# 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.

1,415 questions
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### Stability of a fixed point of a discrete dynamical system

I'm stuck with studying the stability of one fixed point of a discrete dynamical system given in exercise (3) page 44 of Petr Kůrka's Topological and Symbolic Dynamics. Could you please help me? I ...
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### How to explain powers of $(x+1)^{2^n}$ appearing in the Babylonian approximation of $\sqrt x$?

I'm working with this iteration used for approximating square roots and trying to see what I can draw out from it, and in doing so I found something very strange that I can't logically explain. I'm ...
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### System of equations and the Brouwer's Fixed-Point Theorem.

Let's consider the following system of equations: \begin{eqnarray}{ e^{xyz} = \frac{x}{\sqrt{e^{2xyz}+1}}\\ \cos(x+y+z) = \frac{y}{\sqrt{e^{2xyz}+1}}\\ \sin(x+y+z) = \frac{z}{\sqrt{e^{2xyz}+1}} }...
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### Existence of Solution for $Ax+\alpha g(x)=0$

Let $A$ be an $n \times n$ non-singular matrix and $g:\mathbb{R}^n \to \mathbb{R}^n$. I want to prove that for sufficiently small $\alpha$ the equation $\alpha g(x) + Ax = 0$ has a solution. Any ...
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### Fixed point method for non-homogeneous ode and pde

During my course in mathematical methods for physics, my professor introduced us to the "method of fixed point" for studying non-homogeneous systems of odes, odes and pdes. Although I think I ...
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### Intersection of sets in Banach convex space

Let $X$ be uniformly convex Banach space. $f:K\rightarrow K$, such that $\parallel fx-fy\parallel \leq\parallel x-y\parallel\,\,\forall x,y\in K$, with $K$ a nonempty, closed, convex, bounded subset ...
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### How is the triangle inequality being used here?

I'm reading the Banach fixed point theorem (https://en.wikipedia.org/wiki/Banach_fixed-point_theorem) and at one point, this inequality is used: https://puu.sh/AS4uY/78caef5bc6.png However, I'm not ...
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### Example of stable fixed point equation

Schauder's fixed point theorem says that any continuous function $f:K\to{K}$, where $K$ is a nonempty convex and compact subset of a normed linear space $Y$ admits a fixed point. I came across this ...
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### every nonsurjective continuous function from $S^2$ to $S^2$ there exist a fixed point?

can someone please help me to show for every nonsurjective continuous function from $S^2$ to $S^2$ there exist a fixed point? i think since the fuction is not surjective it doesn't contain at least ...
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### Start value for which an iterative function converges

How do i find a start value for which the function: $x = - \exp(x)$ converges. I know how to solve it by making a graph and picking a start value but i don't know how to find the start value for ...
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### how to show an iterative equation converges to a fixed point

let's consider the equation $y'(t) = 16-y^2 = f(t,y)$ for all $t\in [0,1]$ and $y(0) = 0$. How small should the step size $h$ be to ensure the following equation converge for all $i \geq 0$. ...
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### Why application of Sharkovskii's Theorem to Collatz is wrong?

I thought about this years ago but never had it resolved. Sharkovskii's Theorem has a corollary according to Wikipedia that if there exists a 3-cycle on a map on $\mathbb{R}$, then the map has $n$-...
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### Solving multivariable modulo equations?

I am being asked to find fixed points of further iterations of the Cat map, which is defined as: $$C(x,y) = (2x+y , x+y) \pmod 1$$ for $(x,y)$ an element of $[0,1)$ Hence the equation to solve the ...
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### $a_0=0$, $a_{n+1}=a_n-\frac{1}{2}(a_n ^2-a)\rightarrow \sqrt{a}$ using banach's theorem

Let's have a sequence $a_0=0$, $a_{n+1}=a_n-\frac{a_n ^2-a}{2}$. This sequence converges for $0<a<1$ to $\sqrt{a}$. One way of proving this is to prove that $T(x)=x-\frac{1}{2}(x ^2-a)$ is a ...
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### Analysis of function using fixed point

Let $f$ be function definded by $$f(x)=\frac{x^3+1}{3}$$ has 3 fixed point say $\alpha ,\beta ,\gamma$ where $-2 < \alpha < -1$,$0 < \beta < 1$,$1 <\gamma <2$I wanted to show ...
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### If fixed-point iteration has linear convergence, how can Newton's Method have quadratic convergence?

Newton's Method for finding the roots of a function can be considered a type of fixed point iteration of $g(x) = x - \frac{f(x)}{f'(x)}$, since $f(k) = 0 \rightarrow g(k) = k$. But it is well-known ...
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### Prove that there is no fixed point

Let $f: \mathbb R \to \mathbb R$, whereby $f(x):=(x+\sqrt{x^{2}+1})/2$ with $x \in \mathbb R$. Show that $f$ fulfils: $\forall x, y \in \mathbb R$ with $x \neq y$ $d(f(x),f(y))<d(x,y)$, whereby ...
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### Existence of unique fixed point in compact Metric space [duplicate]

Let $(X,d)$ be compact. Show: for a map $f$ that when $\forall x, y \in X$ with $x\neq y$ $d(f(x),f(y))<d(x,y)$ is fulfilled. Then $f$ has a unique fixed point.
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### Banach fixed point theorem ode

I am attempting to do this problem here for studying purposes for an exam I have in a couple months. I was hoping to get some help... For part a) I was trying the following: Given a contraction ...
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### Understanding Selection Theorems

Can someone explain me what is the role of Selection Theorems? Especially; what is the point of the Continuous Selection Theorems for the set valued mappings? Is it just to guarantee a fixed point?
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### How to check for fixed points [closed]

How to check for fixed points in such type of questions
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### Fixed point theorem (At least one fixed point) [closed]

In fixed point theorem, If g is a continuous function g(x) in [a,b] for all x in [a,b], then g has a fixed point in [a,b] i.e. c belongs to[a,b] such that g(c)=c According to this theorem, We have ...
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### Why is Brouwer's fixed point theorem considered a theorem in topology rather than $n$-d real analysis?

Brouwer's fixed point theorem states that if a continuouos function $f$ maps a compact, convex set to itself, then $f$ has a fixed point in that set. All these concepts are topological concepts, ...
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### Geometric Proof of Perron-Frobenius II

The following is proved in these lecture notes. Let $A$ be an $n\times n$ real matrix with all entries positive. Then $A$ has a unique positive eigenvector (up to positve scaling), and the ...
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### General conditions for the Fixed point property on the Quotient topology

Today in my topology class (undergraduate) we introduced the basics for Fixed Point theory for topological spaces (i.e.Fixed point property, retracts, contractability, etc.) and we were discussing the ...
### Common fixed points of two polynomials over $\mathbb{C}$
According to this question (Ritt's classification), given two commuting polynomials $f,g \in \mathbb{C}[z]$, there exists a common fixed point of them. By commuting we mean: $f(g(z))=g(f(z))$. Of ...