Questions tagged [fixed-point-theorems]

A fixed-point theorem is a result saying that a function $F$ will have at least one fixed point (a point $x$ for which $F(x) = x$), under some conditions on $F$ that can be stated in general terms.

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Automorphisms of rationally connected varieties

Let $X$ be a smooth, rationally connected variety over an algebraically closed field of characteristic zero. Denote by $\mathrm{Aut}(X)$ the space of automorphisms of $X$ and for a given $\phi \in \...
Ron's user avatar
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Existence and uniqueness of fixed point in generalized condition of triangular norm

Definition 1) A Menger space is defined as a triple $\left( S,F,T \right)$ where $S$ is a set , $F$ is a collection of distribution functions and $T$ is a triangular norm function $T:[ 0,1 ]\...
nastaran noorivatan's user avatar
2 votes
2 answers
233 views

On fixed point probability in discrete logarithm

Fix an integer $n>2$. Question. What is the probability that, for a given $h\in\Bbb Z_n,$ there is no $$x\in[0,\varphi(n)-1]\cap\Bbb Z$$ such that $h^{x\bmod\varphi(n)}\equiv x\bmod n$?
Turbo's user avatar
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4 votes
1 answer
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Fixed point of a group action

Let $\mathbb{R}^\infty$ be the product of countably many real lines. Assume that a finitely generated group $\Gamma$ acts on $\mathbb{R}^\infty$ (linearly and continuously) and there is a nonempty ...
Anton Petrunin's user avatar
8 votes
2 answers
865 views

Continuity of mapping sending a function to its (brouwer) fixed point

Let $f:[0,1]^n \rightarrow [0,1]^n$ be a continuous mapping. Brouwer's fixed point theorem says that $f$ has a fixed point, i.e., some $x$ such that $f(x) = x$. Suppose we have a continuous family, i....
James Kilbane's user avatar
3 votes
0 answers
109 views

Formal way to prove existence and continuity in an integral equation

In the paper "A Boundary Value Problem Associated with the Second Painlevé Transcendent and the Korteweg-de-Vries Equation" by Hastings and McLeod, the authors study the ODE $$\frac{\mathrm{d}^2 y}{\...
user1337's user avatar
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10 votes
1 answer
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Equivariant Riemann-Hurwitz

The Riemann-Hurwitz formula starts with a genus $g$ algebraic curve $Y$ and a ramified cover $\pi\colon X\to Y$ of degree $N$, with ramification indices $e_P$ and computes invariants of $X$, such as ...
Ben Wieland's user avatar
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Is it possible to find a $\mathbb{C}$-algebra endomorphism of $\mathbb{C}[x,y]$ that fixes a given $w \in \mathbb{C}[x,y]-\mathbb{C}$?

Let $w \in \mathbb{C}[x,y]-\mathbb{C}$ and let $u \in \mathbb{C}[x,y]-\mathbb{C}[w]$. Is it possible to find a $\mathbb{C}$-algebra endomorphism $f$ of $\mathbb{C}[x,y]$ such that $f(w)=w$ and $...
user237522's user avatar
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Random iteration of a set of monotone maps until fixed point

Let $P$ be a poset with a least element $\bot$ ($\forall x \in P.\ \bot \le x$). Let $M$ be a set of monotone maps $P \to P$. Call $x \in P$ reachable if $x = f_1(f_2(...f_n(\bot)...))$ for some ...
Michael Arntzenius's user avatar
13 votes
3 answers
963 views

Closed manifolds with the fixed point property

The real projective plane ${\bf P}^2({\bf R})$ is the only closed surface with the fixed point property: all continuous maps from the plane to itself has a fixed point. The projective spaces ${\bf P}^{...
coudy's user avatar
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1 vote
1 answer
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How to compute the joint spectrum?

Let $(A_{1},A_{2}, \ldots,A_{k})$ be $k$ matrices in $M_{n}(\mathbb{R})$. Is there an algebraic formula, as a generalization of "Determinant" for $k=1$, to compute the joint spectrum of ...
Ali Taghavi's user avatar
9 votes
1 answer
544 views

Is $\{x_n\}$ a Cauchy sequence?

Let $(X,d)$ be a complete metric space and $f$ a mapping of $X$ into itself. Let $\{f^n(x)\}=\{x_n\}$ be the sequence of iterated transforms. Suppose $f$ satisfies that for each $\varepsilon >0$,...
Isra El's user avatar
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7 votes
3 answers
368 views

Does a certain contractive mapping have a fixed point?

Let $f:X\rightarrow X$ be a contractive mapping of a complete metric space satisfying $$d(f(x),f(y))\leq\alpha(d(x,y))d(x,y)$$ where $\alpha:\mathbf{R}^+\rightarrow [0,1)$, and $\alpha(t_n)\rightarrow ...
Isra El's user avatar
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17 votes
4 answers
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What are the major differences between real and complex Banach space?

Most theorems under real Banach space settings have their twin brothers for complex ones, say, the Hahn-Banach theorem. However, some theorems are not valid in complex Banach spaces, and vice versa. ...
Ice sea's user avatar
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6 votes
2 answers
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Reference request: an example of Bott residue formula's usage

Could you give me an example of a clear and beautiful application of Bott residue formula in torus-equivariant cohomology (see below)? I found an example calculating a product of Chern classes on ...
evgeny's user avatar
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spiral forward orbits of analytic functions near repelling fixed points

An anonymous referee informs me that forward orbits near fixed points of analytic functions, such that the members of the forward orbits lie on spirals, are well-known. His citation for this (p. 31 of ...
user88693's user avatar
0 votes
0 answers
269 views

Fixed-point iteration depending on a parameter

Let $f\colon X\times \mathbb{R}\to X, (x,\varepsilon)\mapsto y$, with $X$ open, be a continuous function in both arguments. Consider the following fixed-point iteration \begin{align} x_{k+1} = f(x_k,\...
Ludwig's user avatar
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2 votes
1 answer
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Fixed point theorem for a nonconvex set in a Banach space

Generally speaking, I am looking for a generalization of the Schauder fixed point theorem, which applies to the situation described briefly below. All references I read (e.g. E. Zeider 'Nonlinear ...
jaco's user avatar
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1 answer
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Number of fixed points in Zagier's involution (Fermat's Theorem) [closed]

Zagier's has found a famous one sentence proof for Fermat's theorem on sums of two squares. It centers on the following involution of the set $S= \lbrace (x,y,z) \in N^3: x^2+4yz=p \rbrace $ having ...
Toastgeraet's user avatar
2 votes
1 answer
255 views

characterization of normality by selection theorem

The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...
Kasper Cools's user avatar
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0 answers
105 views

Fixed point shape property

Question: Provide (or prove that it's not possible) a metric compact space which has the fixed point property but not the fixed point shape property. Here is the definition of f.p.s.p.("map" means ...
Włodzimierz Holsztyński's user avatar
3 votes
1 answer
2k views

leray schauder fixed point and schauder fixed point

I have seen these 2 fixed point theorem and I think the condition of Leray Schauder fixed point theorem is very strong and we require to consider the fixed point of $u=\sigma Tu$ $\forall \sigma \in[...
mnmn1993's user avatar
6 votes
2 answers
713 views

Proof of Lefschetz-Hopf Fixpoint Theorem with de Rham cohomology?

Looking for a proof of the Lefschetz-Hopf Fixpoint Theorem with the de Rham Cohomology. (I´m more interestet in the Formula then just the simple statement that if the Lefschetz number is not zero ...
floating's user avatar
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0 answers
203 views

Can the following system of equations be solved analytically/in a closed form?

From a constrained non-linear maximization problem I obtained the following system of equations: $a_1=\frac{1+a_3-\sqrt{a_2a_3}\sqrt{v_1}}{1+\sqrt{\frac{a_3}{a_2}}\sqrt{v_1}}$ $a_2=\frac{2-a_3-\sqrt{...
Paul's user avatar
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3 votes
0 answers
550 views

On a matrix algorithm involving rank-one projections

Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors spanning $\mathbb{R}^n$. Let $p\in [0,1]$ be a rational number and consider the iteration \begin{equation} X_{k+1}=\frac{1}{N}\sum_{i=1}^...
Ludwig's user avatar
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1 vote
1 answer
343 views

Fixed point of quantum operations

A quantum operation is defined as \begin{equation} \varepsilon(\rho)=\sum_{k}M_k\rho M_k^{\dagger} \end{equation} where $\varepsilon(\rho)$ takes an initial state $\rho$ to some final state $\rho'$ ...
Janus's user avatar
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18 votes
1 answer
830 views

Does every automorphism of a separably rationally connected variety have a fixed point?

Let $k$ be an algebraically closed field. Let $X$ be a smooth, projective variety over $k$ that is separably rationally connected, i.e., there exists a $k$-morphism $u:\mathbb{P}^1_k \to X$ such that ...
Jason Starr's user avatar
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8 votes
2 answers
474 views

A property stronger than the fixed point property

Assume that $X$ is a topological space. We say that $X$ satisfies the strong fixed point property if the graph of every surjective continuous self-map intersect the graph of every continuous self-...
Ali Taghavi's user avatar
1 vote
1 answer
186 views

A weak fixed point property

The usual fixed point property can be interpreted in terms of non empty intersection of the graph of all maps with the graph of the identity map. This motivates us to consider the following "weak ...
Ali Taghavi's user avatar
7 votes
2 answers
1k views

Intermediate value for a vector-valued function

Consider a vector-valued function $f: [0,1]^n\rightarrow[0,1]^n$. Write $f(x)=\{f_1(x), ..., f_n(x)\}$ with $x\in[0,1]^n$, where the $f_i: [0,1]^n\rightarrow[0,1]$ are continuous functions with the ...
MthQ's user avatar
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6 votes
0 answers
209 views

A tangential fixed point property for manifolds embedded in Euclidean spaces

Assume that $M$ is a compact orientable manifold which is embedded in some Euclidean space $\mathbb{R}^{N}$ We say that $M\subset \mathbb{R}^{N}$ has the tangential fixed point property if for ...
Ali Taghavi's user avatar
3 votes
2 answers
553 views

Example of measure of non-compactness

I can't understand the following example of measure of non-compactness, which was given in this article. Definition: A non-negative function $\phi$ defined on the bounded subsets of $X$ will be ...
Shinning Star's user avatar
3 votes
2 answers
306 views

Reference request: using integral equations to study asymptotics of ODEs

I was told by my supervisor that one way to study the asymptotic behaviour of solutions to ODEs is "to reformulate them as integral equations, and use fixed-point kind theorems on the resulting ...
user1337's user avatar
  • 463
7 votes
2 answers
462 views

Does a continuous map $f$ from the $n$-ball $B$ into $R^n$ such that $B\subset f(B)$ have a fixed point?

If $f$ is a continuous map from the $n$-ball $B$ into itself, the Brouwer fixed point theorem guarantees a fixed point. What if we assume that $f$ maps $B$ into all $R^n$, and $f(B)$ contains $B$? For ...
Juan Tolosa's user avatar
6 votes
2 answers
410 views

Convergence of Fixed-Point Iteration of a dependent map

Suppose that we have two mappings $T_1(\cdot): Y \mapsto Y$ and $T_2(\cdot,\cdot) : X \times Y \mapsto X$ where both $X$ and $Y$ are compact and convex subsets of the same Euclidean space. Furthermore ...
Rufio's user avatar
  • 61
1 vote
0 answers
52 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 ...
juror's user avatar
  • 43
7 votes
1 answer
421 views

Z/p action on finite contractible complex

Let $p$ be a prime and $X$ a finite contractible CW-complex. Assume $\mathbb Z/p$ acts on $X$. Then it is easy to see that there has to be a fixed point. (E.g. use Lefschetz's fixed point theorem or ...
Jens Reinhold's user avatar
1 vote
1 answer
217 views

Row-stochasticity of the Jacobian matrix of a stationary distribution

Let $P_{\mathbf{p}}$ be a $n \times n$ row-stochastic matrix whose entries are a function of a probability vector $\mathbf{p} \in \mathbf{R}_{> 0}^n$, $\sum_i p_i = 1$ and define the following ...
lum's user avatar
  • 113
11 votes
1 answer
501 views

Does Peano's existence theorem admits a constructive proof?

$$y(t)=y_0+\int_0^t b(y(s))ds$$ $b\in C(R^d)\cap L^\infty(R^d)$ The classical proof for Peano's existence theorem in ODE need use the Ascoli's theorem, so it's not constructive. When $d=1$, in the ...
Guohuan Zhao's user avatar
4 votes
0 answers
76 views

Multi-podal points

Two points $x,y \in \mathbb{R}^n$ are called antipodal if $x = -y$. Stated differently, $x,y$ are antipodal if: They have the same absolute value in each of their $n$ coordinates; Each of their non-...
Erel Segal-Halevi's user avatar
3 votes
1 answer
313 views

Fixed point property for intersection of spaces which are homeomorphic to a disk

The following question is question 9.8 from Miller's paper ``Some interesting problems '': Question Suppose $D_n$ a subset of the plane is homeomorphic to a disk and for every $n\in \omega, D_{n+...
Mohammad Golshani's user avatar
2 votes
0 answers
104 views

Fixed point theorem in ordered spaces

Can someone provide a proof or a source containing a proof of the following theorem Theorem: Let $D$ be a subset of the cone $K$ of partially ordered space $E,$ $F:D\rightarrow E$ be nondecreasing. ...
user71195's user avatar
32 votes
4 answers
5k views

Does the Brouwer fixed point theorem admit a constructive proof?

Wikipedia and a few websites (and a few mathoverflow answers) say there is a constructive proof of the Brouwer fixed point theorem, some others say no. The argument for a constructive proof is always ...
coudy's user avatar
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0 votes
1 answer
93 views

Fixed point property for the projectivization of manifold of fixed rank matrices

Let $M$ be the manifold of all matrices in $M_{n}(\mathbb{R})$ with fixed rank $0<k<n$. The projectivization of $M$ is denoted by $PM$. Does $PM$ satisfy fixed point property?
Ali Taghavi's user avatar
1 vote
1 answer
203 views

Totally non fixed point property

Edit: According to the comment of Pietro Majer, I revise the question Is there a non singleton compact connected Hausdorff topological space $X$ for which the following property hold?: "Constant ...
Ali Taghavi's user avatar
2 votes
0 answers
272 views

Fixed points of self maps

Given $m$ points on $S^n$, is there an explicit polynomial self $1-1$ map of minimum degree $f:S^n\rightarrow S^n$ that fixes only these $m$ points? Can we say something about symmetry group of $f$ if ...
Turbo's user avatar
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1 vote
1 answer
470 views

Existence of a fixed-point free map in a manifold [closed]

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 ...
victor's user avatar
  • 19
5 votes
1 answer
147 views

Continuity of central point operation

Stanisław Mazur and Stanisław Ulam, in their joint paper, characterized the mid-point $\ \frac{a+b}2\ $ in a Banach space in pure metric terms (without algebra). This allowed them to show that any two ...
Włodzimierz Holsztyński's user avatar
1 vote
1 answer
208 views

Lefschetz fixed notation

If $f\colon X\to X$ is a self-map of a nice space with isolated fixed points, then the Lefschetz fixed point theorem relates a global number to local numbers. Some write: $L(f)=\sum_{x\in \mathrm{Fix}(...
Ben Wieland's user avatar
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8 votes
0 answers
227 views

Do commuting homeomorphisms of the $2$-disk have a common fixed point?

Problem 2.20 (attributed to Lima) in Kirby's list of unsolved problems in low-dimensional topology asks: Are there commuting homeomorphisms of the $2$-ball $B^2$ without a common fixed point? ...
Dejan Govc's user avatar