Questions tagged [fixed-point-theorems]

Question 1

Equations of the form $$y = \sin \left(\frac{y^s}{t} - k \right)$$ are surprisingly well approximated as $$y = - \sin k$$ for a large range of $s$ $t$ and $k$ values: Why? I understand why this is so ...

Question 2

I'd like to work out the details of a proof of the Central Limit Theorem that utilizes the Banach Fixed Point Theorem and possibly also entropy. The rough idea is: The average $\bar{X} = \frac{1}{n} \...

Question 3

I have some function from $R^{nk}$ to $R^{nk}$ defined as: $$T(v) = u + f(v) F$$ where u is some vector, f is some non-linear function and F is a nk,nk matrix. i want to prove that there is a unique ...

Question 4

I am trying to disentangle the proof of Brouwer's fixed point theorem via van Maaren's geometry-free Sperner lemma in Eric Schechter's Handbook of Analysis and its Foundations (sections 3.28-3.37). ...

Question 5

For integers $n$, the arithmetic derivative $D(n)$ is defined as follows: $D(p) = 1$, for any prime $p$. $D(mn) = D(m)n + mD(n)$, for any $m,n\in\mathbb{N}$ (Leibniz rule). $D(-n) = -D(n)$. The ...

Question 6

For integers $n$, the arithmetic derivative $D(n)$ is defined as follows: $D(p) = 1$, for any prime $p$. $D(mn) = D(m)n + mD(n)$, for any $m,n\in\mathbb{N}$ (Leibniz rule). $D(-n) = -D(n)$. The ...

Question 7

I'm currently going through Milnor's proof of the Brouwer's fixed-point theorem, which I've linked here. I am able to follow the proof up to theorem 2 - afterwards, I've having a bit of trouble ...

Question 8

I am not sure if this question makes some sense. But I have been thinking for a while now. Let us start with simple problems; let us say that we want to solve the following equation for $x$ $$f(x)=y.$$...

Question 9

This is a rephrasing of the original post in (Interpolation problem with varying nodes) Let $\{f_i\}^{M}_{i=0}$ be a set of real numbers satisfying either $$f_0>f_1<f_2>f_3 \dots$$ or $$f_0&...

Question 10

Given a fixed $n$, I wanted a continuous function $f$ (preferably whose domain is $\mathbb{R}$) such that, applying $f^n$ (i.e. applying it $n$ times) to $x$ gives you back $x$ for any $x$ in the ...

Question 11

Consider the definition of $R$-convergence as given in Definition 9.2.1 of "Iterative solutions of nonlinear equations in several variables" by Ortega and Rheinbolt. Let $A$ be a fixed-...

Question 12

I am studying a problem related to trust propagation / rating in a network. Users rate each other either negatively or positively. From this I want to compute an overall rating vector $\boldsymbol{r} \...

Question 13

Consider this Householder's method : https://mathworld.wolfram.com/HouseholdersMethod.html versus this Halley's method : https://mathworld.wolfram.com/HalleysMethod.html Which method is the most ...

Question 14

I am new to numerical methods and am currently learning Fixed point iteration. I have learned that if you can express $x = g(x)$, and $|g'(x_0)|<1$, then the sequence, $x_{n+1} = g(x_n)$ converges ...

Question 15

One of the evolution equations used in evolutionary game theory is the Replicator type II dynamics $$ x_i(t+1)=x_i(t)\frac{(Ax(t))_i}{x^T(t)Ax(t)} $$ where $A$ is an $M\times M$ payoff matrix with ...

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Questions tagged [fixed-point-theorems]

Why is $y = \sin \left(\frac{y^s}{t} - k \right)$ outstandingly approximated by $y = - \sin k$, even for large $k$?

Central Limit Theorem via Fixed Point Theorem and Entropy

uniqueness of fixed point using a non-linear function

Unclear step in van Maaren's theorem proof by Schechter

Iterating the arithmetic-derivative map $U(n)=n+D(n)−1$

A sequence based on arithmetic derivative that always converges to prime numbers

Question About Milnor's proof of the Brouwer's fixed-point theorem

What are the problems in math that are equivalent to fixed point problems?

Proving the existence of a fixed-point for the interpolation problem

A function $f$ such that repeated application gives you identity? i.e. $f^n(x) = x$ [duplicate]

Why is R-convergence a useful notion?

Convergence of fixed-point iteration $x = \frac{1 + Px}{2 + Px + Nx}$

Householder versus Halley speedtest?

Why fixed point iteration of $x^3 = 1-x^2$ doesn't converge when $x_0 = 0$?

Convergence of discrete replicator equation

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