# 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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### Proof of equality of least fixed point of two continuous function

The question is simple: given a set U, a continuous function (Scott continuity) $f \colon \mathcal{P}(U) \to \mathcal{P}(U)$ and function $g(X) := f(f(X))$, prove that $g$ is continuous and its least ...
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### Reference request for the proof of the Brodskii–Milman fixed point theroem for isometries

Can any one help me to access the paper M.S Brodskii and D.P Milman, On the center of a convex set, Dokl. Akad. Nauk SSSR 59 (1948) 837–840 in Russian? or to prove the theorem If $K$ is a ...
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### Closest fixed point to a convex set

Consider the compact convex sets $Y \subset X \subset \mathbb{R}^n$, and a Lipschitz continuous function $f : X \rightarrow X$. Assume that $f$ has multiple fixed points. (From Brouwer's theorem, $f$ ...
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### About the compactness condition in Schauder fixed point theorem

The theorem is Let $X$ be a locally convex topological vector space, and let $K ⊂ X$ be a non-empty, compact, and convex set. Then given any continuous mapping $f: K → K$ there exists $x ∈ K$ ...
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### Fixed point property in topology

I have a few questions concerning relating the fixed point property for a space $X$ (every continuous map from $X$ to $X$ has at least one fixed point) to some concepts in topology. a). I know that a ...
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### Find out Fixed Points

Consider a set $M$ of all possible square matrices of dimension $k$ over a finite field $F_p$. Consider a map $f$ defined on $M$ as $f(X)=X^2+C$ where $X \in M$ and C is an arbitrary fixed matrix from ...
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### Diffeomorphism and hyperbolic points

Suppose $f$ is a diffeomorphism.Prove that all hyperbolic periodic points are isolated. I tried using the mean value theorem using two diferent periodic points (assuming the periodic points arent ...
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### Firm non-expansiveness in the context of proximal operators

$\newcommand{\prox}{\operatorname{prox}}$ Probably the most remarkable property of the proximal operator is the fixed point property: The point $x^*$ minimizes $f$ if and only if $x^* = \prox_f(x^*)$...
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### $|f(x)-f(y)|<|x-y|$ on a non-empty closed bounded set of real numbers

Let $A$ be a non-empty closed bounded set of real numbers and $f: A \to A$ be a function such that $|f(x)-f(y)|<|x-y| , \forall x,y\in A$ , then how to show that $f$ has a unique fixed point ?
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### If $f^N$ is contraction function, show that $f$ has precisely one fixed point. [duplicate]

If $f$ is a mapping of a complete metric space $(X, d)$ into itself and $f^N$(composite $f$ for $N$ times) is a contraction mapping for some positive integer $N$, then $f$ has precisely one fixed ...
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### Finding the fixed points of a recurrence relation (and systems of) analytically?

How would I go about finding the fixed points of the following recurrence? $$X_n = 2X_{n-1}(2- 3X_{n-1}) + X_{n-1}$$ And therein, determining their stability analytically? Also, how does one find ...
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### Fixed point for a continuous function on a compact set?

If $f:X \rightarrow X$ is continuous and X is compact, will $f$ have a fixed point? We know that a contraction will have a fixed point but I have not come across an example of a continuous function ...
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### Fixed points of the Gamma Function?

I am interested in complex values of $z$ such that $$\Gamma (z) =z$$ Clearly, the one trivial value of $z$ is 1. Also, looking at a graph of the gamma function on the real axis, I can tell that there ...
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### Homogeneous topological space with the fixed-point property

Let $X$ be a topological space. $X$ is said to be homogeneous if for every $x$ and $y\in X$ there is an self-homeomorphism $f$ of $X$ such that $f(x)=y$. Further, $X$ is said to have the fixed-point ...
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### Find the Fixed points (Knaster-Tarski Theorem)

Let $L=\mathcal P(\mathbb N)$ be a complete lattice of subsets of $\mathbb N$. a) Justify that the function $F(X)=\mathbb N \setminus X$ does not have a Fixed Point. I don't know how to solve this. ...
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### Firmly nonexpansive mapping with the fixed point set same as for given nonexpansive mapping

I found PAMS publication vol. 113, no. 3, 1991 by Ryszard Smarzewski called "On firmly nonexpansive mappings". It is written that "to each nonexpansive T on set C one can associate a firmly ...
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### Banach Fixed Point Theorem. Measurable version.

The Banach fixed point theorem has the following statement THEOREM ( Banach contraction principle). Let $(Y,d)$ be a complete metric space and $F:Y\to Y$ be contractive . Then $F$ has a uniqe ...
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### Can the proof of fixed point theorems ever be constructive?

Overall, Brouwer fixed point theorem and Kakutani fixed theorem are non-constructive. Is there any established paper that demonstrates that there exists constructive proofs that do exactly what these ...
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### Continuity of fixed points of an equation

Say I have a function $f(x,\theta)$ such that for each value of $\theta$ the function $f$ has a unique fixed point $x^*(\theta)$, i.e. $$x^{*}(\theta) = f(x^{*}(\theta),\theta)$$ My question is when ...
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### Use a fixed point argument to show there exists a unique solution to the following BVP

Show using a fixed point argument that there exists a unique solution $f\in C[0,1]$ to $$-f''(x)+\sin(f(x))=\sin(x) , x\in (0,1), y(0)=y'(1)=0$$ This is what I have so far: We can show ...
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### Dense subgroup of an extremely amenble group is extremely amenable??

Recall that a topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space $K$ admit a fixed point. i.e there is a point $\xi\in K$ such that $g.\xi=\xi$ for ...
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### Derivative of the Solution to a Fixed Point Iteration

Let $\theta_s$ is the solution to a fixed point equation $$\theta=f(\theta,\lambda)$$ Let $d(\theta,\lambda)$ be another function of $\theta,\ \lambda$. I know $f$ but I have no explicit expression ...
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### Fixed point method where the derivative is one - does it converge

I'm trying to see if the iterative method $x_n=g(x_{n-1})$ where $g(x)=2\sqrt{x-1}$ will converge to $2$, if I take $x_0$ that is sufficiently close to $2$. Indeed notice that $g(2)=2$. and we have a ...
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### Existence of Galerkin approximations to PDE

Given a weak PDE $$\langle b(u),v \rangle +\langle a(\nabla u), \nabla v\rangle=\langle f, v \rangle,\qquad v\in H^1_0$$ where $b\in C(\mathbb{R})$ and $a\in C(\mathbb{R}^n,\mathbb{R}^n)$ grow ...
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### 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 ...
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### Trouble understanding the steps to go from $x^x = 1000$ to the fixed point of $f(x) = \frac{log(1000)}{log(x)}$

I'm following through the Structure and Interpretation of Computer Programs, and in an example, they talk about finding fixed points of function (where $f(x) = x$) They then go on to say, if we want ...
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### Fixed-Points Limit Set

Let $f : \mathbb{R}^n \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ be compact and convex. For all $p \in \mathbb{R}$, consider $A_p \in \mathbb{R}^{n \times n}$, with $p \mapsto A_p$ ...
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I'm learning some applications of the Banach Fixed Point Theorem and I have the following question: Consider the integral equation $\displaystyle x(t)=\int_{0}^{\frac{\pi}{2}}\arctan \left(\frac{x(s)}... 1answer 160 views ### Is there a proof for the Central Limit Theorem via some fixed point theorem? This question arose in my mind when I learned that the Gaussian is a fixed point for the Fourier transform. On the other hand, in e.g. the Banach fixed point theorem we have convergence to a fixed ... 2answers 264 views ### if$f_{n}(x)=f(f_{n-1}(x))$then$f_{10}(x)=x,x\in [0,1]$Define the function$f:[0,1]\to[0,1]$by the following. $$f(x)=\begin{cases} x+\dfrac{1}{2},&0\le x\le\dfrac{1}{2}\\ 2(1-x),&\dfrac{1}{2}<x\le 1. \end{cases}$$ Let$f_1(x)=f(x)$and for ... 1answer 152 views ### Show that$f$has infinitely many fixed points Let$f:ℂ→ℂ$be an entire function. Assume that the equation$f(s)=a$has infinitely many real solutions for all reals$a$. Show that$f$has infinitely many fixed points, i.e., there exists infinitely ... 1answer 301 views ### Continuous function on closed unit ball Take a continuous mapping$f: \bar{B^{n}} \rightarrow \bar{B^{n}}$, where$\bar{B^{n}}$is a closed unit ball in$\mathbb{R}^{n}$. Assume that$f(x) \neq x$for every$x \in \bar{B^{n}}$. Define ... 0answers 66 views ### A basic question about upper hemicontinuity Given a correspondence$f:X\rightarrow 2^X$, suppose X is a closed simplex in$\mathbb{R}^n$, and$f$is compact-valued. We say$f$is upper hemicontinuous if,$\forall x\in X$and every open subset$...
Consider the following equation which holds for all $w$ in some space, $$\langle v(t), w \rangle = \langle v(0), w \rangle - \int_0^t \langle F(s,v(s)), w \rangle$$ where $\langle F(s,v),w \rangle$ is ...