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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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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5 votes
1 answer
315 views

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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2 votes
0 answers
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Is every atriodic nonseparating plane continuum chainable?

There exist atriodic tree-like continua which are not chainable, though the examples that I'm aware of are very complicated. Is every atriodic tree-like plane continuum chainable? This is equivalent ...
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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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2 votes
1 answer
229 views

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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3 votes
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148 views

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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2 votes
1 answer
160 views

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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3 votes
0 answers
69 views

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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3 votes
0 answers
135 views

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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69 votes
2 answers
4k views

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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0 answers
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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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2 votes
2 answers
227 views

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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6 votes
1 answer
288 views

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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0 votes
1 answer
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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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12 votes
2 answers
955 views

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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0 votes
1 answer
132 views

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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1 vote
1 answer
127 views

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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14 votes
2 answers
844 views

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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1 vote
0 answers
49 views

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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2 votes
1 answer
90 views

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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1 vote
1 answer
129 views

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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0 votes
0 answers
85 views

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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2 votes
0 answers
131 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);...
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3 votes
0 answers
64 views

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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2 votes
3 answers
490 views

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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2 votes
0 answers
80 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 ...
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2 votes
1 answer
313 views

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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4 votes
1 answer
429 views

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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7 votes
2 answers
218 views

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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3 votes
0 answers
39 views

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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4 votes
1 answer
321 views

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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4 votes
1 answer
205 views

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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0 votes
1 answer
105 views

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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2 votes
1 answer
107 views

$x '(t) + g (x (t)) = f (t),\quad \forall t\in \mathbb R$ 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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0 votes
0 answers
35 views

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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2 votes
0 answers
121 views

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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2 votes
0 answers
106 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 ...
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  • 1,259
4 votes
2 answers
82 views

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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0 votes
0 answers
47 views

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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3 votes
0 answers
156 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 ...
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1 vote
0 answers
218 views

Is my ansatz for finding $n$-periodic-points of the exponential-function exhaustive?

The following is about getting help for a proof on existence and indexability of periodic points of the exponential-function, here with base $e:=\exp(1)$. Update The question is a complete rewriting ...
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3 votes
0 answers
67 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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-3 votes
1 answer
214 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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2 votes
1 answer
93 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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4 votes
0 answers
284 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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  • 269
2 votes
1 answer
154 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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  • 4,875
4 votes
2 answers
244 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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  • 1,157
3 votes
1 answer
136 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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2 votes
0 answers
139 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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2 votes
1 answer
134 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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