# 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.

1,413 questions
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### Betti sum, cup-length, Lusternik-Schnirelmann category, and critical points

I am trying to make some order in the notions of cup-length, sum of Betti numbers, LS category, critical points of functions. Let $M$ be smooth closed compact manifold. We denote by Crit($M$) the ...
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### Bannach Fixed Point Theorem for rotation and scaling the punctured unit disk

I'm trying to show that a weak contraction is not sufficient for the Bannach Fixed point theorem. The best example I could think of is the function defined on the punctured unit disk to itself given ...
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### Show that $\sqrt{x+2}$ is a contraction

Let $f:X \rightarrow X$ where $X=[0,\infty)$ be defined as $f(x)=\sqrt{x+2}$. I have to show that this mapping is a contraction and find its unique fixed point. The second part is easy: by the CMT, it ...
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### Fixed points of Banach.

Let $S,T : M \hookleftarrow$. By definition. Let $M$ a set. A fixed point of an application $A:M\hookleftarrow$ is an element $\xi \in M$ satisfying $A(\xi)=\xi$. If $TS=ST$ (commute), then what ...
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### Fixed point itteration

Let $x_{n+1}=\sqrt{2+x_n}$ with $x_0=0$ and $y_{n+1}=\sqrt{2+y_n}$ with $y_0=2018$. Do those two sequences have the same limit? I think the answer is "Yes", but I am not really sure. Is my argument ...
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### Convergence analysis for Solving quadratic matrix equation $\mathbf{XBX=A}+\lambda \mathbf{X}$

I seek for a symmetric positive definite (PD) solution $\mathbf{X}$ for the following quadratic matrix equation: \begin{array}{cc} \mathbf{XBX=A}+\lambda \mathbf{X} \tag{1}\label{eq1} \end{array} ...
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### How does PageRank deal with nodes that do not have out-links?

I will use the notation that $A_{ij}=1$ if an arrow exists from $j$ to $i$ and otherwise zero. Just to avoid confusion I use in brackets the standard convention $B_{ij}=1$ when $i$ has a directed ...
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### Proof of fixed point theorem on $D^1$ using the technique used in Brouwer's fixed point theorem.

Let $f : [-1,1] \longrightarrow [-1,1]$ be a continuous function. Then using IVT I have proved that it has a fixed point. Now my question is "Can I prove this result by the technique used in Brouwer's ...
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### Existence of infinite iteration of functions $f_\infty$?

Given a sequence of functions $\{f_n\}$ satisifying an iterated relation such as $f_n(x)=g(x+f_{n-1}(x))$ $f_n(x)=g(xf_{n-1}(x))$ $f_n(x)=g(x/f_{n-1}(x))$ Where $g:=f_1$ is ...
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### Application of Banach contraction principle

Define $T:\mathbb R^3→\mathbb R^3$, $(x,y,z)\mapsto\left(\dfrac12\cos y +1,\dfrac23\sin z,\dfrac34x\right)$. I have checked that this example is a contraction and now I am trying to apply Banach ...
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### initial valued problem

Let $F : [0,a] \times [b-r,b+r] \rightarrow \mathbb{R}$, $a, b, r$ fixed real numbers, satisfying (1) for each $k \in [b-r,b+r]$, $g_k : [0,a] \rightarrow \mathbb{R}$ defined by $$g_k(t) = F(t,k)$$ ...
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### On convergence of a fixed point iteration

I'm stating the problem that I'm stuck with, along with the progress that I've made. Problem : The iteration defined by $x_{k+1}=\frac{1}{2}\left(x_k^2+c\right)$, where $0<c<1$, has two fixed ...
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### Motivation for Banach Space in Global Picard-Lindelöf

I'm looking for the motivation of the core idea to a proof of the global existence version of Picard-Lindelöf. Global Picard-Lindelöf. Let $f:[a,b] \times \mathbb{R}^n \to \mathbb{R}^n$ be ...
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### What sequences in $\Bbb Z[\frac16]$ converge to the fixed point $0$ in $\lvert\cdot\rvert_2$ while being fixed in $\lvert \cdot\rvert_3$?

Suppose some uniformly continuous (under the 2-adic topology) function $f:\Bbb Z[\frac16]\to\Bbb Z[\frac16]$ converges under composition with itself to $0$ (in the 2-adic topology) for all inputs, and ...
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### On applications of the Lefschetz fixed point theorem

I would like to know a little bit about practical applications of the Lefschetz fixed point theorem. I am specially interested in knowing about possibple applications for the study of formal and ...
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### Proving the Bourbaki–Witt Theorem using Recursion.

I am trying to prove the Bourbaki–Witt theorem, which states: Let $\left(X,\leq\right)$ be a partially ordered set and let $f:X\rightarrow X$ be a function. Suppose that $f\left(x\right)\geq x$ ...
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### How to prove the existence of these numbers?

I have to prove that there are two numbers $x_1,x_2 \in (-1,1)$ that are the unique solution of the system $\{3x=\sin(x+y+13), 3y=\cos(x+y)\}$. Considering $x_1,x_2 \in (\frac{-1}{3},\frac{1}{3})$ we ...
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### Coloring the vertices of a decagon

In how many ways can you color, up to symmetry, the vertices of a regular decagon using $q$ colors? (We are talking here about the dihedral group of order $20$). So I was thinking about maybe group ...
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### Continuously chosen points on lines in $\mathbb{R}^{n}$

On each line $l \subset \mathbb{R}^n, n \geqslant 2$ passing through $0$ we choose a point $a(l)$ such that $a(l)$ depends continuously on $l$. I have to prove that there exists a line with $a(l)=0$. ...