# 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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### 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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### 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 ...
1 vote
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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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### 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 ...
1 vote
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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 ...
1 vote
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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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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 $\mathbb ... 2 votes 3 answers 490 views ### How do I apply Brouwer fixed-point theorem in this claim? Let$\zeta:\mathbb{R}\to [0,+\infty)$be a continuous non-negative function such that$\zeta(0)=0$and$\tau\mapsto \zeta(\tau)\tau$is a non-decreasing differentiable function whose derivative is ... 2 votes 0 answers 80 views ### Existence of infinite horizon values in dynamic programming I am working through the book "Foundations of Stochastic Inventory Theory". One of the results in the book is Theorem 11.2. The background to this theorem is as follows. Given finite state ... 2 votes 1 answer 313 views ### What is the fixed-point origin? There is a fixed-point construction used in Anil Gupta & Nuel Belnap, The Revision Theory of Truth, MIT-Press 1993, p. 194: Use only$\wedge, \lnot$and$\forall$as primitive connectives and ... 4 votes 1 answer 429 views ### Real eigenvectors of complex matrices Let$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$to have real eigenvectors ... 7 votes 2 answers 218 views ### Fixed point for a map from$\{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 $$(x_1,x_2,\ldots,x_N)\leq (y_1,\ldots,y_N)\ \text{ if and only if for all }... 3 votes 0 answers 39 views ### Fixed point iteration algorithm when the inputs have dependencies A usual fixed point problem has the form x_{k+1}=f(x_k), and you can efficiently solve it by finding the root to f(x)-x. What if I now have several dependent inputs x_{k+1}=f(x_k, y_k), and y_k=... 4 votes 1 answer 321 views ### Construct closed chain of k-gon around n points-n, k are odd primes number Question 1: I am looking for a proof of the conjectures 1, 2, 3 as follows? Question 2: In conjecture 3, in general case, I can not give a formula of X. But I think, If n, k are odd primes number ... 4 votes 1 answer 205 views ### A closed chain of 2n+1-gon around 2n+1-points I posed a generalization of Theorem 3.2 In my paper Conjecture: Let P_1, P_2,....,P_{2n+1} and O be 2n+2 points in plane. Construct a chain 2n+1 regular {2n+1}-gons A_{1\;1}A_{1\;2}...A_{1\;... 0 votes 1 answer 105 views ### On proving the absence of limit cycles in a dynamical system I'm studying this classical paper in nonlinear dynamics in biophysics and I want to understand it properly. I'm stuck at this two-variable problem which I've struggled with for too long now.$$ \dot M ... 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$\mathbb R$. ... 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 ... 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 ... 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 ... 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$B_{2n}$... 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 $$f^*(x) := ... 3 votes 0 answers 156 views ### Constructive proof of the approximate Brouwer's Fixed Point Theorem for \Delta^n The approximate Brouwer Fixed Point Theorem (aBFPT) for the standard n-simplex is: Let f be a uniformly continuous function from \Delta^n into itself. Then for each \varepsilon>0 there ... 1 vote 0 answers 218 views ### Is my ansatz for finding n-periodic-points of the exponential-function exhaustive? The following is about getting help for a proof on existence and indexability of periodic points of the exponential-function, here with base e:=\exp(1). Update The question is a complete rewriting ... 3 votes 0 answers 67 views ### Convergence of a nonlinear iterative sequence I have the following iterative sequence: \begin{eqnarray*} a_{t+1} &=& (1+\alpha-\beta)^2a_{t} - 2\alpha(1+\alpha-\beta)b_{t} +\alpha^2a_{t-1}+\frac{L}{a_{t}}, \\ b_{t+1} &=& (1+\alpha-... -3 votes 1 answer 214 views ### Function satisfying f(x)^{f^{-1}(x)}=x^2 with f^{-1} is a compositional inverse of f and f:\mathbb{R+}\to \mathbb{R+}? Let f be a function such that :f:\mathbb{R+}\to \mathbb{R+} and f^{-1} is a compositional inverse of f , I have tried to find solution of the following functional f(x)^{f^{-1}(x)}=x^2, I ... 2 votes 1 answer 93 views ### Convergence of estimator given by a fixed point Let X be a non-negative random variable with cdf F and define$$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$X_1,\dots,X_t$... 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 ... 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\...
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 ...