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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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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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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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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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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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$f \in \mathcal{C}(X, X)$ 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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1answer
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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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1answer
31 views

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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26 views

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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3answers
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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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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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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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1answer
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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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1answer
82 views

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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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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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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1answer
29 views

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
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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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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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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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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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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47 views

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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1answer
68 views

Uniformly convex implies strictly convex [closed]

How to prove Uniformly convex implies strictly convex
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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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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 ...
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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 ...
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Is the set of nash equilibria/correlated equilibria convex?

I am curious about the geometry of these sets (assuming compact, convex action space and concave utility function, so the nash must exist). Is there any general argument about when will any solution ...
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3answers
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A question about fixed points and non-expansive map

Let $$K=\{x=(x(n))_n\in l_2(\mathbb{N}):\|x\|_2\le 1\ \text{ and } x(n)\ge 0 \text{ for all } n\in \mathbb{N} \}$$ and define $T:K\to c_0$ by $T(x)=(1-\|x\|_2,x(1),x(2),\ldots)$. Prove : (1) $T$ is ...
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To Prove $T $ is a self map and $T$ have no fixed points

Let $K=\{x=(x(n))\in c_0:0\le x(n)\le 1$ for all $n\in \mathbb{N}\}$. Define $T:K\to c_0$ by $T(x)=(1,x(1),x(2),x(3),...).$ Prove : (a) $T$ is a self map on $K$ and $||Tx-Ty||_\infty=||x-y||_\infty $ ...
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Point-set topological proof of Brouwer's fixed point theorem

I have tried to understand the point-set topological proof of Brouwer's fixed point theorem presented in Cou11. But I couldn't clarify some parts. Here are the theorem and its proof. Theorem: There ...
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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 $$...
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1answer
56 views

Fixed point equation to solve Burgers' equation IVP

Using the equation $u \equiv u ( x , t ) = u _ { 0 } ( x - t u ( x , t ) )$ to compute $u \left( T , x _ { j } \right)$ for the Burgers equation, where the Burgers equation is $u _ { t } + \left( \...
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How to prove the existence of a fixed point of this mapping?

Let $X=\{x_i : i\in I\}\subseteq\mathbf{R}^n$, where $I=\{1,\ldots,m\}$. Then for some initialization $\mu^{(t)}$, and $\pi^{(t)}=\{x\in X : \|x-\mu^{(t)}\|\leq r\}$, $r>0$, we want to prove that a ...
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1answer
41 views

fixed point function (nonlinear equation)

here's the following problem, I'm trying to find a real root by fixed-point iteration method but I can't find a properly $g(x)$ that meets the condition that $|g'(x_0)|<1$. Well, my nonlinear ...
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1answer
23 views

Show that $\exists u_0 \in C : g(u_0) = u_0$, if $g$ is nonexpansive over a Banach subspace.

Exercise : Let $X$ be a Banach space, $C \subseteq X$ compact and convex and $g : C \to C$ a nonexpansive operator. Show that $\exists u_0 \in C : g(u_0) = u_0$. Thoughts : In a previous exercise, ...
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24 views

Is there a way of proving that a function has a particular number of fixed points.

From my understanding, a function is said to have a fixed point if $f(x) = x$. Is there a way for finding how many fixed points a function has?
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1answer
62 views

Question about Banach's fixed point theorem

Let $(x_n) _{n\ge 1}$ be a sequence and $f:\mathbb{R} \to \mathbb{R} $ a contraction. I know that if $x_{n+1} =f(x_n) $ then $(x_n) _{n\ge 1}$ converges to $f$'s unique fixed point by Banach' s fixed ...
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1answer
65 views

Fixed Point Iteration $x^3 - 3 = 0$

I am having trouble solving $x^3 - 3 = 0$ using the fixed point iteration method. It is advised in the problem to put $g(x)$ in a form similar to $g(x) = x + c(x^2 - 5)$ for $x^2 - 5 = 0$ but I am ...
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34 views

I am trying to find the maximum learning rate or stepping rate of steepest descent algorithm in 2 dimensions

Let $f(x,y) = (x-y)^4+2x^2+y^2-x+2y$. I am trying to numerically find the miniumum of $f$. We define a fixed-point iteration scheme \begin{align*} g(x, y) = \vec{x}_{n+1} = \vec{x}_{n} - a\nabla f \...