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 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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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 ...
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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 ...
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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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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 ...
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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 ...
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$x_{n+1}=\frac{2x_n+3f(x_n)}{5}$ 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 $ ...
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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 ...
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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 ...
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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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178 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} ...
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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 ...
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335 views

Schwarz's Lemma, fixed points question

This is from an old qualifying examination question. If f is holomorphic in the unit disk $D$ and $|f(z)|<1$ for all $z\in D$. Suppose also that $f$ has two distinct fixed points in $D$ then ...
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a fixed point theorem

A hiker starts to climb up from base B to the summit S on sat 6am one day,spends the night at S and starts to climb down at 6am the next day.Prove that there is a point on the path B-S (there is only ...
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Can a finite group act freely (as homeomorphisms) on $R^n$

And what for a single finite order element $f$, i.e. $f:R^n\rightarrow R^n$ is a homeomorphism such that $f^d=id_{R^n}$, must $f$ have a fixed point?
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Least common fixed-point

I have been reading a book, "Introduction to Lattices and Order", and I'm trying to solve exercise 8.29 as the following in it: Suppose that $P$ is a complete lattice and let $F$ and $G$ be ...
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Let $0 \leq \alpha < 1$ and let f be a function from $\mathbb{R} $ into $\mathbb{R}$ which satisfies…

Let $0 \leq \alpha < 1$ and let f be a function from $\mathbb{R} $ into $\mathbb{R}$ which satisfies $$ | f(x) - f(y)| \leq \alpha|x-y| \; \forall x,y \in \mathbb{R}.$$ Let $a_{1} \in \mathbb{R}$ ...
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A function over the integers and its fixed points

Define $f:\mathbb{N}\rightarrow\mathbb{N}$ as follows, $f(n)$ is the number of times the digit "1" is needed if we were to write all integers between 1 and $n$ (inclusive) in base 10. So for example ...
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Understanding the Banach fixed point theorem

The Banach fixed point theorem is stated in my book (Applied Asymptotic Analysis by Miller) as Let $\mathcal B$ be a Banach space with norm $\|\cdot\|$. Let $X$ be a nonempty bounded subset of ...
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Question on fixed point

I had some trouble to approach the question above. Especially (2) and (3). I appreciate if you can help!
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399 views

Convergence of fixed-point iteration for convex function

Let $f:[0,1]\to[0,1]$ be a smooth, convex (downward) function satisfying $$ f(0)=f(1)=1,\quad \lim_{x\to 0}f'(x)=-\infty,\quad \lim_{x\to 1}f'(x)=+\infty. $$ I am confident to be able to argue that ...
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fixed point of homeomorphism and compactness of a complete metric space.

I need to know that the following statements if true or false: Every homeomorphism of $S^2\rightarrow S^2$ has a fixed point. Let $X$ be a complete metric space such that distance between any two ...
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Orbit of $\frac{1}{2}$ of the dynamical system defined by $f:[0,1] \to [0,\frac{3}{4}]$ where $f(x)=3x(1-x)$ converges.

I view the orbit of $\frac{1}{2}$ of the dynamical system defined by $f:[0,1] \to [0,\frac{3}{4}]$ where $f(x)=3x(1-x)$ as the sequence $(x_{n})$. So $$x_n = \frac{1}{2}, f\bigg(\frac{1}{2}\bigg), ...
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Brouwer's fixed point theorem for free?

I think I found a proof of Brouwer's fixed point theorem which is much simpler than any of the proofs in my books. One part is standard: Suppose there is an $f:D^n \rightarrow D^n$ with no fixed ...
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187 views

The wedge sum of two circles has fixed point property?

The wedge sum of two circles has fixed point property? I'm trying to find a continuous map from the wedge sum to itself, that this property fails, I couldn't find it, I need help. Thanks
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556 views

Find all the fixed points of the system $f(x,y)=-x+x^3, g(x,y)=-2y$ and use linearization to classify them.

Find all the fixed points of the system $f(x,y)=-x+x^3, g(x,y)=-2y$ and use linearization to classify them. I have found the solutions to be : $x = 0$ or $x = ±1$ and $y = 0 \implies 3$ fixed points ...
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Fixed point of a non linear contraction in a convex set

Hi I'm stuck on the following problem of Haim's functional analysis book. Let $C\subseteq H$ ($H$ a Hilbert space) be a non-empty closed convex subset and let $T:C\rightarrow C$ be a non linear ...
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Fixed Points of a system of diff.equations

I have $f'_1(t)=-af_1(t)f_2(t)+bf_3(t)$ $f'_2(t)=f_1(t)$ $2f'_3(t)=-f_1(t)$ How is it possible to evaluate fixed points of this system of equations and afterwards the stability of these points. I ...
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Periodic orbits

Let $x\in\mathbb R$ be a periodic point with lenght 2 of the recursion $x_{n+1}=f(x_n)$ My book about Dynamic systems says that this recursion has a fixed point now, because a periodic point with ...
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Apply Banach's fixed point theorem

Let $$T:f\mapsto (x\mapsto \frac{2}{5}\int_0^1 (x^2+t^5)f(t) dt + \sin(x))$$ for any $x\in[0,1]$, $f\in C([0,1])$. I want to show that that there is a uniqu $\tilde{f}$ that solves that equation ...
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about fixed point set?

let $K$ be a closed convex subset of a normed space $V$. For any $f: K \to K$ define the fixed-point set of $f$ as follows: $fix(f)=\{x$ belongs to $K$ $|f(x)=x \}$. I have to show that a nonempty ...
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Fixed Point of a complex dynamical spiral system

Last semester I finished my first class on complex variables and of course we had to show that $i^i$ was real. That got me wondering about quantities like $i^{i^i}$ and similar power towers. For my ...
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Conjecture on continuous selection of fixed points of a correspondence

I have the following conjecture to show some sort of a continuous selection of fixed points in a correspondence: Let $S$ and $\Theta $ be non-empty, compact and convex subsets of some Euclidean space ...
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Finding the fixed point

Trying to solve this question, got this answer but have a gut feeling that this might not be the way to do it, by the way this topic is related to fixedpoints The solution that I came up with
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Some homework questions about a Lipschitz function (cauchy sequence)

Do you want to help me with my homework? The exercise is as follows: Consider a Lipschitz function, $h:\mathbb{R}\rightarrow\mathbb{R}$, satisfying for every $x, y$: $$\left| h(x)-h(y) \right| ...
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Distance between fixed points of functions

I'm trying to bound the distance of fixed points of two functions assuming there's some bound on the distance between the functions. Specifically, assume $f_1, f_2:[0,1] \rightarrow [0,1]$ are two ...
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Squaring modulo n — is there a systematic way to generate the fixpoints?

The natural function $$f_k(n) = n^2 \bmod k,\quad f_k(n) \in [0;k-1]$$ has at least the fixpoints $n\in\{0;1\}$ for each $n$, but for some $k$ there are other fixpoints. For instance, $f_{100}$ has ...
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IVP- Has at most one solution

Suppose $f(t, x)$ is nonincreasing with respect to $x$ for all $t \geq 0$ and $x \in \mathbb{R}$. Prove that the IVP problem $$ \left\{ \begin{array}{l} x'(t)=f(t,x) \\ x(t_0)=x_0 \end{array} \right. ...
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215 views

Weissinger's Theorem. How to prove?

Theorem (Weissinger). Let $C$ be a (nonempty) closed subset of a Banach space $X$. Suppose $K : C → C$ satisfies $$\|K^nx − K^ny\| ≤ θ_n\|x − y\|, \quad x,y∈ C $$ with $\sum_n θ_n < ∞$. ...
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Why is the D0L word a least fixed point?

Consider the finite alphabet $\sum=\{0,1,\cdots k-1\}$, the set $\sum^*$ of all words over it, and a mapping $T:\sum\to \sum^*$ such that each $i\in\sum$ is the initial symbol of $T(i)$. Now the ...
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Implicit Function theorem and Bifurcation points

So let us say we have a function $\dot{x} = f(x,r)$ that has some critical point at $(x_0,r_0)$ such that $f(x_0,r_0)=0$. The question now is: when is this a bifurcation point? I understand that ...
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Basic numerical analysis, fixed point iteration

I have just started numerical analysis so this question probably seems trivial. Say I have a function $f(x) = x^2 - x - 3$ I let $g(x) = x^2 - 3$ Then I want to find the roots of $f(x)$ so I have ...
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Continuous Brouwer's fixed point theorem via Stokes's theorem?

Let $B$ denote the closed unit ball in $\mathbf{R}^n$. Brouwer's fixed point theorem states that every continuous map $f:B\to B$ has a fixed point. There is a simple proof using Stokes's theorem, at ...
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No fixed point problem for iterations

Let $f(z)$ be an entire function that is not a polynomial of degree 1 or degree 0 , where $z$ is a complex number. Let $f(z,1) = f(z)$ and let $f(z,n) = f(f(z,n-1))$. Let $g(f,1)$ be the amount of ...
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a fixed point theorem for rectangle in $\mathbb{R}^2$

Well, Could any one tell me how to prove this one or any reference? Let $f$ be a continuos map on $\mathbb{R}^2$, and $S$ be a rectangular region such that as the boundary of $S$ is traversed, the ...
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About fixed point and damping

Assume there is a discrete standard map $$ \begin{cases} x_{n+1} = x_n + y_n,\\[3mm] y_{n+1} = ay_n + b\sin(x_{n+1}) \end{cases} $$ for solving the fixed points, we have $$ (x_{fm}, ...
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Fixed point of continuous function on compact metric space [duplicate]

Possible Duplicate: Prove the map has a fixed point Let $X$ be a compact metric space and $f:X\rightarrow X$ be a continuous function such that $d(f(x),f(y))<d(x,y)$ for every $x,y\in X$ ...
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Continuous function on unit circle has fixed point

The question I have is: Let $f: S^1 \rightarrow S^1$ be a continuous function, where $S^1$ is the unit circle. Prove that if $f$ is not onto, then $f$ must have a fixed point.
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What is the algebraic structure of functions with fixed points?

So I just noticed that the set of functions with a fixed point $$f(x_0)=x_0,$$ are closed under composition $$(f\circ g)(x):=g(f(x)),$$ and with $e(x)=x$, the inverible functions even seem to form ...
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180 views

Brouwer's fixed point theorem implies Sperner's lemma

I'm looking for a proof that Brouwer's fixed point theorem implies Sperner's lemma. The proof should be understandable by an undergraduate. Thanks in advance!