# Tagged Questions

Fixed-point theorem is a result about existence of fixed points, i.e. points fulfilling F(x)=x, under some conditions on the function F. Results of this type appear in many areas of mathematics, e.g. functional analysis (Banach), algebraic topology (Brouwer), lattice theory (Knaster-Tarski, Kleene)...

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### Fixed point, bounded derivative

Let $p\in\mathbb{N}$. Let $f:I\to\mathbb{R}$ differentiable in the closed interval $I$ (bounded or not), with $f(I) \subset I$, and let $g = f\circ f\circ \cdots \circ f = f^p$, where $\circ$ means ...
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### Prove that intervals of the form $(a,b]$, $[a,b)$, $(-\infty,a]$, $[a,\infty)$ do not have the fixed point property.

Prove that intervals of the form $(a,b]$, $[a,b)$, $(-\infty,a]$, $[a,\infty)$ do not have the fixed point property. In the case of open intervals, I can derive that they do not have the fixed point ...
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### Show that the only intervals having the fixed point property are the closed intervals.

By Fixed Point Theorem, I know that it deals with closed interval, for eg, [0,1]. And this theorem will be false if [0,1] is replaced by (0,1). The counter example will be $f:(0,1)\rightarrow (0,1)$ ...
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### Can there be a metric space where no contraction has a fixed point?

We know that: If $X$ is a metric space, then every contraction has at most one fixed point. (Note: if metric space is complete, then we have existence and uniqueness) I wonder if there can be a ...
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### Fixed points of polynomial ring homomorphism

$S=\mathbb R[x+y+z, xy+yz+zx, xyz]$ is the ring of the symmetric polynomials in $\mathbb R[x,y,z]$. Let $\psi\colon S \to R[x,y,z]$ be a ring homomorphism such that \begin{align} x &\mapsto -x,\\...
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### How many fixed points does this function have?

The function is $f :\overline{\Bbb R}\to \overline{\Bbb R}, x \mapsto x^5$. So does it have $3$ or $5$ fixed points ? Thanks in advance !
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### Why is this easy “proof” of Brouwer's Fixed Point Theorem not correct/common?

Brouwer's Fixed Point Theorem states, essentially, that any continuous function on a closed disc to itself has a fixed point. I am familiar with the proof based on the impossibility of a retraction ...
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### Prove that $(Y,\mathcal T_1)$ also has the fixed point property

Let $(X\mathcal T)$ have the fixed point property and let $(Y,\mathcal T_1)$ be a space homeomorphic to $(X, \mathcal T)$. Prove that $(Y,\mathcal T_1)$ also has the fixed point property. I know ...
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The following question comes from Chapter 6.4, Exercise 4 on page 156 in the set of notes Topology Without Tears. Using Exercise 2 and 3, show that while $f: \mathbb R \to \mathbb R$ given by $f(x)... 2answers 32 views ### Where should I begin the study of fixed point theory, especially of multi-valued maps? How should one begin one's study of fixed point theory, especially of multi-valued maps? What background --- in topology, analysis, functional analysis, algebra, and set theory --- should one have? ... 0answers 16 views ### Fixed points and banach spaces [closed] Let$B$be a closed ball centered at$0$in a Banach space$E$and$F:B\to E$be a contractive map such that$F(x)=-F(-x)$for every$x\in\partial B$. Show that$F$has a fixed point. Any ideas? 1answer 29 views ### Function with unique fixed point of all orders Let$f:[a,b]\to [a,b]$be a continuous and monotone function. For each$n\ge 1$, there exists$x_n\in [a,b]$such that $$f^n(x_n) = x_n,$$ where$f^n(x) = f(f(\cdots(f(x))\cdots))$for$n$times. The ... 0answers 15 views ### Using Fixed point iterations for solving system of linear equations Given a system of$n$linear equations $$x_i=\sum_{k=1}^{n}a_{ik}x_k+b_i \quad i=1,2,...,n$$ I'd like to employ the fixed point iteration method to find$x_i. The fixed point iteration define x_i^... 1answer 42 views ### A Bending Buzz Wire Game There is a wire connecting an exit and an entry point. At the entry, the wire has height 0, at the exit, it has height 1. Since the wire is connected, the wire has height 1/2 somewhere, whatever ... 1answer 42 views ### Complete a proof that F(x,y) is contracting. Can anyone fill in the dots in this proof? Let D := [0,\frac{1}{2}]^2. Show there is exactly one (x,y)=(x^*,y^*)\in D such that \begin{align*} x &= \frac{x^3}{2} + y^4 + \frac{1}{4} \,, \... 2answers 57 views ### Prove that a function is contractive I'm stuck with the following. I need to prove that in D:=[0,1]\times[0,1] the function F is contractive, where F:\mathbb{R}^2\rightarrow\mathbb{R}^2 is defined as: \begin{align} F(x,y):=(\frac{... 0answers 15 views ### PDEs - prove continuity of operator Consider the following nonlinear problem \begin{cases} -div(a(u)\nabla u ) =0 & \text{in\Omega$} \\ u=0, & \text{on$\partial \Omega$} \end{cases} $$We can assume \Omega to be a ... 1answer 35 views ### Show there exists C\in\Bbb{R}^n such that |C-A_i|=|B-A_i|+u_i, with A_i,B\in \Bbb{R}^n and u_i close enough to 0 Let A_1,...,A_n,B be vectors in the n-dimensional Euclidean Space, such that they are never on the same affine (n-1)-dimensional subspace. (What? Is that a way to say they span \Bbb{R}^n?). ... 1answer 101 views ### Example of contraction mapping theorem failing for strict metric map Is there an example of f: [0,1] \to [0,1] s.t. |f(x)-f(y)|<|x-y| but a sequence x_0,f(x_0),f^2(x_0)... doesn't converge to its fixed point? where f^n denotes repeated application. Also, ... 0answers 60 views ### Proving a function isn't homotopic to a map to the boundary Let X be a compact Hausdorff space and S a finite dimensional, convex, Hausdorff space. Moreover, let f:X \to S be a closed, surjective map with \tilde{H}^q(f^{-1}(s))=0 for all q\ge 0 and ... 1answer 49 views ### Can there be a limit cycle without a fixed point in 3D space? I am working with a population dynamics model. Basically, I have a nonlinear ODE in R^3 space, (X,Y,Z), and I know that if I start in the an open region (0<X<1,0<Y<1,0<Z<1, ... 0answers 23 views ### Fixed Point in the Space of Rational Functions Let \mathcal R be the space of rational functions and F: \mathcal R \to \mathcal R be a function that transforms a rational function into another rational function. Is there a fixed point ... 2answers 73 views ### Open ball does not have fixed point How we can prove that the open ball in R^n does not have fixed point property (by algebraic topology concepts)? I know D^n -closed ball in R^n- has fixed point property by Brouwer's theorem, but ... 2answers 30 views ### Establish the sufficient condition |g'(x)| < 1 for convergence of an iteration using the Banach fixed point theorem? If x_n = g(x_{n-1}) is an iteration, it converges if g is continuously differentiable and |g'(x)| < 1. The Banach FPT says that if T is a contraction on a complete metric space X then it ... 0answers 50 views ### Fixed Point and Contraction Mapping Consider Tf(x) = \int_0^x e^{-f(s)^2} \; ds for x \in [0,\infty). I want to use the contraction mapping theorem to show that T:C^1([0,\infty)) \to C^1([0,\infty)) has a unique fixed point. From ... 1answer 45 views ### Banach fixed-point theorem for a recursive functional equation I was asked to prove that the functional equation$$ x(t) = \begin{cases} \frac{1}{2} x(3t) + \frac{1}{2},& 0 \leq t \leq \frac{1}{3}\\\ f(t), & \frac{1}{3} < t\leq \frac{2}{3}\\\ \frac{1}{... 1answer 28 views ### Is this an isolated equilibrium point? I've just been learning the definition of an isolated equilibrium point. From my understanding of this definition, I would expect (as an example) the point$x=1$to be an isolated fixed point for the ... 1answer 32 views ### How can I see that we have$\Lambda_a=s\Lambda_a\mathbin{\dot\cup} \big(s\Lambda_a+(1+s)\big)\mathbin{\dot\cup}s\Lambda_b(Silver mean substitution) Consider the (Silver mean) substitution \varrho:\begin{aligned}&a\mapsto aba\\&b\mapsto a\end{aligned}. If we takew^{(1)}=a|a$and$w^{i+1}=\varrho(w^{(i)}), then we get: a|a\... 0answers 24 views ### Topology: every continuous function on \mathbb{R}^2 scales a point The question is simple: Suppose f : \mathbb{R}^ 2 \to \mathbb{R}^ 2 is continuous. Show that there exist \lambda > 0 and x \in \mathbb{R}^2 such that f(x) = \lambda x. So basically, we ... 1answer 77 views ### Statement of Markov-Kakutani fixed-point theorem Markov-Kakutani fixed-point theorem is usually stated as follows: "Let E be a locally convex topological vector space. Let C be a compact convex subset of E. Let S be a commuting family of ... 1answer 51 views ### What's the meaning this DOT notation? I'm reading a chapter in a Model Checking book. I came across this chapter "Symbolic Model Checking", in which the author mentions Fixed Point representation. I don't know how to explain the context, ... 1answer 36 views ### What can we say about the convergence of these fixed-point iterations for \phi:\mathbb{R}\to \mathbb{R} Let \phi: \mathbb{R}\to \mathbb{R} \in C^2(\mathbb{R}) and let x^{*} be a fixed-point of this function. Further assume that |\phi'(x^{*})| \neq 1. We define two sequences \begin{align} &... 3answers 33 views ### Holomorphic map from closed convex domain in hilbert ball into itself has fixed point Let B=\{x \in \ell_2, ||x|| \leqslant 1\} - Hilbert ball X \subset B - open convex connected set in Hilbert ball, \bar{X} - closure of X. F: \bar{X} \to \bar{X} - continuous map that ... 1answer 65 views ### Continuous map in \mathbb{R}^2 has a (scaled) fixed point Let \phi:\mathbb{R}^2\rightarrow \mathbb{R}^2 be a continuous map. How do I prove that there exist a>0 and x\in\mathbb{R}^2 such that \phi(x)=ax? What I know: I thought maybe this can ... 0answers 33 views ### Transform system to polar and sketch phase portrait. Show that (0,0) is an unstable focus. Transform the systemx' = y - x(x^2+y^2-1)y' = -x - y(x^2+y^2-1)$$to polar coordinates, and sketch the phase portrait. Show that it has a unique limit cycle and that all ... 1answer 30 views ### Problem with proof of the upper hemicontinuity of correspondence I have a problem with a proof I found here of the upper hemicontinuity of the best-reply correspondence in the Nash Theorem. Below there is the proof, and here my problems: Problems: Is here ... 1answer 75 views ### Find the fixed points of the system, and sketch the trajectories of the system I am given the following system:$$x' = [(x-1)^2 + y^2]yy' = -[(x-1)^2 + y^2]x \tag{*}$$where x = x(t), y = y(t). I am supposed to Find the fixed points of the system, and ... 2answers 49 views ### Show Existence of Fixed Point in \mathbb{R}^n with Euclidean Metric Consider the closed unit ball in \mathbb{R}^n, B = \{ x \in \mathbb{R}^n : \|x\| \leq 1 \} with the Euclidean metric. Then let g : B \rightarrow B be a function such that$$\|g(x) - g(y)\| \leq \... 0answers 30 views ### IfX$has the FPP then does$X\times I$have the FPP. If$X$is compact subset of$\mathbb{R}^2$and all continuous maps$f:X\rightarrow X$have a fixed point, do all continuous maps$f:X\times I\rightarrow X\times I$have a fixed point? 0answers 45 views ### Uniqueness of solution to integral equation with “endogenous” kernel Let$x,y\in C$and consider the functional equation$T:y\mapsto x$implicitly defined by the following integral equation: $$x(t) = \int_{-\infty}^\infty K(x(t),t,s) y(s) \, ds,$$ with$K$given. ... 0answers 22 views ### Does this functional equation have a (unique) solution? Does the following equation have a (unique) solution for$b_1$? b_1 = \dfrac{1}{2b_1} \left\lbrace -Var(p) + Cov \left(\left[ (p + b_1 + b_0 + \alpha)^2 -4b_1(p + b_0) \right]^{0.5},... 0answers 78 views ### closed unit ball without a fixed point. What I've done so far: Let$B=\{x\in\mathbb{R}^n : \|x\| \leq 1\}$be the closed unit ball in$\mathbb{R}^n$equipped with the standard Euclidean metric. Let$f \colon B \to B$be a function such ... 1answer 14 views ### Find 3 fixed points of function with 2 arguments I'm looking for a way to determine the fixed points of a function with 2 arguments. The specific function is the following:$F:\bigl( \begin{smallmatrix} x \\ y \end{smallmatrix} \bigr) \mapsto \...
In this context, a null field means a field constructed of planar waves $e^{I k_{\mu} x^{\mu}}$ that satisfy the null condition $k_{\mu} k^{\mu} = 0 \implies c^2 k^2 = \omega^2$ Suppose we have a ...