# 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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### The ultrapower of the direct sum is the direct sum of ultrapowers

Currently, I'm reading the paper "Towards the fixed point property for superreflexive spaces" by Andrzej Wiśnicki. In this article, given $X_1,\dots,X_n$ Banach spaces, he defines $(X_1\...

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### Implementable numerical scheme for the equation $a=\text{Erf}\big(z/\sqrt{2N_{a}}\big)$

Let $z>0$ be fixed and $A$ be the set of non-increasing functions from $\mathbb R_+$ to $[0,1]$ with norm $\|\cdot\|:=\|\cdot\|_\infty$. Define by $F$ the operator on $A$ by
\begin{equation*}
F(...

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### Invariant subspaces for matrices via fixed points on Grassmannians

Let $A$ be an $n \times n$ invertible complex matrix. Let $Gr(k)=Gr(k,\mathbb{C}^n)$ be the complex $k$-Grassmannian, $1\leq k \leq n$. Since $A$ is invertible, it maps a $k$-dimensional subspace to a ...

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### Fixed point theorem for the uncountable power of an interval

Does the Brouwer fixed point theorem holds for the uncountable power $[0,1]^\kappa$ of the interval, $\kappa\geq\aleph_1$ ?
That is, does every continuous endomorphism $[0,1]^\kappa\to [0,1]^\kappa$ ...

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### Fixed points of rational continuous piecewise affine maps

Say that a compact convex polytope is rational if is the intersection of half-spaces whose bounding hyperplanes are the zero-sets of affine functions of the coordinates with rational coefficients. Say ...

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### 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 ...

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### Differentiability of the fixed points of a family of contraction maps

Given a general Banach space $B$ and a one-parameter family of contractions $F_t:B\to B$ which is defined for all $t \in (a,b)$. $F_t$ depends continuously on $t$ (in the sense $\lim_{\varepsilon\to 0}...

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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 ...

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### Fixed points cohomology via Lannes T-functor

Is there any reference to the proof of following: let $T$ denote the Lannes functor. Then (see the link above for more details) for any finite $E$-complex $X$ (where $E$ is finite-dimensional $\mathbb ...

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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 ...

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### Has Goedel's Second Incompleteness Theorem been proven using Lawvere's Fixed Point Theorem?

This question is a request for assistance in surveying the existing literature on applications of Lawvere's Fixed Point Theorem (LFPT).
Yanofsky [0] has demonstrated several applications of LFPT to ...

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### Infinite vertex-transitive graph where every automorphism has a fixed vertex

This is a follow-up to the question Connected vertex-transitive graph with the fixed-point property. In particular, it is based on a comment by user bof.
Let $G = (V,E)$ be a graph with $V$ infinite. ...

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### Connected vertex-transitive graph with the fixed-point property

Many connected vertex-transitive graphs $G=(V,E)$ have the property that some of their automorphisms other than the identity have fixed points. To point out two simple examples:
If $G = K_3$ then the ...

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### Reference for unique classical solution to quasilinear uniformly parabolic PDEs

In this post, the author mentioned that "we know there is a unique classical solution (see the references below, for example)". I have tried to read the two references the author provided, ...

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### Two maps into $[0,1]$ are equal at some point

In the paper below, there appears the following theorem:
whose proof is left to the reader. It's not immediately obvious how I would prove this. How about the special case $X=Y=[0,1]$? It seems to be ...

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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 \...

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### The Existence of PDE by Banach vs Leray-Schauder fixed point

Regarding their proof, I deem the Banach fixed point theorem to be more analytical while Leray-Schauder more topological in nature. Owing to this, I am more inclined to use Banach method first, but ...

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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 ...

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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}: ...

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### Does iterating the derivative infinitely many times give a smooth function whenever it converges?

I am a graduate student and I've been thinking about this fun but frustrating problem for some time. Let $d = \frac{d}{dx}$, and let $f \in C^{\infty}(\mathbb{R})$ be such that for every real $x$, $$g(...

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### Fixed point of a contraction map

This question is a continuation of Is this a contraction mapping for small $T$?
Set, for $T, m>0$, $H^m_T:=\{h:[0,T]\to [0,m]:~ h,~h' \mbox{ are both continuous on } [0,T]\}$ endowed with the norm
$...

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### Measure of non-commutativity of two invertible functions

I need to estimate $|x - f^{-1}(g^{-1}(f(g(x))))|$ for various values of $x$ for two smooth invertible functions $f$ and $g$ on $\mathbb{R}$ (actually some other spaces, but $\mathbb{R}$ will do.) Are ...

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### The Tarski-Lindenbaum theorem of the middle value

In Ettore Casari, La Matematica della Verita' (2006), the following theorem (attributed to Tarski and Lindenbaum 1926), is stated, which Casari calls the theorem of the middle value of Tarski-...

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### Can this fixed point theorem generalize to infinite structures?

Suppose that $n$ is a natural number, $X$ is a set, and $S\subseteq X^{2}$ is a subset such that if $x,y\in X$, then there is a unique tuple $(x_{0},\dots,x_{n})$ where $x_{0}=x,x_{n}=y$ and $(x_{i},...

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### A variation of the Ryll-Nardzewski fixed point theorem

Is there a fixed-point theorem that implies the following result?
Let $F$ be a nonempty convex set of functions on a discrete group with values in $[0,1]$. Suppose $F$ is invariant with respect to ...

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### Continuity of Kakutani fixed points

Let $X$ be a compact and convex space and let $T=[0,1]$ be some parameter space. Let $F:X\times T\rightrightarrows X$ be a correspondence that is compact-valued, convex, and upper-hemicontinous. By ...

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### Invariant distributions for iterated random variables (stochastic dynamical systems)

This is related to discrete dynamical systems, with the initial condition $X_1$ being a random variable with a non singular distribution. The system is driven by the iteration $X_{n+1} = g(X_n)$ for ...

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### Is there a square with all corner points on the spiral $r=k\theta$, $0 \leq \theta \leq \infty$?

I've posted this question on Math Stack Exchange, but I want to bring it here too, because 1) the proof seems missing in the literature, although they are some sporadic mentions and 2) maybe it ...

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### Fixed-point theorem for the space of probability flux

Let $\mathcal P_T:=\{\mu=(\mu_t)_{0\le t\le t}: \mu_t\in\mathcal P,~ \forall 0\le t\le 1\}$, where $\mathcal P$ is the space of probability measures on $\mathbb R$. Denote by $\rho$ the metric that is ...

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### Smooth dependence in the fixed point theorem between complete Fréchet manifolds

Let $X,Y$ be complete metric spaces, and let $\Sigma:X\times Y\rightarrow Y$ be a continous mapping which satisfies the following property: there exists a $C<1$, such that for all $x\in X$ and $y_{...

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### Classical fixed-point argument and invertible function

Let $n\in\mathbb{N}$ and $W^{1,\infty}(\mathbb{R}^n)=\lbrace f:\mathbb{R}^n\rightarrow \mathbb{R}^n : \text{ f is bounded and Lipschitz continuous } \rbrace$. Suppose $f\in W^{1,\infty}(\mathbb{R}^n)$ ...

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### Banach fixed point theorem / convergence squeeze

I am trying to prove a convergence result on an iterative scheme which has the initial point defined as
$$x_1 = \frac{1 - s(x_0)}{s(x_0)}$$
where s(x) is some unknown function.
Here is my theorem and ...

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### '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);...

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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 ...

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### How do I apply Brouwer fixed-point theorem in this claim?

Let $\zeta:\mathbb{R}\to [0,+\infty)$ be a continuous non-negative function such that $\zeta(0)=0$ and $\tau\mapsto \zeta(\tau)\tau$ is a non-decreasing differentiable function whose derivative is ...

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### 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 ...

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### What is the fixed-point origin?

There is a fixed-point construction used in Anil Gupta & Nuel Belnap, The Revision Theory of Truth, MIT-Press 1993, p. 194: Use only $\wedge, \lnot$ and $\forall$ as primitive connectives and ...

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### Real eigenvectors of complex matrices

Let $A$ be a nonsingular complex $(3 \times 3)$-matrix (that is, an element of $\mathrm{GL}_3(\mathbb{C})$).
Then what are some of the best-known criteria which guarantee $A$ to have real eigenvectors ...

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### Fixed point for a map from $\{0,1\}^N$ to itself

Let $N\geq2.$ Let $F$ be a function from $\left\{ 0,1\right\} ^{N}$ to itself
dreceasing for the product order defined by $$ (x_1,x_2,\ldots,x_N)\leq (y_1,\ldots,y_N)\ \text{ if and only if for all }...

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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=...

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### Construct closed chain of $k$-gon around $n$ points-$n, k$ are odd primes number

Question 1: I am looking for a proof of the conjectures 1, 2, 3 as follows?
Question 2: In conjecture 3, in general case, I can not give a formula of $X$. But I think, If $n, k$ are odd primes number ...

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### A closed chain of $2n+1$-gon around $2n+1$-points

I posed a generalization of Theorem 3.2 In my paper
Conjecture: Let $P_1, P_2,....,P_{2n+1}$ and $O$ be $2n+2$ points in plane. Construct a chain $2n+1$ regular ${2n+1}$-gons $A_{1\;1}A_{1\;2}...A_{1\;...

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### On proving the absence of limit cycles in a dynamical system

I'm studying this classical paper in nonlinear dynamics in biophysics and I want to understand it properly. I'm stuck at this two-variable problem which I've struggled with for too long now.
$$ \dot M ...

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### $x '(t) + g (x (t)) = f (t),\quad \forall t\in \mathbb R$ have periodic solution $\iff\; \frac 1T \int_0 ^ T f (t) dt \in g (\mathbb R) $

I have a research work concerning the equation: $$x '(t) + g (x (t)) = f (t),\quad \forall t\in \mathbb R$$
f and g are defined and continuous in $\mathbb R$ and with values in $\mathbb R$.
...

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### Terminology: Almost stable states

I have a question about fixed points which are almost stable.
I have an increasing transition function $f:[0,1]\rightarrow[0,1]$ where $f(0)>0$ and $f(1)<1$ but I don't necessarily have ...

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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 ...

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### 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 ...

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### From a point and continuing reflection in $2n+1$ points then midpoint of the end point and the first point is fixed

Given $2n+1$ fixed points: $A_1, A_2,....,A_{2n+1}$ and point $P$. Let $B_1$ is the reflection of $P$ in $A_1$, $B_2$ is the reflection of $B_1$ in $A_2$,...., $B_{2n+1}$ is the reflection of $B_{2n}$ ...

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### Does the following operator have a unique fixed function? Do iterations of the operator converge?

The functions considered are positive real for positive real $x$, and monotonically increasing to infinity. Given such a function $f(x)$, define the (nonlinear) operator $f\mapsto f^*$ by
$$
f^*(x) := ...

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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 ...