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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Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$. ...
Gil Kalai's user avatar
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Converse of Knaster-Tarski's theorem as choice principle

Knaster-Tarski's theorem States that if $(A,\le)$ is a complete lattice, then every monotone function $f :A \to A$ has a fixed point. The proof is carried out in $\mathsf{ZF}$. By $\mathsf{KTC}$ we ...
Average-user's user avatar
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Fundamental groups of reduced subgroup lattices

Let $G$ be a group. Its subgroup lattice, denoted $\Sigma G$, consists of all subgroups of $G$ partially ordered by inclusion. The topology of this poset is quite trivial, since it always has a ...
Vidit Nanda's user avatar
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10 votes
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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 ...
Sam Eisenstat's user avatar
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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
8 votes
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270 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 ...
Włodzimierz Holsztyński's user avatar
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Are bounded groups of thin operators on Hilbert space similar to groups of unitaries?

QUESTION. Let $G$ be a group of bounded operators on $\ell^2$, satisfying $\sup_{x\in G} \lVert x\rVert <\infty$, whose elements are all of the form "identity+compact" (sometimes called &...
Yemon Choi's user avatar
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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
5 votes
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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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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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Does a contractible locally connected continuum have an fixed point property?

I'm surprised that I can't find any research on this topic. Maybe it's too obvious? Kinoshita proved that contractible continuum do not have FPP, but his example is not locally connected. Maybe if we ...
LoliDeveloper's user avatar
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Geometrical meaning of spaces that possess the weak* uniform Kadec-Klee property

What is the geometric meaning or interpretation of spaces that possess the weak* uniform Kadec-Klee property? I am writing the last part of my undergraduate thesis and I would like to add a comment in ...
Tomás Pérez Fernández's user avatar
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Banach's fixed point theorem for quasilinear parabolic PDEs

I have recently started reading into PDE theory, and came across the following question. Consider the PDE $$ \begin{cases} \partial_t \rho = \Delta (\rho + \rho^2) & \text{ on } (0,T) \times \...
Peter Koepernik's user avatar
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Conjecture on convergence of iterated near-matrix square root

Here is a simple problem that has stumped me for some time; sharing with the community, as I suspect it has been solved somewhere, or is immediately implied by the correct theorem. Let $\textbf{diag}: ...
Keith Rush's user avatar
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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 ...
MasM's user avatar
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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
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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) ( \...
Changwang Zhang's user avatar
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Binomial Expectation of Convex Function

Suppose $x$ has a binomial distribution with chance $\alpha$ drawn $k$ times, and let $f(x)$ be a positive convex real valued function. I would like to evaluate $$\frac{\partial}{\partial \alpha} \...
Josh's user avatar
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Fixed point theorem for convex, closed multivalued mapping

There is well-known fixed point theorem theorem for multivalued l.s.c. maps, based on Michael selection theorem: Suppose, that $X$ is compact, convex and metrizable in locally convex Hausdorff ...
Maciej Skorski's user avatar
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Is there a fixed point theorem that applies to $f: \sum_k x_k 10^k \mapsto \sum_k x_k!$?

Let $f:\mathbb{Z}\rightarrow \mathbb{Z}$, $f:x=\sum_k x_k 10^k \mapsto \sum_k x_k!$ where $x_k$ is the $k$-th digit of $x$ in base ten. This function came up in a Project Euler problem. The question ...
Jackson Walters's user avatar
3 votes
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243 views

Is every tree a deformation retract of the disk?

I apologise if this question is not suitable for MathOverflow. We define a graph here to mean a disjoint union of points and copies of $[0,1]$ quotiented so that the endpoints of any interval lie on a ...
Isky Mathews's user avatar
3 votes
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A foliation version of S.Husseini counter example in fixed point theory

In this question we are indirectly inspired by an example by S.Husseini Amer. J.Math 1977 "The Products of Manifolds with the f.p.p. Need Not have the f.p.p" who gave an example of two ...
Ali Taghavi's user avatar
3 votes
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Non-closed trajectories in convex billiards

This is a weak version of this problem, written down in Lviv Scottish Book. I start with necessary definitions. Let $K=-K$ be a centrally symmetric compact convex body in the Euclidean space $\mathbb ...
Taras Banakh's user avatar
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Fixed point iteration algorithm when the inputs have dependencies

A usual fixed point problem has the form $x_{k+1}=f(x_k)$, and you can efficiently solve it by finding the root to $f(x)-x$. What if I now have several dependent inputs $x_{k+1}=f(x_k, y_k)$, and $y_k=...
user3799934's user avatar
3 votes
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193 views

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 ...
ludwigmach's user avatar
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68 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-...
Ruos's user avatar
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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 ...
Ludwig's user avatar
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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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3 votes
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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}{\...
user1337's user avatar
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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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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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3 votes
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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 Picard–...
Sridhar Ramesh's user avatar
3 votes
0 answers
359 views

Convergence to a (unique?) fixed point?

Consider a given $N\times P$ matrix $X$ (full rank with columns ${\bf x}_p$, $p=1,\ldots,P$), a given vector ${\bf y}\in R^N$ and a thresholding function $\eta_\lambda(|x|)=(|x|-\lambda)_+$ with $\...
Sardy's user avatar
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2 votes
1 answer
101 views

Teaching suggestions for Kleene fixed point theorem

I will take over two lectures from a colleague in which we discuss fixed point theory in the context of complete partial orders, and culminates in showing the Kleene fixed point theorem (see f.e. ...
JustVisiting's user avatar
2 votes
0 answers
66 views

Is it known whether a homeomorphism close to the identity of a compact manifold with nonzero Euler characteristic necessarily has a fixed point?

I recently saw in Kirby's list of open problems that it isn't known if two commuting homeomorphisms of a compact manifold close to the identity necessarily share a common fixed point, when the ...
Ahmad Rafiqi's user avatar
2 votes
0 answers
138 views

'Contraction-like' inequality: how to deal with the boundary term?

I am interested in the following problem. Let $D = \mathrm{diag}(d_1, d_2, \ldots, d_n) \in \mathbb{R}^n$ be positive definite, let $B, K \in \mathbb{R}^n$, and let $G\in L^\infty((0, T)\times (0, L);...
user344045's user avatar
2 votes
0 answers
92 views

Existence of infinite horizon values in dynamic programming

I am working through the book "Foundations of Stochastic Inventory Theory". One of the results in the book is Theorem 11.2. The background to this theorem is as follows. Given finite state ...
Tryer's user avatar
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The contraction principle in quasi metric spaces

I am researching contractive mappings and I need the article of I. A. Bakhtin "The contraction principle in quasi metric spaces"(1989) or at least part where explanation is given for ...
Dušan Bajović's user avatar
2 votes
0 answers
152 views

Can Schauder's fixed point theorem apply to a metric space?

I am currently reading the existence proof of Mean Field Game equation, which is a coupled system of Hamilton-Jacobi-Bellman equation and Fokker-Planck equation, see page 42 of the paper here. The ...
kenneth's user avatar
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2 votes
0 answers
150 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 ...
Peter's user avatar
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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}{\...
Debreu's user avatar
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2 votes
0 answers
85 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 $N:S^n\to \mathbb{R}^n$, $...
MasM's user avatar
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2 votes
0 answers
158 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 ...
TJP's user avatar
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131 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. $$\|...
Chazz's user avatar
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2 votes
0 answers
61 views

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
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
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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2 votes
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266 views

Fixed point indices of simplicial maps

Let $K$ be a simplicial complex and let $f:K \to K$ be a simplicial map. Assume the existence of an embedding $\iota: K \hookrightarrow \mathbb{R}^n$ of this complex into Euclidean space and note ...
user24459's user avatar
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0 answers
409 views

Important lines in triangle - reverse problem

It is known that if three numbers $x,y,z$ are the lengths of the edges of some triangle, then there exists a triangle with medians of length $x,y,z$. Also, if $x,y,z>0$ (no condition imposed) there ...
Beni Bogosel's user avatar
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1 vote
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Is the Poincare Birkhoff theorem valid if we change the volume form of the annulus region?

Is the Poincare-Birkhoff theorem valid if we change the volume form of the annulus region? Note: A possible approach could be the following: Is it true to say that the answer is affirmative ...
Ali Taghavi's user avatar