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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Verify (without Banach Contraction Priciple), that the function g(x) = 1 + x - (1/8)x^3 has a unique fixed point

I know how to show that there exist a solution by the intermediate value theorem but I'm not sure how to show that the root is unique?
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How do I show that infinite application of this function gives a constant?

I want to show that g(x) returns the same value independent of x and hence is a constant. $$g(x) = \lim_{n \rightarrow \infty}(\underbrace{f \circ f \circ \cdots \circ f}_{n\text{ times}})(x)$$ ...
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Example of contraction mapping

Please give me some examples of contraction mapping on $(C[0,1]), \lvert \lvert \cdot \rvert \rvert_\infty)$ and $(C[0,1],\lvert \lvert \cdot \rvert \rvert_1) $. Note that : 1. $\lvert \lvert f ...
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41 views

If $f$ is a homeomorphism then any periodic point have period less or equal 2

How can one prove the followiong statment? Let $f:[0,1]\to [0,1]$ be a homeomorphism. If $x\in\operatorname{Per}(f)$ then the period of $x$ can't be greater than $2$, i.e, $f(x)=x$ or $f^2(x)=x$.
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Proof of a fixed point theorem on the disk

There is a very nice fixed point theorem which I'd have liked to give to my students : Let $n$, $m$ be two integers larger or equal to one. Let $B_n$ be the open unit ball in $\mathbb{R}^n$, and ...
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18 views

Simple closed curve with non-zero index implies fixed point.

I'm searching for a proof(sketch) of something similar to the following: Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ be an orientation-preserving homeomorphism and let $\gamma: S^1 \rightarrow ...
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43 views

Find the fixed point of $\cos(x)$ (equivalently of $\cos(\cos(x))$) restricted to $[0,\frac \pi 2]$.

I've proved that $\cos(\cos(x))$ restricted to $[0,\frac \pi 2]$ is a contraction, which imply by Banach's fixed point theorem that it has a unique fixed point on this interval. I've also proved ...
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19 views

Suppose $S^m$ is a contraction on a complete metric space $(X,d)$. I want to show that this implies $S$ has a unique fix-point. [duplicate]

Let $(X,d)$ be a complete metric space and let $S: X \rightarrow X$ be a mapping. Suppose there exist $m \ge 1$ such that $\underbrace {S^m = S \circ S \circ \dots \circ S}_{\text {m times}}$ is a ...
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24 views

A problem about fixed points

We got a real valued continuous function $g$ which is defined on $[0,1]$, $g(0)>0$, $g(1)=1$ and $g'(x) > 0$, $g''(x) > 0$ on $(0,1)$. We need to prove that if $\lim_{x \to 1}g'(x) > 1$, ...
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73 views

Lotka-Volterra First Integral and Fixed Point

I have the following problem that I am dealing with, quite a long time, I must say. Let us assume that we have a predator-prey, Lotka-Volterra system given to us by: \begin{align} & ...
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35 views

Relationship between the Borsuk-Ulam theorem, Brouwer's fixed point theorem (for the ball) and Tucker's lemma

Which of these - the Borsuk-Ulam theorem, the Brouwer's fixed point theorem (for the ball) and Tucker's lemma implies which? I'm a little confused with this. I suspect they may be equivalent. If that ...
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1answer
33 views

Fixed Point Convergence. Finding the interval for which the iteration converges.

I've solved the first part. I think I have something for the second part, but I'm unsure. A) You are given the fixed point problem x=Ax^2 where A>0 is a constant. Compute positive fixed point of the ...
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27 views

Fixed point involved ODE

Vector field $v: R^n \to R^n$ is smooth, and $x\cdot v(x)\geq 0$ when $|x|=1$. Then consider the ODE: $$\dot{x}(t)=-v(x(t)) \ \ t\geq 0 $$ $$x(0)=y$$ For $t>0$ fixed, the map $y\mapsto x(t,y)$ ...
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46 views

Existence fixed point

Let $f: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^n$ and $g:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and compact valued. Consider the function $F: \mathbb{R}^n \times ...
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20 views

Clarification of a passage in Leray-Schauder theorem's proof

I'm looking at the proof of Leroy-Scauder theorems's. This is the statements: If $X$ is a Banach space, $K \subset X$ a convex, close and bounded set, $F:K \rightarrow K$ compact then $F$ has a ...
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33 views

A sufficient condition for the existence of a fixed point for a continuous function.

How did the author use the intermediate value theorem to prove that period $k$ implies period $1$? Please, see the image which explains every thing. The definitions are in the first paragraph. The ...
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14 views

Derivative of a second-iterate map

I have a homework problem I'm working on about the discrete logistic equation: $f(x)=rx(1-x)$ So far, through some experimentation and polynomial division I've dtermined that the fixed poits of ...
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1answer
28 views

Complete metric space, fixed point and ?reverse? fixed point theorem.

Let $(X,d)$ be a complete metric space, let $F: X\rightarrow X$ such that $$\exists L > 1, \forall (x,y)\in X^2, d(F(x),F(y))>L\cdot d(x,y).$$ Show that if $F(X)=X$ then there is exactly one ...
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Fixed point combinator (Y) and fixed point equation

In Hindley (Lambda-Calculus and Combinators, an Introduction), Corollary 3.3.1 on fixed point combinator. In $\lambda$ and CL: for every $Z$ and $n \ge 0$ the equation $$xy_1..y_n = Z$$ can be ...
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67 views

example of a monotone non-continuous map.

Let me start by defining some terminology to be sure I made no errors there. Parts of this are translated freely from my mother tongue so feel free to correct terminology or the definitions themselves ...
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134 views

Use the Contraction Mapping Principle to show that $x=\frac19\sin\left(3x\right) + \sqrt{x}$ has exactly one solution $x\geqslant\frac{8}{9}$

Use the Contraction Mapping Principle to show that $x=\frac19\sin\left(3x\right) + \sqrt{x}$ has exactly one solution $x\geqslant\frac{8}{9}$. I have literally no idea if this is right, please ...
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Existence of a fixed-point free map in a manifold.

I'm having some to proof a question. I have to show that a compact manifold that admits a nowhere vanishing smooth vector field has a smooth map fixed-point free homotopic to the identity map. I know ...
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58 views

Fixed point and extrema

Let $\varphi_{a,b}:\mathbb{R}\ni x \mapsto \cos(ax+b)\in \mathbb{R}$. Show that for every $(a,b)\in (-1,1)\times\mathbb{R}$ there exist exactly one fixed point $s(a,b)$ of $\varphi_{a,b}$. If it is ...
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1answer
32 views

Limit of iterates of discontinuous functions

Suppose I have a function $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ and want to consider the iterates $$f^{(m)}(x_0) = f(\cdots f(f(x_0)))$$ ($m$ times) for some initial point $x_0\in\mathbb{R}^n$. ...
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2answers
80 views

Finding a Möbius Transformation given constraints

I am trying to solve this problem, but am running into very complicated solving, and think that there is a simpler approach that I am missing. Find a Möbius transformation $M(z)$ that satisfies ...
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1answer
53 views

Why does the fixed point theorem hold for every lambda term?

Can someone give a clear and simple answer for why the fixed point theorem holds for every $\lambda$-term, in contrast with the fact that not all numerical function have a fixed point?
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3answers
259 views

Fixed point and fractional iteration: if $F(k)=k$ then $F^{1\over n}(k)$ is another fixed point of $F$

My knowledge of the fixed points and iteration equals zero, same for the notation and terminology but I really need to know if this deduction has trivial errors or is really as nice as it seems. I ...
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1answer
90 views

Blackwell's condition for a contraction: Why is boundedness neccessary?

I'm trying to understand the proof that certain operators $T$ are a contraction if they fulfill Blackwell's sufficient conditions. In particular, I try to understand why the operator $T$ has to map ...
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53 views

find out fixed point of a function

Using mathematical calculus, I wand to determine all the fixed points of the function $f^3$ where $f$ is given by: $$ f:[0,1]\rightarrow[0,1];\;f(x)=4x(1-x) $$ and such that those fixed points are not ...
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Fixed point free Involution over topological space with infinite connectivity

Is there a topological space with infinite connectivity with fixed point free involution over it?
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$f:\mathbb R \to \mathbb R$ is a differentiable function such that $f'(x)\le r<1 $ , does $f$ necessarily have a fixed point ? [duplicate]

Let $f:\mathbb R \to \mathbb R$ be a differentiable function . If $\exists r \in \mathbb R $ such that $|f'(x)|\le r<1 , \forall x \in \mathbb R$ then using Lagrange's theorem one can show $f$ is a ...
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eventually constant maps

Let $f:[0,1]\to [0,1]$ be a continuous function with a unique fixed point $x_{0}$ Assume that $\forall x\in [0,1], \exists n\in \mathbb{N}$ such that $f^{n}(x)=x_{0}$. Does this implies ...
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2answers
88 views

Banach Fixed Point Theorem problem contradiction

I have a following problem. Let $X = R$, $d(x,y) = |x-y|$, $T(x) = \sqrt{x^2 + 1}$ but $T$ does not have a fixed point. Does this contradict Banach's Fixed Point Theorem? I know that if $X$ ...
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56 views

When does a continuous function defined on a non-compact closed and bounded convex set has a fixed point?

Is there any result in fixed point theory which will give the existence of a fixed point for a continuous function defined on a non-compact, closed and bounded convex set?
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An equation with multiple solutions: finding the maximum of the function of the solutions.

Possibly, this is a bad (stupid) question, but sometimes some discussion helps. I have a fixed point equation (involving $\tanh$). I would like to derive the dependency of some function of the fixed ...
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A problem similar to Banach fixed point theorem

a) Let $(X,d)$ be a complete metric space and let $T: X \to X$. Prove that if there exists a natural $n$ such that $T^n(x)$ (composition of $T$ $n$ times) is a contraction then $T(x)$ has a unique ...
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1answer
40 views

Metric Geometry determining fixed points

Let $f\colon\mathbb{R}\to\mathbb{R}$ be given by $f(x)= e^{-x}$. Show that $f$ has a fixed point and determine what it is.
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Proof that a continuous function from the unit ball to itself without fixed points implies existence of retract from unit ball to unit sphere

Assume $f:B_{1}\to B_{1}$ (where $B_{1}$ is the closed unit-ball in $\mathbb{R}^{n}$ ) is a continuous function that has no fixed points I need to construct a function $g:B_{1}\to B_{1}$ which ...
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86 views

Use the Mean Value Theorem to show that if $|f'(x)| ≤ C<1$, then $f$ has at most one fixed point

Use the Mean Value Theorem to show that: if $|f'(x)| ≤ C < 1$ $\forall x$, then $f(x) = x$ has at most one solution. So using the Mean Value Theorem I know that $$-1<-C\leq ...
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Maximum diameter for the preimage of a point when the degree is not 1 or -1

When the degree of a map $S^n \rightarrow S^n$ isn't 1 or -1, are there always two points that map to the same point and whose distance from each other can be bounded below by a function of the ...
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prove that $x \mapsto \mathrm e^{-x}$ has a unique fixed point on R

Can anybody prove $x \mapsto \mathrm e^{-x}$ has a unique fixed point on R using the fixed point iteration theorem?
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31 views

Characterizing conditions for $\tanh{(kx-b)}=x$ to have 1/2/3 fixed points.

I am trying to understand what are the conditions for $\tanh{(kx-b)}$ to have 1 or 2 or 3 fixed points. That is I am trying to characterize conditions on $k$ and $b$ for which equation ...
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1answer
45 views

contractive and find its limit

The real sequence $\{P_n\}$ is defined as $p_1$=2, $p_{n+1}$=$\frac{2}{1+p_n}$, n $\in N$. prove that $\{p_n\}$ is contractive, deduce that it converges, and find its limit. first we need to show that ...
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Corollary of Banach fixed-point theorem

Let $(X, \left\lVert\cdot\right\rVert)$ be a Banach space. Let $A:X\to X$ be a linear map and $\nu\in \mathbb{N}$ such that $A^k:X\to X$ is a contraction for every $k>\nu$. Is it true that for ...
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1answer
21 views

Undamped Forces

I want to make sure I am doing this problem correctly, especially when it comes to drawing the potential function V(x). Consider the system of differential equations: $$\dot {x}=y$$ $$\dot ...
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71 views

The set of all fixed points of a continuous function $f:[0,1] \to [0,1]$ , satisfying $f \circ f=f$ , is a non-empty interval?

Let $f:[0,1] \to [0,1]$ be a continuous function such that $f \circ f=f$ on $[0,1]$ , then is it true that the set $\{x \in [0,1] : f(x)=x \}$ is a non-empty interval? I can show that it is ...
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1answer
39 views

give a counter example that $T^n$ is contraction will not imply that $T$ is contraction.

Let $T$ be a contraction map then $T^n$ is contraction.. We can prove this result by induction on n.. But the converse is not true... help me to give a counter example that $T^n$ is contraction will ...
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1answer
42 views

Fixed Point Iteration - Numerical Analysis

please help me solve the following question. Qsn: Solve the following system by Fixed Point Iteration. $$ x^2-2x+y^2-2y=3$$ $$x+y=-1$$ Progress: So I know that we have to assume one of the ...
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38 views

When does a continuous function defined on a closed and bounded convex set has a fixed point?

For a function $f$ defined from a domain $K$ to itself, we have a point $x$ in $K$ is said to be a fixed point of $f$ if $f$ maps $x$ to itself. When the domain K is a compact convex set with some ...
3
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1answer
89 views

Holomorphic function with a unique fixed point

Let $\omega \subset \mathbb C$ be a simple connected set and $\,f:\omega \to A$ is an analytic function where $A \subset \omega$ is compact. Show that $f$ has an unique fixed point. I think we can ...