# 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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### Constructive proof of the approximate Brouwer's Fixed Point Theorem for $\Delta^n$

The approximate Brouwer Fixed Point Theorem (aBFPT) for the standard $n$-simplex is:
Let $f$ be a uniformly continuous function from $\Delta^n$ into itself.
Then for each $\varepsilon>0$ there ...

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55 views

### Convergence of a nonlinear iterative sequence

I have the following iterative sequence:
\begin{eqnarray*}
a_{t+1} &=& (1+\alpha-\beta)^2a_{t} - 2\alpha(1+\alpha-\beta)b_{t} +\alpha^2a_{t-1}+\frac{L}{a_{t}}, \\
b_{t+1} &=& (1+\alpha-...

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188 views

### Function satisfying $f(x)^{f^{-1}(x)}=x^2$ with $ f^{-1}$ is a compositional inverse of $f$ and $f:\mathbb{R+}\to \mathbb{R+}$?

Let $f$ be a function such that :$f:\mathbb{R+}\to \mathbb{R+}$ and $f^{-1}$ is a compositional inverse of $f$ , I have tried to find solution of the following functional $f(x)^{f^{-1}(x)}=x^2$, I ...

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87 views

### Convergence of estimator given by a fixed point

Let $X$ be a non-negative random variable with cdf $F$ and define
$$G(s) = E[\max(0,u(X)-sX)],$$ where $u$ is some real function.
Let $s_0$ be the unique fixed point of $G$.
Now let $X_1,\dots,X_t$ ...

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208 views

### On some characteristics of continuous maps $S^n \to \mathbb{R}^n$

I've asked this question about two month ago in math exchange but there were no answer to it.
Any information or paper relating to this question is appreciated.
By the Borsuk-Ulam theorem we know ...

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137 views

### Dual fixed point

Let $E$ be a Banach space, let $T:E\to E$ have norm $1$ and let $\nu\in E^*\setminus\{0\}$ be such that $T^*\nu=\nu$. Under which conditions there is $e\in E$ such that $Te=e$ and $\langle e,\nu\...

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168 views

### Contactomorphisms have in general no fixed points

Let $(V, \xi = \ker \alpha)$ be a cooriented contact manifold. A contactomorphism of $(V, \xi)$ is a diffeomorphism $\phi$ which preserves the contact structure $\xi$ and its coorientation. In other ...

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123 views

### Is this sum of nonexpansive maps itself nonexpansive?

For Hilbert spaces $\mathcal{H}_X$, $\mathcal{H}_Y$ and $\mathcal{K}$, consider a linear map $f\colon \mathcal{K} \oplus \mathcal{H}_X \to \mathcal{K} \oplus \mathcal{H}_Y$ that is given as a matrix
$...

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120 views

### Browder's Fixed Point Theorem in uniformly convex Banach spaces with non-identical image

This is in fact an exercise from Dirk Werner's book "Funktionalanalysis", but I do think that the result is quite interesting and up to now, I can only partly solve this problem. From the point of my ...

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109 views

### Is there a fixed-point index theorem that treats the fixed points on the boundary?

Suppose $f$ is a continuous function from a unit cube $[0,1]^n$ to itself, then $f$ has at least a fixed point. Further suppose $f$ is smooth, $0$ is a regular value of $f(x)-x$, and the fixed points ...

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166 views

### “Determinant” rather than “trace” in the alternative formula “Lefschetz number”

For a self map $f$ on a topological space $X$ we replace "trace" with "determinant" in the alternative Lefschetz formula $$\Lambda(f)=\sum(-1)^i trace(f^*)|H^i(X,\mathbb{Q})$$
So we have
$$\...

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420 views

### Fixed points under a finite group action on projective variety

Let us have an algebraic action by a finite group G on a complex projective variety $X=\bigcup\limits_{i=1}^N X_i$, whose irreducible components $X_i$ are all smooth and of the same dimension $d$, and ...

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53 views

### Sufficient and necessary condition for the global uniqueness of fix-points

https://www.ams.org/journals/proc/1976-060-01/S0002-9939-1976-0423137-6/S0002-9939-1976-0423137-6.pdf
This paper gives a sufficient condition for the uniqueness of SCHAUDER fix point. I wonder if ...

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69 views

### Existence of a fixed point for this operator

I'm looking for some mathematical results that I might be able to apply to see if an operator I'm considering has a fixed point.
In particular consider,
$$ Ag(x) = \Big\{ \xi(x) + K(g(x))^\frac{1}{\...

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77 views

### $n$-point map $S$ from $k$-manifold $X$ to $\mathbb{R}^k$

It is known that for every continuous map $M:S^1\to \mathbb{R}$ there are infinitely many double points. Also by the Borsuk-Ulam theorem, this is true for each continuous map $N:S^n\to \mathbb{R}^n$, $...

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153 views

### Why is $\widetilde{W}$ closed?

We consider $(x _{n})$ a sequence of almost fixed points for $T$ in $C _{0}$. Since $C _{0}$ is weakly compact, we can assume that $(x _{n})$ is weak compact. Also, since the problem of the fixed ...

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84 views

### When are quadratic integer programs “easy to solve”?

Let $N_i=\{0,1,\dots,\bar{n}_i\}$ and define $N=N_1\times \dots \times N_I$. I want to maximize $f$ on $N$. $f$ has the following form
$$
f(n) = \sum_i A_i n_i -\sum_i \sum_{j\neq i} B_{ij} (n_i-n_j)^...

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134 views

### $B _{\ell ^{2}} ^{+}$ with the norm $\lVert\lvert \cdot \rvert\rVert _{\sqrt{2}}$ doesn't have normal structure

$\newcommand\binorm[1]{\lVert#1\rVert}\newcommand\trinorm[1]{\lVert\lvert#1\rvert\rVert}$Consider the space $\ell ^{2}$ with the standard norm
\begin{align*}
\binorm x_{2} = \left( \sum _{i =1} ^{\...

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659 views

### How is this fixed point theorem related to the axiom of choice?

I'm hoping the answer to this is well-known.
Let $X$ be an ordered set (i.e. poset). An inflationary operator $f$ on $X$ is a function $f: X \to X$, not necessarily order-preserving, such that $f(x) ...

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199 views

### On the existence of fixed points of a matrix iteration

Let $A\in\mathbb{R}^{n\times n}$ be a Hurwitz stable matrix (i.e. all eigenvalues of $A$ have negative real part). Let $\succeq$ denote the standard partial order in the cone of positive semidefinite ...

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318 views

### Does the Banach space $( \ell ^2 \oplus \ell ^2 )$ have F.P.P?

The space $( \ell^2 ,\lVert \cdot \rVert _2 )$ is a Hilbert space. The space
$X=(\ell^2 \oplus \ell^2 , \lVert \cdot \rVert_\infty )$ is a Banach space. Does X have fixed point property? (For any ...

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72 views

### Takahashi minimization theorem for lower pseudo-continuous functions on complete metric spaces

Takahashi minimization theorem says : Let $(X,d)$ is a complete metric space, $f:X\to \mathbb{R}\cup\{+\infty\}$ is a proper(not constantly +$\infty$) lower semi continuous function, which is bounded ...

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136 views

### Property of Fixed Point Function

Given an operator $\mathcal{T}$ that maps from a function $f: \mathbb{R}^d\rightarrow \mathbb{R}$ to another function $f': \mathbb{R}^d\rightarrow \mathbb{R}$, we are interested in the fixed point $f^*...

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150 views

### On the notion of multiplicity of a fixed point [closed]

I am trying to understand the notion of multiplicity of a fixed point of a map $f: M \to M$, say, $M$ being a smooth closed manifold, and $f$ being a smooth diffeomorphism.
There is a notion of ...

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148 views

### Betti sum, cup-length, Lusternik-Schnirelmann category, and critical points

I am trying to make some order in the notions of cup-length, sum of Betti numbers, LS category, critical points of functions.
Let $M$ be smooth closed compact manifold. We denote by Crit($M$) the ...

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302 views

### On Lefschetz theorem and sum of Betti numbers as lower bounds for fixed points

Let $M$ be a closed manifold with holomorphic cell decomposition (if it is complex), or at least with only even cohomology. In particular, its Euler characteristic is equal to the sum of its Betti ...

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123 views

### Integrand in equivariant Atiyah-Patodi-Singer index theorem

I am trying to determine the integrand in the equivariant Atiyah-Patodi-Singer index theorem by Donnelli ("Eta invariants for G-spaces") for spin manifolds if one only has isolated fixed points. In "...

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138 views

### Fixed-point properties for affine actions of topological groups

T. Mitchell [Illinois J. Math. 14 (1970) 630--641] defined four properties of a topological semigroup, and in particular of a topological group $G$.
Two of them are:
(F2) Every jointly continuous ...

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388 views

### Existence of Solution, System of Equations

Suppose $P(\lambda, i)$ is the probability that a Poisson random variable with average $\lambda$ is equal to $i$, i.e. $\frac{\lambda^i}{e^{\lambda}i!}$
I think the following system of equations ...

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261 views

### Is it possible to continuously select a probability distribution over fixed points in Brouwer's fixed point theorem?

According to Brouwer's fixed point theorem, for compact convex $K\subset\mathbb{R}^n$, every continuous map $K\rightarrow K$ has a fixed point.
However, these fixed points cannot be chosen ...

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192 views

### Question on K.Gobel's paper 1969

Let $X$ be uniformly convex Banach space. $f:K\rightarrow K$, such that $\parallel fx-fy\parallel \leq\parallel x-y\parallel\,\,\forall x,y\in K $, with $K$ a nonempty, closed, convex, bounded subset ...

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108 views

### prove the singularity of a matrix as solution of a non-linear equation

Let $B$ ($n \times n$) and $R$ ($m \times m$) be two square matrix with $n>m>0$ who satisfie:
$B=(I-KH)B(I-KH)^T+K RK^T$
with $K=BH^T(HBH^T+R)^{-1}$ and $rank(H)=m$
I would like to prove $...

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405 views

### Solution of an equation with Jacobi theta function

I have been struggling with this equation for some time and I do not seem to find any conclusive answer (it's from my research, not a homework).
It has to do with the real solutions $x$ to the ...

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243 views

### Brouwer fixed-point for open ball and bijective uniformly continuous function?

Let $B^-$ be the open unit ball in $\mathbb{R}^n$ for $n\in\mathbb{N}$, and let $f$ be a uniformly continuous function from $B^-$ to itself, with a uniformly continuous inverse $f^{-1}$.
Under these ...

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156 views

### A particular measure of noncompactness?

I am working on an article based mainly on the notion of Measure of non-compactness, to study a particular type of fixed point theorems.
Let $\mathcal M $ to be the family of all nonempty bounded
...

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128 views

### Banach spaces with unconditional basis have w-FPP

A Banach space $X$ is said to have w-FPP (weak fixed point property) if for every non-empty, weakly compact and convex subset $K\subseteq X$; every non-expansing mapping $T:K\longmapsto K$ i.e.
$$\|...

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382 views

### Base of topology for metric-like space

Let $X$ be a nonempty set and $p:X\times X\rightarrow\mathbb{R}^+ $ be a function satisfying the following conditions for all $x,y,z\in X$: \begin{align} &1)\enspace p(x,y)=0\implies x=y \\ &2)...

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177 views

### Can we internalize topological fixed point theorems in an effective topos?

Reflective oracles are a kind of Turing oracle that give stochastic answers about the outputs of Turing machines. This works in a self-referential way, where they can answer queries about Turing ...

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89 views

### 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 \...

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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 ]\...

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144 views

### On fixed point probability in discrete logarithm?

Given integer $n>2$ 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 $h^{x\bmod\varphi(n)}\equiv x\bmod n$?

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519 views

### 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 ...

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408 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....

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### 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}{\...

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462 views

### 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 ...

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280 views

### 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 $...

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137 views

### 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 ...

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784 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}^{...

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206 views

### 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 ...

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382 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 $\varepsilon >0$,...