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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If $f$ maps a complete metric space $S$ onto $S$ and $f^2=f \circ f$ is a contraction mapping, show $f$ has a unique fixed point.

If $f$ maps a complete metric space $S$ onto $S$ and $f^2=f \circ f$ satisfies the fixed point theorem given below, show $f$ has a unique fixed point. The following is the fixed point theorem: If f ...
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Given $\frac{dy}{dx}=x^2+y^2$ and initial condition $\varphi (0)=1$, find the first 6 terms in the Taylor expansion solution $y=\varphi (x)$

Given $\frac{dy}{dx}=x^2+y^2$ and initial condition $\varphi (0)=1$, use the method of reduction to an integral equation and successive approximation to find the first 6 terms in the Taylor expansion ...
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Using a fixed point theorem.

Let $x,y \in[0,1] $, consider the following system of equations: $$ ((x+y)/2)^n-x=0 $$ $$ {x^n \over x^n+y^n+1}-y=0 $$ where $ n \in N $ a) Transform the system of equations into equivalent fixed ...
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Show using the fixed point theorem that if $f(x)=\frac{1}{4}[1-x-\frac{1}{10}x^5]$ is defined on $I=\{x|0 \leq x \leq 1\}$ then it has a zero in $I$

The main idea here is to apply the fixed point theorem to $g(x)=f(x)+x$, in order to show that f has a zero in $I$. If $g$ has a fixed point (i.e. $g(x_0)=x_0$), then $f(x_0)=0$. I just don't see how ...
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21 views

Question about function compositions

Let us have $m,n$ positive integers, and suppose, that $ f o f ... f(m$ times$)$ and $f o f... f(n$ times$)$ have an $x$ fix point. For what $(m,n)$ positive integers will it be true, that $x$ is a ...
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73 views

What is the role of fixed point theorems in modern mathematics?

About Fixed Point Theorems, Wikipedia says: Results of this kind are amongst the most generally useful in mathematics. This seems an accurate statement: indeed, there are many journals ...
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41 views

Proof of Banach's homeomorphism theorem without the contraction map principle.

Let $E$ a Banach's space and $X\subset E$ open. The Banach's homeomorphism theorem tells us that if a function $F:X\to E$ is a contraction on $X$ then $(I+F):X\to E$ is a homeomorphism of $X$ onto ...
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48 views

How to get the proper fixed point iteration function?

When we find the approximated root of a function $f(x)$ in an interval $[a,b]$ from the fixed point iteration method, we derive a new function $g(x)$ which has a fixed point as a root of $f(x)$. Is ...
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31 views

So i read that kannan maps are caristi maps, how do I prove it?

Let $(X,d)$ be a complete metric space and $f:X \rightarrow X$ be a function such that $d(f(x),f(y)) \leq k(d(x,f(x))+d(y,f(y)))$ for $k \in[0,\frac{1}{2})$ I have tried to prove the following ...
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60 views

Condition for maximizer of convex combination to be expansion mapping

I have $\Pi_n:\mathbb R^{n+1}\rightarrow \mathbb R$ and $F_n:\mathbb R^2\rightarrow \mathbb R$ with $$F_n(x,a)=\Pi_n(x,...,x,a)$$ $$f_n(x)=\operatorname{ArgMax}_{a\in\mathbb R}\{F_n(x,a)\} $$ such ...
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24 views

Fixed point location for functions

How are fixed points calculated? Are intersections of $ y = f(x) ,y = f^{-1} (x) $ graphs give real fixed points for all $f$ ?
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305 views

Can't there be more than one fixed points in a contraction? or none?

I was going through the contraction mapping theorem in my book where it says, that if $\phi: G\to G$ is a contraction, then $\phi$ has a unique fixed point $\alpha$ on $G$. Sequence {$x_n$}, $x_{n+1} ...
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32 views

Deriving the fixed point for $\omega$ (i.e. $\lambda x.xx$) and proving it to be so

I am studying the simply typed $\lambda$-calculus, and I am struggling a bit with really understanding fixed-points and the $\mathbf Y$ combinator. I have read or skimmed all the questions on here ...
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93 views

Example of a function, so that $g(x)\neq x$

I'm trying to find an example of a function $g:\mathbb{R}\to \mathbb{R}$ (or $g:[1,\infty) \to \mathbb{R}$), so that $$|g(x_1)-g(x_2)|<|x_1-x_2|$$ for all $x_1, x_2\in \mathbb{R}$ ( or $x_1,x_2\in ...
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21 views

Unclear passage of a theorem concerning compact operators (Schauder fixed point theorem)

I'm looking at this proof of Schauder theorem and I am struggling with a passage. This is my problem: Let $X$ be a Banach space, $K \subset X$ a convex, close and bounded set and $F:K \rightarrow ...
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64 views

Which of the followings have a fixed point?

Consider the following sets : $$S=\left\{(x,y)\in \mathbb R^2:x^2+y^2=1\right\}.$$ $$D=\left\{(x,y)\in \mathbb R^2:x^2+y^2\le 1\right\}.$$ $$E=\left\{(x,y)\in \mathbb R^2:2x^2+3y^2\le 1\right\}.$$ ...
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28 views

Atiyah-Bott fixed point formula; signs

In classical paper by Atiyah-Singer on page 16 (or 560) stated formula $(3.1)$. It should give classical Lefschetz fixed-point formula if the operator is $d + d^* : \Omega^{even} \rightarrow ...
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27 views

Noisy contraction mapping

I am trying to describe the behavior of the iterates of a contraction mapping when noise is added. Given a real valued random variable $X_{0}$ a sequence $\{Z_{n}\}_{n=0}^{\infty}$ of i.i.d real ...
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37 views

$f (x)=\exp(x/2)−25x^2$. Show that f on $ (4\log(20), \infty)$ has exactly one root.

Let $$f (x)=\exp(x/2)−25x^2$$ Show that $f$ on $ (4\log(20), \infty)$ has exactly one root $x^*$. (Note that log the natural logarithm) I'm struggling with this question, we were given a hint, ...
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20 views

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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29 views

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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46 views

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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59 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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19 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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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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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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25 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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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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42 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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43 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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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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47 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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22 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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1answer
36 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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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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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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137 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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45 views

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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2answers
59 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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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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85 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 ...
3
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1answer
55 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
262 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
134 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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2answers
55 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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18 views

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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87 views

$f:\mathbb R \to \mathbb R$ 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 ...