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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Brouwer's fixed point theorem (for unit balls) and retractions
Let $X$ be a generic Banach space over $\mathbb R$ (also infinite dimensional), and let $B$ be the unit ball in $X$.
I want to prove that the following proposition
$B$ is a fixed-point space if ...
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1answer
52 views
How to prove that the following iteration process converges?
I have the following iteration process: $$ p_{n+1} = \frac{{p_{n}}^3 + 3 a p_{n} }{3 {p_{n}}^2 + a } , $$ where $a > 0$.
Q1: How to prove that this iteration process converges for every number ...
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25 views
Cartwright-Littlewood Theorem
I have some question in the proof of the following theorem. I pose the question mark
on each statment that i want to know. I would be so grateful if someone can help me.
Thanks in advance
An ...
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3answers
97 views
Prove that $f$ has a fixed point .
For $f:[a,b]\rightarrow [a,b]$ is a continiuous . Prove that $f$ has a fixed point . Is that true if we chane $[a,b]$ by $[a,b)$ or $(a,b)$.
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1answer
28 views
Brouwer transformation plane theorem
Can somebody show that BPTT, version 2 is deduced from BPTT, version 1
[BPTT, version 1]
Let $h$ be a fixed point free orientation preserving homeomorphism of $\mathbb{R}^2$ Every point ...
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65 views
periodic point of homeomorphism of plane
Let $h$ be a homeomorphism of $\mathbb{R}^2$ onto itself such that $h(K)=K$ for some compact subset $K$ of $\mathbb{R}^2$. Show that $h$ has a periodic point in $\mathbb{R}^2$.
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algebraic or homotopical proof for Kakutani fixed point theorem
As Kakutani fixed point theorem is a genral case of Brouwer fixed point theorem, and one can read the proof from homotopy theory books. I wonder if there is any proof for the Kakutani using homotopy ...
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1answer
40 views
Show that sequence approaches fixed point of a function
Problem
Let $f(x)$ be a differentiable function on $\Bbb R$ with $\left|\,f ' (x)\right| \leq r < 1$, where $r$ is constant. Then consider the sequence $\{x_n\}$ such that $x_1 = 0$, $x_{n+1} = ...
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0answers
32 views
Fixed Point Iteration Scheme
I have been asked to "Find a fixed point iteration scheme for minimising $f(x) = e^{cos (x)}$".
Does anybody know what a fixed point iteration scheme actually is? I know it's not Fixed Point ...
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0answers
65 views
Properties of the derivative $\frac{\delta F(\phi(x))}{\delta\phi(y)}$ , where $ \phi(x) = F(\phi(x)) $ is a fixed point problem.
Dear experts I have a fixed point problem of the type:
$ \phi(x) = F(\phi(x)) $, where $x \in \mathbb R^3 $.
$\phi(x)$ is differentiable non-negative function on a given domain. I am trying to find ...
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1answer
22 views
Is there any space with normal structure but not uniform normal structure?
It is clear that uniformly normal structure implies normal structure. But, is there any space that has normal structure but it doesnt have uniformly normal structure?? Would you mind to give me some ...
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1answer
38 views
A smooth or analytic function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer
In A continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer, I asked about continuous function. This time, I would like to ask the ...
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2answers
44 views
A continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer
Can we find a continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ for every integer $k$ and every prime number $n$? And other points other than these are not fixed ...
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1answer
28 views
Chaos/Fixed points. I was reading a book by Strogatz and I encountered this.
Now, I always thought a fixed point implied $f(x)=x$, so somebody tell me, what is he talking about here?
Thank you.
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1answer
56 views
How to show a function taking the form as $F(x,y) = 0$ is a contraction mapping?
Let $\Phi (x)$ be the cumulative distribution function of the standard normal distribution. Given $x_0$, $x_1 = \Phi(x_0-x_1)$.If $x_n$ is given, $x_{n+1} = \Phi(x_{n}-x_{n+1})$(By drawing a graph, ...
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2answers
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Using the Banach Fixed Point Theorem to prove convergence of a sequence
Use the Banach fixed point theorem to show that
the following sequence converges. What is the limit of this
sequence?
$$\left(\frac{1}{3},
\frac{1}{3+\frac{1}{3}},
...
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1answer
78 views
Question about the infimum of $|f(x)-x|$ , where $f(x)=x$, $x$ is a fixed point of the nonlinear equation.
I am trying to check if the following property holds for fixed points:
Suppose:
$ f(x)= x $ is given, with solution $x = \theta \gt 0 $
I would like to show :
$ \forall \epsilon \in (0,1), ...
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function to recover
I have the following problem
$
f(x,y,z) = g(x,y) + \int \int f(x,y',z')dy'dz' + \int \int \int f(x',y',z')dy'dz' +
$
Where $g$ is known, $x,y$ and $z$ are continuous variables taking values in ...
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2answers
55 views
Showing a contraction without a fixed point
Suppose $f: [1, \infty) \to [1, \infty]$ defined by $f(x) = x + \frac{1}{x}$ for all $x \geq 1$. I want to prove that:
\begin{equation}
|f(x)-f(y)| < |x-y|
\end{equation}
except when $x=y$, but ...
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1answer
77 views
How many fixed points can a differentiable function have?
Let $f$ be a differentiable function on $\mathbb{R}$. Then could any one tell me which of the following statements are necessarily true?
If $f'(x)\le r<1$ for all $x$ then $f$ has at least one ...
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0answers
47 views
Connection between codata and greatest fixed points
It's easy to see how inductively-defined data types correspond to least fixed points. Let's take the natural numbers as an example, with constructors are $0 : \mathbb N$ and $s : \mathbb N \to \mathbb ...
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1answer
77 views
Generalization of Banach's fixed point theorem
I wanted to show that if $f:X->X$ is a function from a complete metric space to itself and if $f^k$ is a contraction, then $f$ has a unique fixed point (say $p$) and for any $x$ in $X$ $f^n(x)-> ...
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Unique solution to $x^4 + 7x -1 = 0 $ on $[0,1]$ (Banach's fixed point theorem)
I want to show that $x^4 + 7x -1 = 0 $ has a unique solution on $[0,1]$.
The idea is to use Banach's fixed point theorem. However, I see a problem with this as the statement of the theorem says that ...
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1answer
63 views
Functions $f:X\to X$ with no fixed points, for $X$ a punctured disk or a sphere.
$X$ is the punctured closed unit disc $D^2-\{0\} = \{(x, y) \in \mathbb{R}^2: 0 \lt x^2+y^2 \le 1\}$
Is the answer that $f$ maps all $(x,y)$ to $0$ which is not included in the unit disk and so ...
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1answer
72 views
Stokes' and Green's Theorem Integral Setup
a) For the vector :
$$v= 4y\hat{x}+x\hat{y}+2x\hat{z}$$
evaluate,
$$\int(\nabla \times v) \cdot da$$
over the hemisphere represented by the upper half plane of
$$x^2 + y^2 + z^2 = a^2$$
(this is ...
3
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1answer
102 views
Application of Banach fixed-point theorem
I am looking for a function $f:[0,1]\rightarrow\mathbb R$ which satifies $\int_{0}^{1}\frac{\sin(f(t)-y)}{2}\,\mathrm dy=f(t)$ for $t\in[0,1]$.
The first thing I do is to define a function ...
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1answer
30 views
Is antipodal symmetry really necessary for Tucker's Lemma?
Tucker's Lemma is here.
Let's stay within the 2D case for now. A standard proof is constructive:
(1) Pick an arced edge on the boundary of the circle. Note its labeling (for example, (1, 2)).
(2) ...
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+50
Why is convexity a requirement for Brouwer fixed points? Shouldn't “no holes” be good enough?
Brouwer's fixed point theorem:
Every continuous function $f$ from a convex compact subset $K$ of a Euclidean space to $K$ itself has a fixed point.
I am wondering why the word "convex" is in ...
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1answer
97 views
Is there a simple proof of Borsuk-Ulam, given Brouwer?
(Brouwer) Any continuous function from a convex compact subset K of a Euclidian space to itself has a fixed point.
Given this lemma, is there a simple proof of:
(Borsuk-Ulam) Any continuous ...
5
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2answers
106 views
How to show something is a contraction?
If we let $X$ be a complete metric space, and let $S:X\to X$ be a map, such that $S^m$ is a contraction. We now want to show, that $S$ has a unique fixed point
This is what I've thought so far:
Due ...
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2answers
57 views
Limit of a sequence of fixed points also a fixed point?
Suppose I have a continuous function $f : [0,1]^n \rightarrow [0,1]^n$ (maybe $n$ is infinite). Suppose I have a sequence $\{a_n\}_{n=1}^\infty$ of points in $[0,1]^n$ where each $a_n$ is a fixed ...
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1answer
40 views
Name a stable output of a function taking 2 arguments
$\mathbb{C}$ is a fixed finite set, a fair chaotic sequence $(c_n \in \mathbb{C})$ is defined such that $\forall c \in \mathbb{C}, \exists n_0 \in \mathbb{N}, n > n_0 \wedge c_n = c$. That means ...
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3answers
85 views
Prove the sequence $x_n = \frac{x_{n-1}}{2}+\frac{A}{2x_{n-1}} \rightarrow \sqrt{A}$ as $n \to \infty$
Show that if A is any positive number, then the sequence defined by:
$$x_n = \frac{x_{n-1}}{2}+\frac{A}{2x_{n-1}}$$
for any $n \geq 1$ converges to $\sqrt{A}$ whenever $x_0 > 0$.
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votes
2answers
230 views
Convergence of fixed point iteration for polynomial equations
I'm looking for the solution $x$ of
$$x^n+nx-n=0.$$
Thoughts: From graphing it for several $n$ it seems there is always a solution in the interval $[\tfrac{1}{2},1)$. For $n=1$, the ...
4
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1answer
115 views
Iteration of $x/\log x$
Consider $f(x)= \frac{x}{\log x}$ iterated n times for given x in $Z^+.$ $f^2 = f \circ f.$
Let $x_1 = x^2.$
What I would like to show (or disprove) : $\exists ~\alpha = x_n - e > 0 $ such ...
1
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1answer
91 views
Lipschitz condition on nonlinear ODE
Suppose we have the ODE $$x''=-a\sin{x}.$$
Then let $$x'=y$$ and $$y'=-a\sin{x}.$$ So $$\mathbf X = \begin{pmatrix} y \\ -a\sin{x} \end{pmatrix}.$$ Im confused about how to show a Lipschitz ...
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2answers
141 views
Finding a functor satisfying a recursive equation
Say I want to find a functor satisfying something like: \[FA = 1 \sqcup (A \times FA).\]
Equivalently, I want to find for each $A$ a fixed point of: \[GB = 1 \sqcup (A \times B)\] (I'll worry about ...
3
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1answer
68 views
Brouwer's fixed point theorem for infinite dimensional real space in subsystems of second order arithmetic
$\text{WKL}_0$ proves Brouwer's fixed point theorem for continuous functions on $\lbrack 0,1 \rbrack^n$, when $n$ is finite. What subsystem of second order arithmetic is needed to prove Brouwer's ...
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1answer
47 views
Show the following using fixed point theorem
Show that if A is s pxp real matrix such that $||I_p-A|| <1$, then for any choice of b$\in\mathbb{R}^p$ and $u_0\in\mathbb{R}^p$ the vector $u_{n+1}=b+(I_p-A)u_n$ converge to a solution x of the ...
1
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2answers
87 views
Show that a function $f(x)$ maps to a set of points.Fixed point theorem
Show that the function $f(x)=\frac{1+x^2}{2}$ maps the set of points $0\leqslant x\leqslant 1$ into itself and has a fixed point in that interval even though there does not exists a positive ...
4
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2answers
157 views
Fixed points in category theory
Is there a usual way to express the concept of fixed points in category theory? What would I say to express that a morphism has a fixed point? Thank you!
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70 views
The equation $2 \cosh(3.1786803659501505 z) = z$?
Let $a$ be a positive real number and $z$ a complex number.
I was wondering about the equation $2 \cosh(a z) = z$ where we solve for $z$.
Clearly if $z$ is a solution than so is its conjugate.
It ...
2
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3answers
125 views
Where does the iteration of the exponential map switch from one fixpoint to the 3-periodic fixpoint cycle?
In the answering of another question in MSE I've dealt with the iteration of $x=b^x$ where the base $b=i$. If I reversed that iteration $x=log(x)/log(i)$ then I run into a cycle of 3 periodic ...
3
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1answer
68 views
$x_{n+1}=\frac{2x_n+3f(x_n)}{5}$ showing $f$ has a fixed point
Let $f: \Bbb{R} \rightarrow \Bbb{R}$ be a differentiable function , and suppose that there is a constant $A<1$ such that $|f'(t)|\le A$ for all real $t$. Define a sequence $\{x_n\}$ by $
...
4
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1answer
85 views
fixed-point iteration
Are there functions $f$ of one real variable $x\in X$ that are not contracting maps on the set $X$ but for which, given the starting point $x_0$, the fixed-point iteration $x_n=f(x_{n-1})$, for ...
2
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2answers
76 views
does it have unique fixed point?
$p:C[0,1]\rightarrow C[0,1]$ defined by $p(f(x))=\int_{0}^{x} (x-t)f(t)dt$, well, I am getting all constant functions are fixed points, but the answer says that it has unique fixed point. I got ...
2
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1answer
106 views
Homotopy equivalence an retractions
I have some questions about homotopy. Before starting here a definition:
A topological space $X$ is called contractible if $X$ is homotopy equivalent with a one-point-space
Suppose $X$ a toplogical ...
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0answers
91 views
Fixed-point iterations global or local convergence
I have the following function $f(x) = \frac{x}{2}-\sin(x)+\frac{\pi}{6}-\frac{\sqrt3}{2}$
and I have the iteration function $\phi(x) = \frac{x}{2}+\sin(x)-\frac{\pi}{6}+\frac{\sqrt3}{2}$. Then for the ...
2
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1answer
111 views
Fixed point theorem on graphs?
I have a graph $G=(V,E)$ where to each vertex $v$ I have associated a value, $\hat{v}$ (ie I have a "network" in the terminology here http://snap.stanford.edu/snap/index.html ).
Let $\phi : \hat{V} ...
2
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3answers
95 views
Is this function necessarily a contraction?
If $(X, d)$ is a compact metric space satisfying $d(f(x), f(y)) < d(x, y)$ for all $x, y \in X$ such that $x \ne y$, is $f$ necessarily a contraction?
I know an analogue of the Banach Fixed Point ...
