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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A Fixed Point Theorem in Complete Metric Spaces for Self-Maps Satisfying This Contractive Condition
Let $(X, d)$ be a complete metric space, and let $f \colon X \longrightarrow X$ be a self-map of $X$ for which there exists a real constant $h \geq 0$ such that
$$
d \big( f(x), f(y) \big) \leq h \max ...
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Iterative procedure for graph
Let us have a simple connected undirected graph $G_n$.
Name two vertices "associated" if they have at least one common neighbor
(its adjacency lists intersected) regardless of whether they ...
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Can you prove the unique readability of concatenation of terms with the Kleene Fixed-point Theorem?
Suppose $\mathcal L$ is a language consisting of constant symbols, function symbols, and relation symbols. In Sets, Models, and Proofs by Ieke Moerdijk and Jaap van Oosten, the set $T$ of $\mathcal L$-...
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Coincidence points and intersection with the diagonal
Let $X$ and $Y$ be smooth manifolds, and let $f,g : X \to Y$ be continuous maps.
In coincidence theory, it is standard to reduce the study of coincidence points$\{ x \in X | f(x) = g(x) \}$ to the ...
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Why is $y = \sin \left(\frac{y^s}{t} - k \right)$ outstandingly approximated by $y = - \sin k$, even for large $k$?
Equations of the form $$y = \sin \left(\frac{y^s}{t} - k \right)$$ are surprisingly well approximated as $$y = - \sin k$$ for a large range of $s$ $t$ and $k$ values:
Why? I understand why this is so ...
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Central Limit Theorem via Fixed Point Theorem and Entropy
I'd like to work out the details of a proof of the Central Limit Theorem that utilizes the Banach Fixed Point Theorem and possibly also entropy. The rough idea is:
The average $\bar{X} = \frac{1}{n} \...
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uniqueness of fixed point using a non-linear function
I have some function from $R^{nk}$ to $R^{nk}$ defined as:
$$T(v) = u + f(v) F$$
where u is some vector, f is some non-linear function and F is a nk,nk matrix.
i want to prove that there is a unique ...
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Unclear step in van Maaren's theorem proof by Schechter
I am trying to disentangle the proof of Brouwer's fixed point theorem via van Maaren's geometry-free Sperner lemma in Eric Schechter's Handbook of Analysis and its Foundations (sections 3.28-3.37). ...
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Iterating the arithmetic-derivative map $U(n)=n+D(n)−1$
For integers $n$, the arithmetic derivative $D(n)$ is defined as follows:
$D(p) = 1$, for any prime $p$.
$D(mn) = D(m)n + mD(n)$, for any $m,n\in\mathbb{N}$ (Leibniz rule).
$D(-n) = -D(n)$.
The ...
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A sequence based on arithmetic derivative that always converges to prime numbers
For integers $n$, the arithmetic derivative $D(n)$ is defined as follows:
$D(p) = 1$, for any prime $p$.
$D(mn) = D(m)n + mD(n)$, for any $m,n\in\mathbb{N}$ (Leibniz rule).
$D(-n) = -D(n)$.
The ...
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Question About Milnor's proof of the Brouwer's fixed-point theorem
I'm currently going through Milnor's proof of the Brouwer's fixed-point theorem, which I've linked here. I am able to follow the proof up to theorem 2 - afterwards, I've having a bit of trouble ...
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What are the problems in math that are equivalent to fixed point problems?
I am not sure if this question makes some sense. But I have been thinking for a while now. Let us start with simple problems; let us say that we want to solve the following equation for $x$
$$f(x)=y.$$...
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Proving the existence of a fixed-point for the interpolation problem
This is a rephrasing of the original post in (Interpolation problem with varying nodes)
Let $\{f_i\}^{M}_{i=0}$ be a set of real numbers satisfying either $$f_0>f_1<f_2>f_3 \dots$$ or $$f_0&...
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A function $f$ such that repeated application gives you identity? i.e. $f^n(x) = x$ [duplicate]
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 ...
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Why is R-convergence a useful notion?
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-...
