# Questions tagged [fixed-point-theorems]

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) etc.

1,413 questions
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### Only one fixed point for $f:\bar{\mathbb{D}}\rightarrow\bar{\mathbb{D}}$ on the boundary.

We know for Brouwer theorem that $f$ (continuous bijective function) have a fixed point. My questions are: 1) Is there a function with only one fixed point $x_0\in Int(\bar{\mathbb{D}})$ (open disk)?...
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### Must a holomorphic function from $D(0,1)$ to $D(0,1)$ have a fixed point?

Must every holomorphic function $f:D(0,1)\longrightarrow D(0,1)$ have a fixed point? I know that any holomorphic function with two fixed points is the identity: $f=Id$, but I can't find out an ...
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### In a complete lattice every monotone function has a fixpoint (Knaster–Tarski Theorem)

L is a complete lattice, so every subset has a supremum and infimum. In addition, there exists a function $f:L \rightarrow L$ such that $a \leq b$ implies $f(a) \leq f(b)$. Prove that there exists ...
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### Is there a fixed point theorem I could use to solve this problem?

let $E = C([0,1]),\,\,$ $K : E \to E, \,\, (Kf)(x) = \int_0^1K(x,y)f(y)dy$ also $\|K\| \leq a < 1$ I want to prove that there for $g \in E$ there exists a unique $f_g \in E$ that satisfies the ...
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### How to show something is a contraction?

If we let $X$ be a complete metric space, and let $S:X\to X$ be a map, such that $S^m$ is a contraction. We now want to show, that $S$ has a unique fixed point This is what I've thought so far: Due ...
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### Show that there exists $\xi\in [a,b]: f(\xi)=\xi$.

Let $a,b\in\mathbb{R},~a<b$ and consider $f\colon[a,b]\to [a,b]$ continuous. Show that $f$ has a fixed point. i.e. that there exists a $\xi\in [a,b]$ with $f(\xi)=\xi$. My idea is to consider ...
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### Existence of infinite iteration of functions $f_\infty$?

Given a sequence of functions $\{f_n\}$ satisifying an iterated relation such as $f_n(x)=g(x+f_{n-1}(x))$ $f_n(x)=g(xf_{n-1}(x))$ $f_n(x)=g(x/f_{n-1}(x))$ Where $g:=f_1$ is ...
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### Local Truncation Error of Implicit Euler

The LTE of an implicit Euler method is $O(h^2)$ because the method has order $O(h)$, but I'm not sure where to get started in proving this arithmetically. Any help would be appreciated. Thank you!
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### Function on a Power Set

Let $f\colon \mathcal{P}(A)\mapsto \mathcal{P}(A)$ be a function such that $U \subseteq V$ implies $f(U) \subseteq f(V)$ for every $U, V \in \mathcal{P}(A)$. Show there exists a $W \in \mathcal{P}(A)$ ...
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### Fixed Point of a complex dynamical spiral system

Last semester I finished my first class on complex variables and of course we had to show that $i^i$ was real. That got me wondering about quantities like $i^{i^i}$ and similar power towers. For my ...
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### Proving Talagrand's contraction lemma for Gaussian processes with the Banach fixed-point theorem

I've done the standard proof of Talagrand's contraction lemma for Gaussian processes (see Exercise 7.2.13 in Vershynin's High-Dimensional Probability) using the Sudakov-Fernique inequality as ...
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### Reference request: equivalence between formulas in fixed point and first-order logic

I'm looking for materials on the relationship between first-order and fixed-point logics, specifically on the condition for a formula in some sort of fixed-point logic to have an equivalent first-...
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### Why does the fixed point theorem hold for every lambda term?

Can someone give a clear and simple answer for why the fixed point theorem holds for every $\lambda$-term, in contrast with the fact that not all numerical function have a fixed point?
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### Showing a sequence $x_{n+1} = Tx_n$ forms a contraction mapping

I want to show that the sequence given by $$x_{n+1} = Tx_n = x_n-\frac{(x_n^2-2)}{x_n+x_{n-1}}$$ forms a contraction mapping. That is $$|Tx_1-Tx_2|\leq c|x_1-x_2|.$$ Where $c$ is to be determined. I ...
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### Convergence to a fixed point [duplicate]

Let $f : [a,b] \rightarrow [a,b]$ be a continuous function s.t. $f'(x)$ is defined on $(a,b)$ and $\left\lvert f'(x)\right\rvert \leqq t$ where $0<t<1$. Prove that for any point $x_0$ in $[a,b]$ ...
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### 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 ...
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### Prove $(x_n)$ defined by $x_n= \frac{x_{n-1}}{2} + \frac{1}{x_{n-1}}$ converges when $x_0>1$

$x_n= \dfrac{x_{n-1}}{2} + \dfrac{1}{x_{n-1}}$ I know it converges to $\sqrt2$ and I do not want the answer. I just want a prod in the right direction. I have tried the following and none have ...
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### How to explain powers of $(x+1)^{2^n}$ appearing in the Babylonian approximation of $\sqrt x$?

I'm working with this iteration used for approximating square roots and trying to see what I can draw out from it, and in doing so I found something very strange that I can't logically explain. I'm ...
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### generalization of Banach fixed-point theorem on short maps?

If $\ T:X \longrightarrow X \$ is contraction, then using Banach fixed-point theorem we know that the fixed point exists and all other points converge to that point. But what happens if $T$ is not ...
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### Is $f : [0,\infty]\rightarrow [0,\infty]$ which is continuous and bounded has a fixed point

Question is to check if : $f : [0,\infty]\rightarrow [0,\infty]$ which is continuous and bounded has a fixed point. I have first of all considered boundedness. So, $f(x)$ should not have $x$ as ...
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### Fix point of squaring numbers mod p

Take the set of integers $\{0, 1, .., p-1\}$, square each element, you get the (smaller) set of quadratic residues. Repeat until you get a fix point set. The size of this set is a function of $p$. ...
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### Is there any way to give sense to a geometric/visual proof?

Suppose one is given the following visual proof that $$\lim\limits_{n \to \infty} \sum_{k=1}^n \frac{1}{2^k} = 1$$ which is the following construction over $[0,1]\times[0,1]$ What this is ...
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### Fixed point property of Cayley plane

I want to know whether the Cayley plane has fixed point property or not. I think it is so but I am not able to prove this. It certainly does not admit maps of period two without fixed points. By fixed ...
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### Continuous on the unit ball – odd on the unit sphere – does it have a fixed point?

For $n\in\mathbb N$, let \begin{align*} B^n\equiv&\;\{\mathbf x\in\mathbb R^n\,|\,\lVert \mathbf x\rVert\leq 1\}\text{ and}\\ S^{n-1}\equiv&\;\{\mathbf x\in\mathbb R^n\,|\,\lVert \mathbf x\...
The Banach theorem states that if a (self) map on a complete metric space is Lipschitz with ratio $< 1$, it has a unique fixed point. What about modifying the hypotheses to say that the only ...
Does a continuous mapping $f\colon \mathbb R \to \mathbb R$ which satisfies $f(f(x))=x$ for each $x \in \mathbb R$ necessarily have a fixed point?