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, ...

learn more… | top users | synonyms

8
votes
0answers
170 views

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 ...
7
votes
0answers
222 views

Fixed points in computability and logic

I asked this question on CS.SE, too: http://cstheory.stackexchange.com/questions/27322/fixed-points-in-computability-and-logic I would like to understand better the relation between fixed point ...
7
votes
0answers
450 views

Fixed point: linear operators

I ask my question in two parts: though the topic is similar, I would like to distinguish linear and general cases since methods may be too different while my questions are broad. Consider a space $X$ ...
6
votes
0answers
262 views

Closest fixed point to a convex set

Consider the compact convex sets $Y \subset X \subset \mathbb{R}^n$, and a Lipschitz continuous function $f : X \rightarrow X$. Assume that $f$ has multiple fixed points. (From Brouwer's theorem, $f$ ...
5
votes
0answers
120 views

Reference request: equivalence between formulas in fixed point and first-order logic

I'm looking for materials on the relationship between first-order and fixed-point logics, specifically on the condition for a formula in some sort of fixed-point logic to have an equivalent ...
5
votes
0answers
81 views

Limit of a Discrete Dynamical System, Part 2

In my previous post (i.e., Limit of a Discrete Dynamical System) the following system was considered: ...
5
votes
0answers
135 views

Converse to Lefschetz fixed point theorem (counterexample)

The celebrated Lefschetz fixed point theorem, in its simplest form, says (following Wikipedia) that if $f\colon X \to X$ is a continuous map of a compact triangulable space $X$ to itself, then $f$ has ...
5
votes
0answers
140 views

Showing a sequence $x_{n+1} = Tx_n$ forms a contraction mapping

I want to show that the sequence given by $$x_{n+1} = Tx_n = x_n-\frac{(x_n^2-2)}{x_n+x_{n-1}}$$ forms a contraction mapping. That is $$|Tx_1-Tx_2|\leq c|x_1-x_2|.$$ Where $c$ is to be determined. I ...
4
votes
0answers
25 views

Literature on the convergence of $x_{n+1} = f(x_n)$ in general

When faced with a recurrence of the form $x_{n+1} = f(x_n)$, my toolbelt for proving convergence is very limited: if $f$ isn't $k$-lipschitzian with $k<1$, and/or if I can't find some complete set ...
4
votes
0answers
113 views

A question on fixed point property

Assume that $0<k<n-1$, Note that $\mathbb{C}P^{k}$ can be considered as a closed subset of $\mathbb{C}P^{n}$, in a natural way. We collapse $\mathbb{C}P^{k}$ to a point. The resulting space is ...
4
votes
0answers
513 views

Fixed point of a shrinking map. Proof.

Could you please verify my proof? Let $f:X \to X$ be a shrinking map on a compact metric space $(X,d)$. In other words: $d(f(x),f(y)) < d(x,y)$ for all $x,y \in X$. Prove: $f$ has a unique ...
4
votes
0answers
152 views

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 ...
4
votes
0answers
429 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 ...
4
votes
0answers
214 views

Vector valued contraction

I am familiar with the Banach Fixed Point Theorem and I have used it to prove existence and uniqueness of functional equations in Banach spaces like $C(X)$, the space of bounded continuous function ...
3
votes
0answers
72 views

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 ...
3
votes
0answers
106 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 ...
3
votes
0answers
134 views

Reference request: generalized Lefschetz-Hopf fixed point theorem

Let $X$ be a smooth projective variety over $\mathbb{F}_p$. A basic (and very important) theorem is that we have equality $$ \# X(\mathbb{F}_{p^n}) = \sum_{i=0}^{2\dim X} (1)^i ...
3
votes
0answers
221 views

On the Proof of the Perron-Frobenius Theorem.

The Perron-Frobenius theorem states that a square matrix with nonnegative entries has a real nonnegative eigenvalue. One possible proof uses the Brouwer fixed point theorem, and every proof I've seen ...
3
votes
0answers
163 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 ...
3
votes
0answers
141 views

Fixed point: general case

This is the second part of the question Fixed point: linear operators. Here I would like to ask you about the general case. A lot of concepts can be described or even defined as a solution of a ...
2
votes
0answers
13 views

Solution Space for a Fixed Point Problem

Hi I need to find the criteria for which the following has a solution: $$X= K_1 (a_1-b_1X)^{c_1} (a_2-b_2 X)^{c_2} (a_3-b_3X)^{c_3} X^{c_4}$$ where $K_1>0; a_1>0; c_1>1; (b_2 b_3)\leq0; ...
2
votes
0answers
34 views

Equilibrium points of $\dot x(t)=-2\cdot x^3(t)$

The following differential equation is given: $$ \dot x(t)=-2\cdot x^3(t)\qquad x(t)\in\mathbb R $$ I am asked to find the equilibrium points of the system. By definition, the equilibrium points are: ...
2
votes
0answers
40 views

Edelstein Fixed Point Theorem

Let $(M,d)$ be a metric space, $M$ compact. If $f:M \to M$ is continuous and weakly contractive (i.e. $d(f(x), f(y)) < d(x,y) , \forall x,y \in M$), then $\exists x_0 \in M $ unique s.t ...
2
votes
0answers
47 views

Proof that the function on $C[0,b]$ is a contraction

Question : Let $a$,$b$ be real numbers with $0\lt b\lt 1$ . Consider the subset $X\subset C[0,b]$ consisting of the functions $f$ s.t $f(0)=a$ .Then $X$ is closed in $C[0,b]$. ...
2
votes
0answers
61 views

Example of Measure of non-compactness?

I can't understand the following example of measure of non-compactness, which was given in a research article. Definition: A nonnegative function $\phi$ defined on the bounded subsets of $X$ will ...
2
votes
0answers
42 views

Convergence of continuation scheme for fixed-point via homotopy

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a non-expansive map, i.e. $$\|f(x) - f(y)\| \leq \|x - y\|$$ for all $x,y\in\mathbb{R}^n$. Further, assume $f$ has at least one fixed-point $x^\star$. ...
2
votes
0answers
55 views

Can we find an example?

I have this theorem, and i want to know if i can find an expression of the operator $A$ which satisfy this theorem:
2
votes
0answers
29 views

Show that F can have at most two fixed points

Consider a function $F:\mathbb{R}^n\to\mathbb{R}^n$, where $F=(F_1,...,F_n)$. Suppose that $F$ is strictly quasi-concave, and that for all $i=1,...,n$ the function $F_i:\mathbb{R}^n\to\mathbb{R}$ is ...
2
votes
0answers
59 views

What are conditions for Tarski's fixed point theorem so that greatest and least fixed points are not the same?

Tarski's fixed point theorem claims that for any monotone function on a complete lattice there are least and greatest fixed points (or more generally, that the set of fixed points is a complete ...
2
votes
0answers
30 views

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 ...
2
votes
0answers
32 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 ...
2
votes
0answers
29 views

Proof of equality of least fixed point of two continuous function

The question is simple: given a set U, a continuous function (Scott continuity) $f \colon \mathcal{P}(U) \to \mathcal{P}(U)$ and function $g(X) := f(f(X))$, prove that $g$ is continuous and its least ...
2
votes
0answers
54 views

Reference request for the proof of the Brodskii–Milman fixed point theroem for isometries

Can any one help me to access the paper M.S Brodskii and D.P Milman, On the center of a convex set, Dokl. Akad. Nauk SSSR 59 (1948) 837–840 in Russian? or to prove the theorem If $K$ is a ...
2
votes
0answers
45 views

Dense subgroup of an extremely amenble group is extremely amenable??

Recall that a topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space $K$ admit a fixed point. i.e there is a point $\xi\in K$ such that $g.\xi=\xi$ for ...
2
votes
0answers
58 views

Identification of (Centers of) Cycles in a Discrete Time Dynamical System

I am studying dynamics on nonlinear Discrete Time Dynamical System of the form $$ \vec{X}_{t+1} = D(\vec{X}_t), $$ where D is some nonlinear function. I was looking for a (relatively) quick ...
2
votes
0answers
60 views

Fixed-Points Limit Set

Let $f : \mathbb{R}^n \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ be compact and convex. For all $p \in \mathbb{R}$, consider $A_p \in \mathbb{R}^{n \times n}$, with $p \mapsto A_p$ ...
2
votes
0answers
57 views

A basic question about upper hemicontinuity

Given a correspondence $f:X\rightarrow 2^X$, suppose X is a closed simplex in $\mathbb{R}^n$, and $f$ is compact-valued. We say $f$ is upper hemicontinuous if, $\forall x\in X$ and every open subset ...
2
votes
0answers
112 views

fixed point iteration $x_{k+1}=f(x_k)$ with multiple fixed points

I have a fixed point iteration problem in my research, as below: $$x_{k+1}=f(x_k)$$ where $$f(x)=\frac{\lambda r}{g}\sum_{i,j}\pi_if_{ij}\left[1- \exp\left\{\displaystyle ...
2
votes
0answers
79 views

What is the method for solving this class of problem?

$\frac{dx}{dt} = y$ $\frac{dy}{dt} = -\partial y - \mu x - x^2$ Find the fixed points and discuss stability. I'd at least like to know what I should be googling, any help is appreciated.
2
votes
0answers
82 views

Proving $\frac{k-1}{k}$ is an attractor of the logistic map $kx(1-x)$.

Consider the logistic map $f(x) = kx(1-x)$ defined on $\mathbb{R}$. We already know $\frac{k-1}{k}$ is a fixed point of $f$, but my issue is showing it's an attractor when $k \in [1,3]$. There is an ...
2
votes
0answers
138 views

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 ...
1
vote
0answers
24 views

The conjugate of a complex function

This may be a stupid question. I am trying to explain/prove the following: Let $f$ be a complex rational function of degree $d ≥ 2$ with fix point $z_0 \neq 0$. It is always possible to conjugate ...
1
vote
0answers
27 views

Convergence of autonomous system / time-scale separation

Let $x \in \Delta_n$, where $\Delta_n = \{u \in \mathbf{R}^n_{\ge 0} \mid \sum u_i = 1\}$ is the probability simplex. Suppose that I have an (autonomous) dynamical system $$ \dot{x} = A(x) \cdot x $$ ...
1
vote
0answers
61 views

Proof Attempt of Brouwer (via Separating Hyperplane Theorem)

In part motivated by the discussion here, I have been playing with trying to prove Brouwer's theorem appealing as minimally as possible to topology. In the 1-dimensional case I believe one can ...
1
vote
0answers
16 views

multidemensional fixpoint iteration

We want to solve the following system of equation, which shows an intersection between a circle and an ellipse . $x^2+y^2=5$ $\frac{x^2}{16}+y^2=\frac{5}{4}$ We can expres this system as a fixpoint ...
1
vote
0answers
22 views

Theorem about number of crossing-points between a function and a line

Assume $f(x)$, with $x \in [a,b]$. Take $u$ so that $f(a)<u<f(b)$. By the Intermediate value theorem, we know that $f(x)$ crosses $u$ at least once. My question is, given some extra information ...
1
vote
0answers
31 views

Show that the fix points of a function couldn't be in the interior

I want to solve the following problem: Show that the fix points of a function $f:\mathbb B^n\rightarrow \mathbb B^n$ could possibly not be in the interior. By this, Show that the Brouwer fixed-point ...
1
vote
0answers
83 views

Fixed point theorem for multivalued functions on b-metric space

Hy guys! I'm studying some fixed point results for multi-valued function on a b-metric space $(X,d,s)$. I'm looking for the proof of a theorem wich is just a generalization of Nadler's results for ...
1
vote
0answers
63 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 ...
1
vote
0answers
38 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 ...