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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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
86 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 $ ...
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115 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
143 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 ...
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1answer
179 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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1answer
154 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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282 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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Intersection point complete proof [closed]

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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2answers
74 views

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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1answer
206 views

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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137 views

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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3answers
115 views

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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1answer
163 views

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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113 views

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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1answer
313 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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67 views

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), ...
2
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2answers
372 views

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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1answer
145 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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404 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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1answer
107 views

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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1answer
97 views

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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175 views

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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184 views

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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1answer
98 views

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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1answer
187 views

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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73 views

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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271 views

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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2answers
95 views

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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3answers
63 views

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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1answer
53 views

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. ...
2
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1answer
152 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 $\sum_n θ_n < ∞$. ...
2
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1answer
72 views

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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1answer
179 views

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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827 views

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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2answers
343 views

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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1answer
95 views

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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Three questions about fixed points

Pick out the true statements. Let $f : [0, 2] \to [0, 1]$ be a continuous function. Then, there always exists $x \in [0, 1]$ such that $f(x) = x$. Let $f : [0, 1] \to [0, 1]$ be a continuous ...
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0answers
60 views

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 $x,y\in X$ ...
5
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3answers
672 views

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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1answer
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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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1answer
166 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!
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1answer
102 views

Topology Fixed Point Theorem

suppose the wind is blowing on the surface of the earth in a constant and continuous fashion. Suppose also that at every point on the equator, the wind is blowing directly east, so the wind doesnt ...
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296 views

Equivalence of Brouwers fixed point theorem and Sperner's lemma

I'm looking for a proof that Brouwer's fixed point theorem implies Sperner's lemma. All proof I've found just prove that there must be at least one completely colored n-simplex, not that there must be ...
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129 views

Fixpoint of monotone operators

Let $X$ be some set and let $F$ be the set of all functions with a domain $X$ and a range $[0,1]$. We consider $F$ to be a partially ordered set with $f\leq g$ if and only if $f(x)\leq g(x)$ for all ...
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2answers
505 views

Proof $\lim\limits_{n \rightarrow \infty} {\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}=2$ using Banach's Fixed Point

I'd like to prove $\lim\limits_{n \rightarrow \infty} \underbrace{\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}_{n\textrm{ square roots}}=2$ using Banach's Fixed Point theorem. I think I should use the ...