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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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Fixed point property for the total space and base space of a principal bundle

Is there a principal bundle $P\to X$ such that $P$ has the fixed point property but $X$ does not have? Is there an example of this situation where $P,X$ and the fibers are compact smooth manifolds?(...
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Lawvere - Conceptual Mathematics - Retract of a Fixed Point Space has a Fixed Point

This is from Lawvere's Conceptual Mathematics in a section about fixed points: Suppose that $A$ is a retract of $X$, i.e. there are maps $r\colon X \to A$ and $s\colon A \to X$ such that $r \circ s = ...
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Do iterative formula only solve for one positive root?

For all the questions I have done using an iterative formula, the quadratic equation only ever solves to one root, I am not sure is this always the case, or is there a method to find multiple roots? ...
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Attracting and repelling fixed points and cycles

Consider the iteration which produces the Mandelbrot set: $f(z) = z^2+c$. At $c=0$, this iteration has an attractive fixed point. At $c=-1$, it has an attractive 2-cycle. As $c$ varies from $0$ to ...
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Must a holomorphic function from $D(0,1)$ to $D(0,1)$ have a fixed point?

Must every holomorphic function $f:D(0,1)\longrightarrow D(0,1)$ have a fixed point? I know that any holomorphic function with two fixed points is the identity: $f=Id$, but I can't find out an ...
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How will fixed point change if the mapping changes?

Generally, the question is we have a contraction mapping $M:\mathbb{R}^n\rightarrow\mathbb{R}^n$. Define a new mapping $\tilde{M}:\mathbb{R}^n\rightarrow\mathbb{R}^n$ where $\tilde{M}:Mx+c$ where $c$ ...
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Understanding and applying Banach's contraction principle.

Consider the following operator $$K(x)(t)=\int_\limits{0}^{1}x^2(s) \ ds + At^2 \quad \forall \ t\in[0,1], \tag1$$ acting on the Banach space $C([0,1])$ of continous functions $||x||=\...
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Let $f$ be a continuous function on the unit circle. Then there exists a fixed point $p$ such that $f(p) = p$.

Let $S = \{p = (x, y) \in \mathbb R^2 : x^2 + y^2 = 1\}.$ Let $f : S \to S$ be a continuous function. Then, there always exists $p \in S$ such that $f(p) = p.$ My try:- $S = \{p = (x, y) \in \mathbb ...
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Elchanan Mossel’s Dice Paradox using fixed point updating

The problem: You throw a dice until you get 6. What is the expected number of throws (including the throw giving 6) conditioned on the event that all throws gave even numbers. The following is the ...
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How do I use Runge-Kutta Iterative method to produce a table of values for a function?

Describe how the fourth-order Runge-Kutta method can be used to produce a table of values for the function $$f(x)=\int_0^x e^{-t^2}\ \mathsf dx$$ at $100$ equally spaced points in the unit ...
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Convergence of sequence $U_n$ such that $U_0=2$ and $U_{n+1}=2+\frac{1}{U_n}$

Take the sequence $U_n$ such that $U_0=2$ and $U_{n+1}=2+\frac{1}{U_n}$ Prove that this sequence converges and find its limit. I can't find if the sequence is increasing or the opposite.
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Fixed point theorem on disjoint interval

I am to prove that there exists $x^*\in [0,1] \cup [2,3]$ for a function $f$ from $[0,1] \cup [2,3] \to [0,1] \cup [2,3]$ s.t $f(x^*) = x*$ basically the fixed point theorem for the above disjoint ...
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On the convergence of Newton's square root iteration

This is inspired by my answer to Let $x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n})$. Prove that $x_{n+1} < x_{n}$ which I was unsatisfied with. This is undoubtedly not new but I haven't seen these ...
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Brouwer fixed-point theorem on a function

For $f:\mathbb R^n \rightarrow \mathbb R^n \quad \exists R \gt 0 $ that $f$ is continous on $\bar B_R(0):= \{ x \in \mathbb R^n : \Vert x\Vert_2 \leq R \} \subset \mathbb R^n$ . Let also $\langle f(x)...
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Are the properties of not having fixed points and being its own inverse enough to characterise the antipodal map?

While trying to solve a problem I ended up with a smooth map $f:S^n \to S^n$ which I know is fixed point free and such that $f \circ f $ is the identity map of the sphere. I would like to conclude ...
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Prove that $T$ is not contraction and $T^2$ is contraction

$T$ is not contraction and $T^2$ is contraction"> For (a) $||(Tf)(t)-(Tg)(t)||= ||\int ^t_0 f(s)ds-\int ^t_0 g(s)ds||\\ =||\int ^t_0(f(s)-g(s))ds||\\=\sup_{0\le t\le 1}|\int ^t_0(f(s)-g(s))ds| \\ \...
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How to determine convergence rate

An iterative method has been used to solve a non-linear equation f$(x)=0$. The table below show the iterations $x_k$ at $k$. $$\begin{array}{c|c|} & \text{} & \text{} \\ \hline \text{k}...
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fixpoint iteration to solve $y'(t)=y(t), y(0)=1$

Solve the initial value problem $y'(t)=y(t)$, $y(0)=1$ on the interval $[0,1]$ with a fixpoint iteration of the operator $T: Y\to Y, (Ty)(t):=y_0+\int_0^t f(s,y(s))\, ds$. Begin with $y_0(t)=0$ and ...
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Does there exist a homotopy between identity function and any continuous function?

(My question is related to the Brouwer fixed-point theorem.) Let $B$ be a closed ball of $\mathbb{R}^n$. Q 1. If $f : B \rightarrow B$ is a continuous function, then is there a homotopy between $...
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Proof of a fixed point theorem on the disk

There is a very nice fixed point theorem which I'd have liked to give to my students : Let $n$, $m$ be two integers larger or equal to one. Let $B_n$ be the open unit ball in $\mathbb{R}^n$, and ...
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How to find the fixed point of a monotonic but not Scott-continuous function over a complete lattice?

The Knaster–Tarski theorem states that a monotone function $f: L \mapsto L$ on complete lattice $L$ has fixed points, which also form a complete lattice. The theorem itself doesn't specify a way to ...
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Does this iterated sequence always end in a finite number of steps to a number which is divisible by a perfect number?

I posted this question at MathOverflow, but then I realized that maybe it is more appropriate to ask it here: Let $f$ be a multiplicative arithmetic function which maps $\mathbb{N}$ to itself, such ...
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Books about fixed-point theory

I am looking for book recommendations about Fixed-Point Theory. I found this post: Book Recommendation for Iterated Functions? recommending a book by Shashkin. However, I cannot find who Shashkin is,...
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$f \in \mathcal{C}(X, X)$ on metric space with $\sum_{n=1}^\infty d(f^n(x), f^n(y)) < \infty$ has a fixpoint

Let $X$ be a complete metric space and $f : X \to X$ continuous such that $$\sum_{n=1}^\infty d(f^n(x), f^n(y)) < \infty $$ for all $x, y \in X$, where $f^n$ means $f \circ \ldots \circ f$ $n$-...
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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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Godel's Diagonalization Lemma As Application Of Lawvere's Fixed Point Theorem

I've read through this paper with applications of Lawvere's fixed point theorem. On the diagonalization lemma, they say the following: For one thing, how can $f$ and $\Phi_{\cal E}$ be well ...
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Convergence to a fixed point [duplicate]

Let $f : [a,b] \rightarrow [a,b]$ be a continuous function s.t. $f'(x)$ is defined on $(a,b)$ and $\left\lvert f'(x)\right\rvert \leqq t$ where $0<t<1$. Prove that for any point $x_0$ in $[a,b]$ ...
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Typo confusion: What theorem to prove?

I'm given the following problem: Let $f: \mathbb{R}^N \rightarrow \mathbb{R}^N$ be a contraction with contraction-constant $\lambda \in [0, 1)$, i.e. we have for all $x, y \in \mathbb{R}^N$ that ...
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Contraction mapping of the logistic map

I wish to find the values for which the logistic map behaves as a contraction map $$x_{n+1}=rx_n(1+x_n)\equiv F(x;r)$$ i.e, I wish to find for which $r$, the mapping above admits a unique fixed ...
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Why must a directed complete partial order with an increasing self map F contain a “roof” with respect to F?

I'm trying to solve exercise 8.20 in Davey & Priestley's "Introduction to Lattices and Order". The problem asks me to prove the third CPO fixpoint theorem: If $P$ is a directed complete partial ...
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Lefschetz number of a biholomorphism of $\mathbb{P}^n_\mathbb{C}$

How can I prove that each biholomorphism with non degenerate fixed points $F :\mathbb{P}^n_{\mathbb{C}}\to \mathbb{P}^n_\mathbb{C}$ has exactly $n+1$ fixed points ? The idea is that $\...
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Converting 1-D Fixed point iterator into n-D in Matlab

I wrote a function, which takes as input an iteration function, a starting value, an error tolerance, and a maximum number of iterations. The output is the final value, whether the error bound was ...
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Why does the fixed point theorem hold for every lambda term?

Can someone give a clear and simple answer for why the fixed point theorem holds for every $\lambda$-term, in contrast with the fact that not all numerical function have a fixed point?
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Approximate the limit of quadratic recurrence by applying Banach fixed-point theorem to its bounds

I have a recurrence of the form $u_0=0, u_1=50, u_n=-a_{n-1}+u_{n-1}+50$, where $a_{n-1}$ is a probabilistic amount which I can not describe in a simple formula. But I can set bounds for the ...
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a question in Nonlinear Equations and Their Solution by Iteration

I'm reading the book "Theoretical Numerical Analysis(A Functional Analysis Framework)" by Kendall Atkinson and Weimin Han. Please help me with the exercise 5.4.2 (Page 241). Consider the nonlinear ...
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1answer
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Show root using Banach Fixpoint

I'm required to show that: $f(x) = e^x - 4x$ has a root in $(0,1)$ using the Banach Fixpoint theorem. The fact that $f((0,1)) \neq (0,1)$ confuses me. How do I proceed without knowing that $f$ isn't ...
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Find Smallest contraction coefficient

I have been given the following function $f:[-1,1]\to \mathbb{R}$: $$ f(x)=\ln(x+2)-x $$ And I have been asked whether it is a contraction or not, and if it is, I have to find the smallest contraction ...
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Solving $f(x) = \frac{x^2}2 +x - \int_0^x f(t)dt, x\in[0,1] $ with Iteration Method

I have problem solving the following integral equation : $$f(x) = \frac{x^2}2 +x - \int_0^x f(t)dt, x\in[0,1] $$ using the iteration method with initial approach $f_0(x)= \frac{x^2}2 +x$ I applied ...
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A conjectured identity for the fractional iterates of $\sqrt{x}+2$

Deeply impressed by the Fractional iterates of $x²-2$, a problem by Ramanujan post one month ago, I decided to investigate a little by myself on the topic and discovered that using Carleman matrices ...
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Variant of the Contraction Mapping Theorem

Let (C, ||.||) be a closed subset of a Banach space, and let f: C -> C be a mapping such that ||f(x) - f(y)|| < ||x - y|| for all x, y in C. Must there exist a fixed point in C that f maps to ...
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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 $X$ $f^n(x)\...
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Show $f(x)=f_{0} + \int_{0}^{x} F \circ f$ has a unique solution.

I have an analysis exercise and I have literally no idea where to start or even a solid comprehension of what the question is. The exercise is as follows: Now, if I am correct, $\left| \Phi(f)(x)-\...
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Brower fixed point theorem vs conformal maps

I was reading a proof of the following theorem. If $U$ is the open unit disc in $\mathbb{C}$, and $f:\mathbb{C}\to\mathbb{C}$ is analytic, then $d(f(x),f(z))<d(x,z)$ if $x\not=z$ (here $d$ is the ...
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Why does Fixed Point Iteration work?

I have searched online for an answer, but everyone gave the method, and no one explained why is it working. I'll first write what I do understand. Let $f(x)$ be a continuous function at $[a,b]$. ...
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Find a function with 4 unstable fixed points

I need to find a function that has 4 fixed points, and all of them are unstable. I don't know how to proceed in this kind of problem, but i know how to find the fixed points in a function, and i know ...
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What would be the $g(x)$ of fixed point iteration method for the equation $f(x)=x\sin(x)+\cos(x)=0$ which satisfies the condition $|g'(x)| < 1?$ [closed]

I've tried finding the $g(x)$ for the equation $f(x)=x\sin(x)+\cos(x)=0$ by squaring or multiplying, but nothing seems to fulfil the condition of $|g'(x)|<1.$
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Show that $T^N$ contraction and $T$ has a unique fixed point in $X$

Let $(X,d)$ be a complete metric space and $T:X\to X$ be a mapping such that for some sequence $(\alpha_n)\in (0,\infty), d(T^nx,T^ny)\le \alpha _n d(x,y) $, for $x,y\in X$. If $\liminf_{n\to \infty} \...
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1answer
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Finding if a fixed point is attractor or repulsor without differentiation.

Given the function $F(x)=\frac{\pi}{2}\sin(x)$. Find the fixed points and, if they exist, determine if the points are attractors or repulsors without differentiation. I already found the fixed points ...
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57 views

Closed graph and fixed points

I’m currently trying to understand the following Proposition from a paper i’m reading: Prop.: Let $X$ and $Y$ be two Hausdorff topological linear spaces. Let $H:X \times Y \rightarrow Y$ be a ...
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27 views

Conditions to apply Banach fixed-point theorem

Say that we have a recurrence of the form $u_{n+1}=f(u_{n})$, where $f:R \to R$, then, what is the conditions on $f$ to have a convergent series $u_{n}$. I have the following questions: Is it enough ...