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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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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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220 views

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
39 views

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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65 views

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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1answer
60 views

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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1answer
32 views

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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626 views

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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1answer
51 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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25 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 ...
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1answer
18 views

I want to understand the triangle inequality about contraction operator

I read the Chow's paper, Multigrid algorithms and complexity results, https://dspace.mit.edu/handle/1721.1/14254 I have a question on page 42. Let me write the Lemma 2.4.3 on the page. Lemma 2.4.3 ...
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12 views

Suppose $k_n\subset[0,\infty)$ with $k_n\to1$ as $n\to\infty$ and $\alpha_n$ is in $[0,1)$ is $\sum\alpha_n(k_n(M+1)-1)<\infty$ where $M>1$

we am working on a mapp to show that the Mann iteration converges. At the end of the computation we came up the series $\sum\alpha_n(k_n(M+1)-1)$, from previous work it is known that the series $\sum\...
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1answer
57 views

Fixed-point iterations for quadratic function $x\mapsto x^2-2$

Let $f(x)$ be $x^2-x-2$. I want to find the root using FPI in an interval where it will converge. I have chosen $g(x)=x^2-2$ and so $g'(x)=2x$. The convergence condition, $|g'(x)|<1$ is ...
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484 views

Fixed Point Convergence. Finding the interval for which the iteration converges.

I've solved the first part. I think I have something for the second part, but I'm unsure. A) You are given the fixed point problem $x=Ax^2$ where $A>0$ is a constant. Compute positive fixed ...
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Proving Talagrand's contraction lemma for Gaussian processes with the Banach fixed-point theorem

I've done the standard proof of Talagrand's contraction lemma for Gaussian processes (see Exercise 7.2.13 in Vershynin's High-Dimensional Probability) using the Sudakov-Fernique inequality as ...
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11 views

Fixed point iteration with the same power coefficient

I have a function where $f(x) = x^3cos(x)-x^3/10$, with that said, how do i find the fixed point iteration formula for it. I have tried adding an unknown to it and get the $x$ but it does not converge ...
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1answer
33 views

Convexity is crucial in Schauder-Tychnonov fixed-point theorem.

The following Theorem is well-known: Schauder-Tychnonov fixed-point theorem: Let $K$ be a compact convex subset of a Banach space, $E$. If $T:K\to K$ is continuous, then $T$ has a fixed point. I'm ...
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The map $f:\bar{B}\to \bar{B}, $ continuous such that $ \bar{B}\subseteq C[0,1],$ unit closed, need not have a fixed point.

The map $f:\bar{B}\to \bar{B}, $ continuous such that $ \bar{B}\subseteq C[0,1]$, need not have a fixed point. Know about the Brouwer fixed point Theorem on $\mathbb{R} ^n$ which states that if $ \...
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1answer
101 views

Prob. 7 (a), Sec. 28, in Munkres' TOPOLOGY, 2nd ed: A contraction of a compact metric space has a unique fixed point

Here is Prob. 7, Sec. 28, in the book Topology by James R. Munkres, 2nd edition: Let $(X, d)$ be a metric space. If $f$ satisfies the condition $$ d\big( f(x), f(y) \big) < d(x, y) $$ for ...
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Sufficient and necessary condition for the global uniqueness of fix-points

https://www.ams.org/journals/proc/1976-060-01/S0002-9939-1976-0423137-6/S0002-9939-1976-0423137-6.pdf This paper gives a sufficient condition for the uniqueness of SCHAUDER fix point. I wonder if ...
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73 views

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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1answer
68 views

Uniformly convex implies strictly convex [closed]

How to prove Uniformly convex implies strictly convex
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26 views

Existence of fixed points for this Markov operator.

Perhaps math overflow is a better place to put this but I'm looking for some mathematical results that I might be able to apply to see if an operator I'm considering has a fixed point. In particular ...
6
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1answer
94 views

How to show the optimization/ODE fixed point iteration steps converge?

I have $\vec{C} = G(\vec{\beta})$ by solving a system of ODE numerically. Thanks for the help of Robert the ODE can be found in this link: Solving a system of ODE Also $\vec{\beta}$ should satisfy $$...
3
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1answer
58 views

Rigorous proof of a unique solution using Banach's Fixed Point Theorem

I would like to have feedback on the overall quality of the following proof. Question: Prove that $x^5+7x-1=0$ has a unique solution in $[0,1]$. Proof: Let $f(x)=\frac{1-x^5}{7}$ and note that any ...