The fixed-point-theorems tag has no usage guidance.

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### 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 ...

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118 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 ...

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195 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 ...

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votes

**1**answer

122 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 ...

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31 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 ...

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votes

**1**answer

298 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 ...

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vote

**1**answer

122 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 ...

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**1**answer

210 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 ...

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58 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 ...

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77 views

### Alternating projections and trace preserving maps

Let $\{\Pi_i\}_{i=1}^N$, $\Pi_i\in \mathbb{C}^{n\times n}$ be a set of orthogonal projections. By the Von Neumann alternating projection theorem, it holds that
$$
...

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223 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, ...

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91 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. ...

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1k 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 ...

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**1**answer

62 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?

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**1**answer

92 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 ...

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201 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 ...

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**1**answer

98 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 ...

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**1**answer

109 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 ...

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**1**answer

144 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 ...

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129 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?
...

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**1**answer

78 views

### Fixed points of finite order isometries of metric spaces

I would like to show the following:
Let $X$ be a complete metric space that is uniquely geodesic (i.e. each two distinct points are connected by a unique geodesic segment) and $\phi\colon X\to X$ an ...

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**1**answer

95 views

### Almost fixed point property

Let $X$ be a Hausdorff topological space with the following property:
For every continuous function $f:X\to X$, there is a finite subset $S\neq \emptyset$ of $X$ with $F(S)\subset S$
...

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119 views

### Noncommutativization of fixed point theory

What papers or references have been devoted for a noncommutativization of "Fixed point theory". Here the terminology Noncommutativiztion, as usual, indicates to that famous table with 2 columns: ...

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226 views

### A problem on chains of squares — can one find an easy combinatorial proof?

Consider the unit square $ S = [0,1] \times [0,1] $. For each $ n \in \mathbb{N} $, we can tessellate $ S $ by the collection
$$
A
= \left\{
\left[ \frac{i}{n},\frac{i + 1}{n} \right] \times
...

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295 views

### Homeo-Fixed point property

Edit: According to comment of Michał Kukieła I revised the question
A topological space $X$ satisfies "Homeo-fixed point" property if every homeomorphism $f$ on $X$ possess a fixed point.
...

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95 views

### Prove a complicated function (in epidemic spreading search) to be convex

When analyse epidemic spreading, I came across to prove that a complicated function $f(x)$ is convex when $0 \leq x \leq 1$.
\begin{equation}
f(x)=\frac{b_1g'(x) f_1(x)^{n-2}+g'(1) \gamma}{g'(1) ( ...

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210 views

### What do you call a fixed point theorem for a mapping from a subset of a space to the whole space?

There are a number of fixed point theorems in which we have a map from some subset of a (metric, topological, ...) space to the whole space. (Usually, there is some condition regarding the behavior ...

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**1**answer

136 views

### Reference request for proof of Brodskii-Milman theorem “On the center of a convex set”

Can anyone 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 ...

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141 views

### Fixed points and universal maps for posets

In a recent post about f.p.p. for poset products, @M.Mirabi brought back an old-standing problem about the fixed point property of the product of two arbitrary posets which already enjoy the fixed ...

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939 views

### Is the fixed point property for posets preserved by products?

Recall that a partially ordered set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point.
Theorem. Suppose $P$ and $Q$ are posets ...

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### Brouwer fixed points via flow

Let $B$ be a closed ball in $n$-dimensional space, let $f$ be a "sufficiently smooth" map from $B$ to $B$, and let $p$ be a point on the boundary of $B$.
It seems to me that something like the ...

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**1**answer

187 views

### Fixed point relation $\ Fix\ $ for pairs of manifolds

First the classical definition: a topological space $X$ has the fixed point property (fpp) $\ \Leftarrow:\Rightarrow\ $ for every continuous $\ f : X\rightarrow X\ $ there exists $\ p\in X\ $ such ...

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869 views

### A question on fixed point theory

I asked this question in MSE, but I did not received any answer, so I repeat it here:
http://math.stackexchange.com/questions/858238/a-question-on-fixed-point-property
Assume that $0<k<n-1$, ...

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259 views

### Existence of an integral equation (Faedo-Galerkin, Banach fixed point, Picard-Lindelof)

Let $m \geq 2$ and let $m'$ be its conjugate. Let $w_j$ for $j=1, ..., k$ be a basis of $H_1 \cap L^{m'}$. The task is to show that there is a $u(t) \in \text{span}(w_1, ..., w_k)=:A$ such that
...

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838 views

### Is Schauder's Conjecture Resolved?

Schauder's Conjecture: "Every continuous function, from a nonempty
compact and convex set in a (Hausdorff) topological vector space into
itself, has a fixed point." [Problem 54 in The Scottish ...

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268 views

### Fixed point problem with a monotone vector as a fixed point?

Suppose $F : [0,1]^n \to [0,1]^n$ is continuously differentiable and $0 < \frac{\partial F_1}{\partial x_i} \leq \dots \leq \frac{\partial F_n}{\partial x_i} < \beta < 1$ for all $i ...

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249 views

### A variation of the Banach fixed-point theorem

Let $(X, d)$ be complete metric space, $q \in [0, 1)$ be a real number, and $f$ be a map that satisfies $$d(f(x), f(y)) \leq q \cdot d(x, y)$$ for all $x, y \in X$. Then, Banach fixed-point theorem ...

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256 views

### Fixed points and their continuity (2)

Yesterday I asked a question about fixed point. Here is the link.
In summary, the question was,
Let $f : I^2 \to I$ be a continuous map, where $I := [0,1]$ is the unit interval. It is a basic ...

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**1**answer

198 views

### Fixed points and their continuity

Let $f : I^2 \to I$ be a continuous map, where $I := [0,1]$ is the unit interval. It is a basic fact that for each $y\in I$, the function $x \mapsto f(x,y)$ admits a fixed point. I want to ask whether ...

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**1**answer

627 views

### What is the relation between Lefschetz fixed point theorem and Poincare-Hopf theorem on vector fields?

In Dubrovin/Fomenko/Novikov Modern geometry--Methods and applications, Part II, the (Poincare-)Hopf theorem is treated in section 15.2 (see theorem 15.2.7 on page 129), while the Lefschetz theorem on ...

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191 views

### Seemingly ill-founded recursion and the recursion theorem

The following line well-defines a family of subsets $\{S_i\}_{i\in\mathbb N}$ of $\mathbb N$:
$n\in S_i$ iff $n=2i$ or $\exists j<i$ such that $n\in S_j$.
The following line does not:
...

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**1**answer

943 views

### Periodic Orbit property

A topological space $X$ satisfies "Periodic orbit property", briefly POP, if for every continuous map
$f:X \to X$, there exist a natural number $n$ and a point $x_{0}\in X$ such that ...

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51 views

### A fixed point problem in infinite dimensions for monotonic algebras

Let $A$ be an algebra over $[0,1]$, whose operations are all unary monotone (increasing or decreasing) bijections, except that $A$ also includes the infimum operation over finite or countably many ...

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286 views

### Fixed point for a self-mapping on subset of C[0,1]

Let $f_1$ and $f_2$ be arbitrary self-mappings on $C([0,1])$ with $f_2 > f_1$. Define set $F = \{f \in (C[0,1])| f_1 \leq f \leq f_2 \mbox{ and } f \mbox{ is increasing}\}$. Is it true that every ...

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**1**answer

314 views

### A weak fixed point property for Grassmannian

Let $f$ be a continuous function on complex Grassmannian $G(k, 2n+1)$. Is it true to say that there is a $k$-plane $Y$ such that $Y$ has nontrivial intersection with $f(Y)$?
A motivation for ...

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236 views

### Infinite-dimensional hex

Suppose $n$ players take turns selecting vertices of the grid $[k]^n = \left\{0, 1, 2, \ldots, k-1\right\}^n$. Each player is assigned a pair of opposite faces of the grid, and wins the game if they ...

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261 views

### An approximate infinite-dimensional fixed point theorem

Given $\epsilon > 0$ and $f : [0, 1]^{\omega} \rightarrow [0, 1]^{\omega}$, can we find $x$ such that $x \in \textrm{Conv}\left( \left\{f(y) : ||y - x||_{\infty} < \epsilon\right\}\right)$?
In ...

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98 views

### Fixed point theorm that does not require the hemi-continuity of the set valued map?

All of the fixed point theorem I have seen (like Kakutani and Brower, Browder ) required the set valued map to be hemi-continuous (lower). Is any fixed point theorem that can assure the existence of ...

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59 views

### can we say fixed point existance of a set valued map over a compact set is homotopy invariant?

Consider two set valued maps over different compact sets as $F(\mathbf{x}):D\rightarrow\rightarrow D$, $G(\mathbf{x}):E\rightarrow\rightarrow E$ where $D,R\subset Y$. Assume there is a homotopy pair ...

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265 views

### A Fixed point Theorem that does not need the convexity of set valued map?

I am looking for a fixed point theorem for set valued maps that does not assume the set valued map should be convex valued.
Something like contractiblity or other properties can be replaced with ...