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 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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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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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
86 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
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
264 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
153 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
57 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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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 ...
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187 views

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
93 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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1answer
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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1answer
61 views

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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2answers
62 views

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
41 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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1answer
68 views

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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88 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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105 views

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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2answers
54 views

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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1answer
32 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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73 views

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
22 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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1answer
73 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
47 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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51 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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40 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 ...
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1answer
92 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 ...
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1answer
29 views

Proving Theorem: subspace of polynomials of degree two or less?

How can I prove that the set $S$ of polynomials of degree $2$ or less, whose coefficients sum to zero, is a subspace of all polynomials with degree $2$ or less? I know I need to show that $a+b+c=0$ ...
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Why use the Lefschetz Zeta function?

Given a compact, triangulable space $X$ and a continuous function $f: X\rightarrow X$, then we define the Lefschetz number $\Lambda_{f}$ by $$\Lambda_{f} = \sum_{k\geq0}(-1)^{k}Tr(f_{\ast}\vert ...
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1answer
82 views

Unique Fixed Point for an Analytic Function

If $\Omega \subset \mathbb{C} $ is open and connected and $ f:\Omega \to \mathbb{C}$ is an analytic function, such that $\,\,K=\overline{\text{Im}\, (\,f)} \subset \Omega$ is a compact set. Prove that ...
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1answer
76 views

Proof of Tarski's self-reference lemma

In http://www.math.hawaii.edu/~dale/godel/godel.html, Tarksi's self reference lemma is mentioned but the proof is omitted. Tarski's Self-Reference Lemma. For any formula $p(x)$ in an adequate ...
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3answers
64 views

What are the conditions to apply Brouwer fixed-point theorem (one dimentional case)?

Does this theorem work for a continuous function $f : ]a,b[ \rightarrow ]a,b[$ with $a, b \in \mathbb{\bar{R}}$ ? Thanks in advance.
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1answer
121 views

Determine $p$, $q$ and $r$ so that the order of the fixed point iteration for computing $a^{1/3}$ becomes as high as possible

So I'm given the following equation for computing $a^{1/3}$ $$x_{k+1}=px_k + \frac{qa}{x_k^2} + \frac{ra^2}{x_k^5}$$ and I have to find the p, q, and r so that this equation converges to $a^{1/3}$ as ...
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57 views

Nullclines for differential equations

Consider the system of differential equations $$\dot {x}=y-x^2$$ $$\dot {y}=x-y$$ a. Determine the fixed points (1,1) (0,0) b. Determine the nullclines and the signs of $\dot {x}$ and $\dot {y}$ ...
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1answer
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Only one fixed point for $f:\bar{\mathbb{D}}\rightarrow\bar{\mathbb{D}}$ on the boundary.

We know for Brouwer theorem that $f$ (continuous bijective function) have a fixed point. My questions are: 1) Is there a function with only one fixed point $x_0\in Int(\bar{\mathbb{D}}) $ (open ...
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1answer
34 views

Differential equations, stability of fixed points

Consider the differential equations: $$\dot{x}=x^2-9$$ $$\dot{x}=x(x-1)(2-x)=-x^3+3x^2-2x$$ a. Find the stability type of each fixed point. (I am not sure about the stability of the points. Do I ...
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1answer
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Fixed Point Equivalences of Axiom of Choice

Axiom of Choice has many known equivalences. Also there are many known fixed point theorems (unproved statements) which provide useful information about existence of fixed points for particular ...
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1answer
51 views

Contraction vs. iterated function convergence

Let $X$ be a Banach space with norm $|x|_X$. (For example $X=\mathbb R$) We assume that a function $F: X \rightarrow X$ is Lipschitz continuous but not a contraction map, hence: $$ | F(x) - F(y) |_X ...
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1answer
38 views

Computing the fixed point for $cosx$

While studying about Compiler Design I came with the term 'fixed point'.I looked in wikipedia and got the definition of fixed point but couldn't get how fixed point is computed for $cosx$ as said in ...
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1answer
31 views

Fixed Point Summary

I know someone has given me resources for this before but I can't seem to find them... Would someone please summarize stable vs unstable, attracting vs repelling, and node, saddle,etc fixed points? I ...
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1answer
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Fixed points and stability of them

Find the fixed points and classify them for the system of equations: $$x'=v$$ $$v'=-x+wx^3$$ $$w'=-w$$ $$0=v$$ $$0=-x+wx^3$$ $$1=wx^2$$ $$0=w$$ is the only fixed point (0,0,0)?? jacobian: ...
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1answer
90 views

System of differential equations, phase portraits and stability of fixed points

Consider the system of differential equations: $$x'=-x-y+4$$ $$y'=3-xy$$ a. Find the fixed points. $x'=-x-y+4$ $x+y=4$ $x+3/x=4$ x=3,x=1 $y'=3-xy$ $y=3/x$ fixed points: (1,3), (3,1) b. ...
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77 views

History of fixed point theory

I am looking for encyclopedic references for fixed point theory and its applications. What is the best reference for this subject? thank you.
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66 views

System of differential equations, phase portrait

Consider the system of differential equations: $$x'=-2x+y-x^3$$ $$y'=-y+x^2$$ a. Determine the fixed points. (Am I correct in thinking that to determine the fixed points, I must set x' and y'=0? I'm ...
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1answer
83 views

Show that operator T is a contraction mapping

I want to check whether the operator T defined as: $Tf(x) = \beta \left\lbrace \sum_{\theta} \mu_\theta \left[ h_\theta(x) + \int f(x') Q_\theta(x,dx') \right]^\alpha \right\rbrace^{1/\alpha} $ is a ...