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, ...

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Fixed-Point Applications

Let $1<a<e^{1/e}$, and define $f(x)=a^x$ (b) Show that if $x_1$ is any point in the interval $(1,e)$ and $p$ is a fixed point of $f(x)$ in the interval $(1,e)$, then $$|f(x_1) - p | <|x_1 - ...
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“Comparing” fixed point Theorems.

In class, I saw Banach's (Picard) fixed point theorem: Given a complete metric space and a contractive mapping, it admits a unique fixed point. And Brouwer's: Given a continuous function in ...
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Free $\mathbb{Z}_{2}$ action on the plane

Motivated by the following question we ask: Is there a free action of $\mathbb{Z}_{2}$ by homeomorphism on $\mathbb{R}^{2}$? Lie groups with no free $\mathbb{Z}/2\mathbb{Z}$ action
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Application of Banach Fixed Point Theorem to sequences

I have the following question Show that there is unique real bounded sequence $(a_n: n \in \mathbb{N})$ such that $$a_n = \frac{n+1}{n}+\sum^\infty_{m=1}\frac{\sqrt{3a^2_{m+n}+1}}{4m^2}$$ for all $n ...
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What is the relationship between eigenvector and computing PageRank?

I read several papers about PageRank and didn't get stuck in understanding the idea of PageRank because it is simple, but I got stuck in computing PageRank, those papers talk about some mathematical ...
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Every finite group of congruences of the $n$-dimensional Euclidean space has a fixed point

Is there an "absolute" proof of the fact that if $G$ is a finite group of congruences of the $n$-dimensional Euclidean space, then $G$ has a fixed point?
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Second order DE with Banach fixed point theorem

Let's consider such a problem: $$\left\{\begin{matrix} &u''-u=f(u)+g(x) & \\ &u'(0)=u'(1)=0 & \end{matrix}\right.$$ for $x\in(0,1)$, where $u=u(x)$ isn't known and $f \in ...
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Fixed Point Iteration Method - Starting Point

If $g(x):=21^{1/2}x^{-1/2}$, then $21^{1/3}$ is a fixed point of $g$. Question: Using the Fixed Point Iteration Method, are there conditions on the starting point $x_0$ in order for the method to ...
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Prove that there is a unique $\phi \in C([a,b])$ such that $\phi (x) = A(x) + \int_a^xK(x,y)\phi(y)dy$

Contraction Map Question I was reviewing another question on here (linked above), and I wanted to expand on it with another question (I hope this is the correct way to do such a thing). Here is my ...
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Set of all contraction maps

Consider $(X, d)$ be a compact metric space, and let $Con(X)$ denote the set of all contraction maps on $X$. We shall define the “distance” between two maps $f, g ∈ Con(X)$ as follows, $$d_{Con(X)}(f, ...
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Contractive composite function

A question arises when we are dealing with a function that is not known to be contraction but the composites of that function to itself is a contraction (it is called eventually contractive). More ...
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Is fixed point property a topological property?

Is fixed point property a topological property? I already came up with some examples, and think the answer may be yes. But I don't know theoretical proof to get that.
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Fixed point in a special continuous mapping

Show that a continuous mapping $f:[0,1]\to [0,1]$ which satisfies $f(f(x))=x$ for each $x \in [0,1]$, and for which $f(x)$ does not equal x for at least one x in [0,1], must have exactly one fixed ...
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Contraction Map (Fredholm integral type)

I have been stuck on this problem for a little while. I think the proof might be similar to that of proving the Fredhom integral of the second type, however I am not sure. I am a little confused on ...
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How can I find the roots of x^2+px+q=0 by using contractive mapping theorem?

Basically, Contractive mapping theorem guarantees that if F is contractive on some interval, then there exists a unique fixed point. However, $x^2+px+q=0$ has two different solutions as long as ...
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Continuous mapping and fixed points

Does a continuous mapping $f\colon \mathbb R \to \mathbb R$ which satisfies $f(f(x))=x$ for each $x \in \mathbb R$ necessarily have a fixed point?
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Diffeomorphism on $S^2$ to $S^2$ with at least two fixed points

Let $f:S^2\to S^2$ diffeomorphism orientation-preserving such that for every fixed point of $f$, $1$ is not an eigenvalue of $df_p$. Then $f$ admits at least two fixed points. I'd to ...
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Convergence of autonomous system / time-scale separation

Let $x \in \Delta_n$, where $\Delta_n = \{u \in \mathbf{R}^n_{\ge 0} \mid \sum u_i = 1\}$ is the probability simplex. Suppose that I have an (autonomous) dynamical system $$ \dot{x} = A(x) \cdot x $$ ...
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Convergence of fixed point iteration when $g'(p)=1$.

I am dealing with a function $f(x)=e^{-\frac{1}{x^2}}$, which has a root $p$ of infinite multiplicity at 0. I am struggling with the convergence rate of the resulting standard Newton fixed point ...
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Inverse function theorem - good proof

I am looking for a reference which give a full demonstration of the inverse function theorem (let say in Banach spaces) where we can have estimates of the bounds of the neighbourhoods that we build to ...
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Is this map defined from an ODE a contraction?

Let $f:\mathbb{R}^+\to \mathbb{R}$ be a $T$ periodic function, i.e. $f(t+T) = f(t)$ for all $t\in \mathbb{R}^+$. Let $\sigma\in \mathbb{R}^+$. Define a weighted $\mathcal{l}_2$ norm, $\left\| \cdot ...
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Fixed point problem with matrix

I am looking for guidance for the following fixed point problem. $A$ is an $n\times n$ matrix. The rows of $A=(A_{i})_{i\in N}$ are in the set $A_{i}\in\mathcal{A}$, where $\mathcal{A}$ is the set ...
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Fixed Point equivalence proof

Given two finite-state LTSs $L$ and $M$ (where $M$ is a "specific" subgraph of $L$) and the Hennessy-Milner-Logic monotone interpretation function $\Theta_F(S)$, where $F$ is the HML formula and $S$ ...
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If $X$ is a compact space and $A:X\rightarrow X$ is such that $\rho(Ax,Ay)<\rho(x,y)$, $x\neq y, \Rightarrow A $ have only one fix point on X.

I have problems with this demostration can anybody help me please? If $X$ is a compact space and $A:X\rightarrow X$ is such that $\rho(Ax,Ay)<\rho(x,y)$, $x\neq y, \Rightarrow A $ have only one ...
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Compare two fixed points component-wisely

Tarski fixed point theorem conclude that a isotone from a complete lattice to itself have a largest fixed point. Now suppose I have two isotone on defined on the same lattice, and one isotone is ...
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Polynomials in two variables over $\mathbb{F}_p$ fixed by $\operatorname{SL}_2(\mathbb{F}_p)$

Let $A=(a_{ij})\in\operatorname{SL}_2(\mathbb{F}_p)$. Consider the ring map $A:\mathbb{F}_p[x,y]\to\mathbb{F}_p[x,y]$ defined by $$A(x)=a_{11}x+a_{21}y$$ $$A(y)=a_{12}x+a_{22}y$$ and extended ...
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Prove that the equation $\cos x - x -1/2 = 0$ has a unique real solution

Prove that the equation $\cos x - x -1/2 = 0$ has a unique real solution. My solution: I was starting with a function $F(x) = \cos x - 1/2$ and the interval $[0,\pi/4]$ and trying to show that the ...
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3 fixed points proof

The problem: Suppose $g(x)$ is continously differentiable and has exactly 3 distinct fixed points $r_1 < r_2 < r_3$ with $\lvert g \prime (r_1) \rvert = .5$ and $\lvert g \prime (r_3) \rvert = ...
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Proof that the function on $C[0,b]$ is a contraction

Question : Let $a$,$b$ be real numbers with $0\lt b\lt 1$ . Consider the subset $X\subset C[0,b]$ consisting of the functions $f$ s.t $f(0)=a$ .Then $X$ is closed in $C[0,b]$. ...
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Given $y_0\in(0,b)$ and $y_{k+1}=\frac{1}{2}y_k(3-y_k^2)$ converges. Find $b$.

Given $y_0\in(0,b)$ and $y_{k+1}=\frac{1}{2}y_k(3-y_k^2)$ converges to $1$. Find $b$. I know the sequence converges quadratically. But I have no idea how to find $b$.
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Finding roots by Fixed Point Iteration

How to know or how to find the root of the equation by Fixed Point Iteration? In FPI is there any definition/theorem of when root exists? Or is it correct that when ...
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Help me find a fixed point.

I want to find a fixed point of $1/(x+1)$. I set: $p = 1/(p+1)$ $p^2+p-1=0$ $p= (-1\pm\sqrt(5))/2$ But I plug in $f((-1 + \sqrt(5))/2)$ and $(-1 - \sqrt(5))/2$ but I don't get the same output to ...
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If $G<S_n$ is transitive, calculate $1/|G| \cdot \sum_{g \in G} f(G)$

$G<S_n$ is transitive calculate $1/|G| * \sum_{g \in G} f(g)$ where $G<S_n$ and $f(g) = |\{ 1 \le i \le n | g(i) = i \}|$ I tried to use the orbit stabiliser theorem but didn't get anywhere ...
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If $f:S^2\to S^2$ is homotopic to the identity does it have a fixed point? [duplicate]

Question: Let $f:S^2\to S^2$ be a continuous map that is homotopic to the identity. Does $f$ necessarily have a fixed point? I thought about this question after learning the proof of Brouwer ...
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Fixed point property for the total space of the canonical line bundle over $\mathbb{C}P^{2n}$

It is well known that the even dimensional complex projective pace $\mathbb{C}P^{2n}$ has the fixed point property. What about the total space of the canonical line bundle over $\mathbb{C}P^{2n}$? ...
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Finding the condition number of an iterative method.

I'm trying to find the condition number on the function $A$ for the iterative method below however I'm struggling to begin. $$p_{n+1}=p_n-A(p_n)\frac{f(p_n)}{f'(p_n)}=g(p_n)$$ In particular, the ...
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Example of Measure of non-compactness?

I can't understand the following example of measure of non-compactness, which was given in a research article. Definition: A nonnegative function $\phi$ defined on the bounded subsets of $X$ will ...
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How does banach fixed point theorem related to matrix analysis?

As stated by Banach fixed point theorem, a contraction mapping has only one fixed point. In plain words it means that the contraction mapping T has only one solution that satisfy $Tx = x$. A ...
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Fixed point iteration, finding g(x)

I have struggle on finding this function g(x). Assume function $f(x) = 5x^3 -20x + 3$ and it is specified to find root in [0, 1]. So I guess, first thing is to find function g(x). $$g_1(x) = ...
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Does the fixed point in the Brouwer's Fixed-Point Theorem have to be an interior point?

As the question suggests, does the fixed point in the Brouwer's Fixed-Point Theorem have to be an interior point? Many thanks in advance. To be clear, I'm using the statement of Brouwer's Fixed-Point ...
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Exam question on fixed point iteration

I am solving the following exam problem. Problem: An iterative scheme is given by $$ x_{n+1}= \frac{1}{5}\left(16-\frac{12}{x_n} \right).$$ Such a scheme with suitable initial approximation $x_0$ ...
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A question on the Banach Contraction Mapping Principle

The BCMP states that in a complete metric space $X$, a contraction mapping $T$ on $X$ has a unique fixed point, i.e. if $T$ satisfies $d(Tx, Ty) \le k d(x,y)$ such that $0 \le k < 1$, then $T$ has ...
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If $d(f(x), f(y)) \leq kd(x, y)$ for all $x, y \in X$. Then $f$ is continuous and has exactly one fixed point

Let $d$ be a complete metric for $X$. Let $f: X \to X$ be a function. Suppose there is a number $k$, with $0 < k < 1$, such that $d(f(x), f(y)) \leq kd(x, y)$ for all $x, y \in X$. Then $f$ is ...
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If two continuous maps of an interval commute, then they agree at some point

Let $f,g:[0,1] \rightarrow [0,1]$ be continuous functions such that $f\circ g =g\circ f$. Prove that there exists $x \in [0,1]$ such that $f(x)=g(x)$
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Clarification on the difference between Brouwer Fixed Point Theorem and Schauder Fixed point theorem

From Zeidler's Applied Functional Analysis Brouwer The continuous operator $A:M \to M$ has a fixed point provided $M$ is compact, convex, nonempty set in a finite dimensional normed space ...
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Can some help me understand Zeidler's intuitive proof of Brouwer Fixed Point theorem

On pg53, Zeidler gives the Brouwer's Fixed Point Theorem The continuous operator $A: M \to M$ has a fixed point provided $M$ is a compact, convex and nonempty set in a finite dimensional normed ...
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Problem in Banach Fixed Point Theorem for a functional equation

I was recently presented this within the context of topological spaces: I am asked to show that there exists a unique continuous function $ f\colon \left[0,\frac{1}{2}\right] \rightarrow \Bbb R $ ...
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$A^2$ has a fixed point implies $A$ has also a fixed point

Let $X$ be a Banach space and $A:X\to X$ a bounded operator. Assume that $A^2$ admits a fixed point in $X$ i.e. there exists $x_0\in X$ such that $A^2x_0=x_0$. Does this mean that $A$ has also a fixed ...
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A question about fixed-point iteration sequence of a two times continuously differentiable function

I am stuck at this problem: Let $g:[a,b]\to[a,b]$ be a 2 times continuously differentiable function that satisfy: for all $x\in [a,b]$, $g''(x)\neq 0$ And let $s$ be an arbitrary fixed-point of ...
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If $f$ is Lipschitz continuous on a closed interval $[a,b]$ such that $f([a,b])\subseteq [a,b]$ then it has a unique fixed-point

I am stucked at this problem: Prove or give a counter-example for the following sentence: If $f:[a,b]\to\Bbb{R}$ is Lipschitz continuous on a closed interval $[a,b]$ and $f([a,b])\subseteq [a,b]$ ...