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Questions tagged [fixed-points]

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function. That is to say, c is a fixed point of the function f(x) if and only if f(c) = c. A set of fixed points is sometimes called a fixed set.

I think it can be merged to https://math.stackexchange.com/tags/fixed-point-theorems/info

590 questions
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I have found a number. Google and OEIS come up blank: 0.696340872970033948754981...

It kind of looks like $\log(2)$, but it isn't. This is a fixed point of a map $f:\Bbb{R}\to\Bbb{R}$ with $f(x) = (1-x)^{1-x}$. When you iterate the map it misbehaves at $0$ and $1$, so I started at $1/2$. Convergence is geometric, so it's not hard…
Emanuel Landeholm
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If $f: \mathbb R \to \mathbb R$ and $f$ is continuous and for every real value of $x : f(f(x))=x$ , prove that there is a $c$ where $f(c)=c$

If $f: \mathbb R \to \mathbb R$ and $f$ is continuous and for every real value of $x : f(f(x))=x$ , prove that there is a $c$ where $f(c)=c$. I know how to prove that a continuous function like $f: [a,b] \to [a,b]$ has a fixed point, but I don't…
Emily
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Fixed point question: $|f'(x)|=1$ case

I have a fixed point question. Let's say we have the following recursive equation: $x_{n+1}=f(x_{n})$. According to Wikipedia, $x_n$ converges to a fixed point $x_0$ as long as $|f'(x)|<1$ in an open neighbourhood of $x_0$. However, I've read on the…
FluffyCat
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Global fixed point (function)

So suppose that we have an operator $T:C[0,\infty)\to C[0,\infty)$ such that for all $M\in\mathbb{R}^+$ the restriction of $T|_{C[0,M]}$ maps into $C[0,M]$ and has a unique fixed point, then is that enough to deduce that $T$ itself has a fixed…
Melody
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Set theory, fixed points

Let $f : \mathcal{P}(B) \to \mathcal{P}(B)$ be a monotonic function and $I$ - its least fixed point. Prove that: if $I \subseteq A \subseteq B$ and $f_A : \mathcal{P}(A) \to \mathcal{P}(A)$, $f_A(X) = A \cap f(X)$ for all $X \subseteq A$, then $I$…
A. Mason
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Finding unknowns of natural log

I am given two points $(1.5, 3.1)$ and $(6.5, 1.45)$ and I have to find a function that cuts through these points. I have chosen to use an $\ln(x)$ function in the general form: $y=a\ln(x-h)-k$ I know that an additional point is needed to find the…
Deep Patel
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fixesd point of strictly monotonic continuous function

Let $f$ be a strictly monotonic continous real valued function defined on $[a,b]$ such that $f(a)b$. Then, which one of the following is true? a) There exists exactly one $c \in (a,b)$ such that $f(c)=c$ b) There exist exactly two…
dipali mali
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A problem on fixed points in complex

Prove that every holomorphic function on the closed disk $\overline{\Delta}(0,1)$ with $|f(z)|<1$ when $z\in \overline{\Delta}(0,1)$ has at least one fixed point in $\Delta (0,1)$. I was thinking that I could use Maximum Modulus Principle for…
Quang123
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Existence of recurrence wrt an equivalence relation implies existence of recurrence wrt equality

Assume I have $f : A \to B$, $g_a : A \to A$, $g_b : B \to B$ and $R$ an equivalence relation on $B$. If I know $f(a)$ is $R$-related to $g_b (f (g_a (a)))$ for all $a\in A$, is it the case that there must exist an $h$ where $h(a) = g_b(h(g_a(a)))$?…
Michael Norrish
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Disprove existence of fixed-point of endofunction constructor

This might be quite a naïve question, but I seem to be unable to find a good answer. The set-valued functional $$ F(X) = (X \to X) \cup \{ \bot \} $$ is not monotonic: $F(\{t\}) = \{\bot,\mathrm{id}_{ \{ t \} }\} \not \subseteq F(\{t,f\}) = \{…
Sebastian Graf
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Fixed Point $f : \mathbb{R} \setminus \{ \tfrac{\pi}{2} +k\pi : k \in \mathbb{Z} \} \to \mathbb{R} : x \mapsto \sqrt{x^{2}+1}-\tan{x} $

I have to show that $f : \mathbb{R} \setminus \{ \tfrac{\pi}{2} +k\pi : k \in \mathbb{Z} \} \to \mathbb{R} : x \mapsto \sqrt{x^{2}+1}-\tan{x} $ has a fixed point in the interval $[0,1]$. Meaning $\exists \xi \in [0,1]: f(\xi) = \xi$ I know how to…
jaki
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$f,g:[1,2]\to[1,2]$ such that $fg=gf$ but for no $x\in[1,2]$ we have $f(x)=g(x)=x$

Does there exist mappings $f,g:[1,2]\to[1,2]$ such that $f\circ g=g\circ f$ but for no $x\in[1,2]$ we have $f(x)=g(x)=x?$ I am trying to figure out such mapping for quite a long time but failed. For all such pair I found a common fixed point. Please…
My Math
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given $x,y$ and $\theta$, calculate its distance to origin $(0,0,0)$

Apologies if this question sounds really dumb, but I am faced with a set of points: $x_1,y_1,\theta_1$ $x_2,y_2,\theta_2$ .. and I am looking to work out the distance between each row $(x_1,y_1,\theta_1)$ to the origin $(0,0,0)$. I have not…
AJW
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How to find fixed point without graph approach?

The function tan(x)=x has 1. Unique fixed point 2. No fixed point 3. Infinite many fixed points 4. More than one but finitely many fixed points. My attempt: I have solved this problem by graphical approach. The intersecting points of the function…
Mathforjob
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Classifying fixed points

So I am working with the following set of non-dimensionalized differential equations: $ dx/dτ = (xy - x) $ $dy/dτ= (-xy - ay + b)$ Where a is always positive and b can be positive or negative. My next step is to find and classify fixed points. I…
umz
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