of convergence of this (fixed point) iterative method

This is a a previously asked question in my university. Find the order of convergence for the iterative method $x_{n+1}=g(x_n)$ to solve $f(x)=0$ where, $$g(x)=x-\frac{f(x)}{f'(x)}-\frac{f''(x)}{2f'(...
Nothing special's user avatar
1 vote
2 answers
98 views

Fixed point of a cone over an $n$-point space

I'm trying to answer the following question: Let $CX$ denote the cone over an $n$-point space $X=\{1,\dots,n\}$. Show that every continuous map $f:CX\to CX$ has a fixed point. Any help is appreciated. ...
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Is this application of Brouwer fixed point theorem correct?

I recently read a paper using Brouwer fixed point theorem in finite dimensional Hilbert space as follows: $X$ is a finite dimensional Hilbert space with inner product $(.,.)$ and norm $||.||$. $f:X \...
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Quadratic Convergence of Newton's Method for Matrix Inverse Calculation

This discussion centers on Newton's method as an iterative approach for computing the matrix inverse. Let $A$ be a non-singular $n \times n$ matrix, and consider the non-linear function $f(X) = A - X^{...
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Question About Banach's Fixed Point Theorem.

Let $f:[3,\infty)\to \mathbb R$ defined as $x\mapsto \sqrt x$. Then for $ x,y \in [3,\infty)$ we have $$ \left|\sqrt x-\sqrt y \right| =\left|\frac{x- y}{\sqrt x+\sqrt y } ~\right|\leq \frac 1 {2\...
Meet Patel's user avatar
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1 answer
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Confusion on Blackwell's condition for a contraction: $T: B(X)\to B(X)$

Blackwell's condition for a contraction: Why is boundedness neccessary? (Theorem: Blackwell's sufficient condition for a contraction.) Let $X \subset \mathbf{R}^l $ and let $B(X)$ be a space of ...
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Probabilistic analog of Banach fixed-point theorem

Are there probabilistic analogs of Banach fixed-point theorem? For example, is there a notion of probabilistic contractive mappings that gives rise to a probabilistic fixed-point theorem? Sorry for ...
fool's user avatar
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General method for finding invariant subsapces of a nonlinear system

Suppose we are given a system: $$\dot{x_{1}} = f_{1}(x_{1},...,x_{n})$$ $$...$$ $$\dot{x_{n}} = f_{n}(x_{1},...,x_{n})$$ And are interested in finding subspaces of the vector space that are invariant ...
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2 answers
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There exists a function that satisfies $\sum_{n=1}^\infty |f^{[n]}(x) - f^{[n]}(y)| < \infty$ but is not a contraction?

There exists a function $f: \mathbb{R} \to \mathbb{R}$ that satisfies $$ \sum_{n=1}^{\infty} |f^{[n]}(x) - f^{[n]}(y)| < \infty \quad \forall x,y \in \mathbb{R} $$ where $f^{[n]}(x) = f(f(f...(f(x))...
Átila Luna's user avatar
2 votes
2 answers
84 views

Must a converging sequence converge to a fixed point?

I am trying to show that the infinite tetration $\lim_{n \to \infty}$$^{n}x$ converges for real values of $x$ within $[e^{-e},e^{\frac{1}{e}}]$. To do so, I examined the fixed points of the following ...
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2 votes
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Brouwer's fixed-point theorem

I'm trying to understand the cohomological proof of this theorem, but I'm stuck at a point, which brings me to ask a more general question: let $i:Z\rightarrow X$ be the inclusion of a closed set into ...
CarloReed's user avatar
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Is there a Julia fractal that contains uncountable many copies of itself?

We know that the Mandelbrot fractal contains a countable number of copies of itself. See : Does the Mandelbrot fractal contain countably or uncountably many copies of itself? Where that is explained. ...
mick's user avatar
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Banach's Fixed Point Theorem application

I'm trying to solve the following question: Consider $M$ a complete metric space, $k > 1$ and $f: M \to M$ a surjective function, satisfying $d(f(x),f(y)) \geq k d(x,y)$, for every x,y $\in M$. ...
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Let the endomorphism $f$ be given by $f: \mathbb{R^3}\to \mathbb{R^3}, a \to \begin{pmatrix} 1 & 1 & 2\\ 0 & 1 & 0 \\ 3 & 0 & 1 \end{pmatrix} \cdot a$

Let the endomorphism $f$ be given by $f: \mathbb{R^3} \to \mathbb{R^3}, a \to \begin{pmatrix} 1 & 1 & 2\\ 0 & 1 & 0 \\ 3 & 0 & 1 \end{pmatrix} \cdot a$. Find a basis for Fix($f$...
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1 vote
1 answer
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Find the set of convergence of iterative sequence

Qustion Suppose $f(x)=\frac{1}{4}+x-x^2$. Show that the iterative sequence $0,f(0),f^2(0),\cdots$ converges to some $L$. Find all $x\in \mathbb R$ such that $x,f(x),f^2(x),\cdots$ also converges to $...
SuperSupao's user avatar
1 vote
1 answer
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Spectral Radius at the Unique and Globally Attractive Fixed Point of a Specific Type of Mapping

Consider a mapping $f: \mathbb{R}_+^n \rightarrow \mathbb{R}_{++}^n$. If it is monotonic and strictly subhomogeous, then it is contractive under the Thompson’s metric (See Lemma 2.1.7). Here, ...
maphado fan's user avatar
2 votes
0 answers
36 views

Convergence of Two interrelated Sequences

Consider two sequences described below: $$\alpha_{t+1} = (1-\beta_t^2)\alpha_t,$$ $$\beta_{t+1} = (1-C \alpha_t\alpha_{t+1})\beta_t,$$ where $C>0$ and we know $0 <\beta_1<1$ and $0<\...
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28 votes
2 answers
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Problem in an inductive proof of Brouwer's fixed point theorem.

I've been reading this article that proposes an inductive proof of Brouwer's fixed point theorem just using "basic" topology, avoiding the usage of homotopy groups as the usual proof goes. ...
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