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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.

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2answers
580 views

Show that a function $f:P(X)\to P(X)$ preserving the subset relation has a fixed point

We have a map $f:P(X)\to P(X)$, where $P(X)$ means the part of $X$ and the function is monotone (by considering inclusion "$\subseteq$"). So $\forall \space A\subseteq B $ we have $f(A)\subseteq f(B)$...
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unique fixed point problem

Let $f: \mathbb{R}_{\ge0} \to \mathbb{R} $ where $f$ is continuous and derivable in $\mathbb{R}_{\ge0}$ such that $f(0)=1$ and $|f'(x)| \le \frac{1}{2}$. Prove that there exist only one $ x_{0}$ such ...
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1k views

Showing that $f$ has exactly one fixed point

Let $\gamma$ be the circle $\{z \in \mathbb{C}: \lvert z\rvert=1 \}$. Suppose $f$ is a function analytic on an open set containing $\gamma$ and its interior and that $\lvert\, f(z)\rvert<1$ for ...
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Brouwer's fixed point theorem for free?

I think I found a proof of Brouwer's fixed point theorem which is much simpler than any of the proofs in my books. One part is standard: Suppose there is an $f:D^n \rightarrow D^n$ with no fixed ...
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636 views

Banach fixed point theorem and $\sum d(T^n(x),T^n(y))<\infty$

Let $T:X \to X$ be a map on a complete non-empty metric space. Assume that for all $x$ and $y$ in $X$, $\sum_n d(T^n(x),T^n(y))<\infty$. Then $T$ has a unique fixed point. guess: I assume that the ...
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172 views

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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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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414 views

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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824 views

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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276 views

No fixed points imply no periodic points

Let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a smooth injective function with $\operatorname{det}[f'(x)]\not=0 $ for all $x\in\mathbb{R}^n$. Moreover assume that $f$ has no fixed points. Can $f$ have a ...
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158 views

Is the limit of a recursively defined sequence always a fixed point?

Let $(x_n)$ be a sequence of real numbers such that $x_{n+1}=f(x_n)$ for all natural numbers $n$, where $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$. And suppose that $(x_n)$ converges to ...
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4k views

Why does Fixed Point Iteration work?

I have searched online for an answer, but everyone gave the method, and no one explained why is it working. I'll first write what I do understand. Let $f(x)$ be a continuous function at $[a,b]$. ...
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99 views

How to generalize this version of Tarski’s Fixed Point Theorem?

I could prove the following result from my Real Analysis course: Let $f:[0,1] \rightarrow [0,1]$ be an increasing mapping. Then it has a fixed point. I understand that this is a very baby version ...
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226 views

Contractive Operators on Compact Spaces

Suppose that $T: M \to M$ is a compact contractive Operator on a nonempty compact subset $M$ of a complete metric space $X$. Show that $T$ has a unique fixed point. Further show that the sequence ...
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910 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{...
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156 views

Path Connectedness and fixed points

We have the following given to us, Let $α, β \colon [0, 1] \to [0, 1]$ be (not necessarily continuous) functions such that $α(x) ≤ β(x)$, for all $x ∈ [0, 1]$. The set $K = \{\,(x, y); α(x) ≤ y ≤ β(...
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88 views

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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147 views

An application of the Banach Fixed Point Theorem to a system of equations

I am solving a question whose first item is to demonstrate the Banach Fixed Point Theorem, and the second item is as follows: Show that for any parameter $t \in \mathbb{R}$ the system $$ \begin{cases}...
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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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Proof that the solution to cosx = x, is the limit of a recursive sequence.

So I've got this question. Exists a sequence $a_n$ such that: $$a_0 = \frac \pi4, a_n=\cos\left(a_{n-1}\right)$$ Prove that $\lim_{n\rightarrow\infty} a_n = \alpha$ Where $\alpha$ is the solution to $...
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1answer
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Continuous bijections from the open unit disc to itself - existence of fixed points

I'm wondering about the following: Let $f:D \mapsto D$ be a continuous real-valued bijection from the open unit disc $\{(x,y): x^2 + y^2 <1\}$ to itself. Does f necessarily have a fixed point? I ...
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168 views

System of equations $a(1 - b^2) = b(1 -c^2) = c(1 -d^2) = d(1 - a^2)$

Given a positive real number $t$, find the number of real solutions $a, b, c, d$ of the system $$a(1 - b^2) = b(1 -c^2) = c(1 -d^2) = d(1 - a^2) = t$$ I have a solution Let $f(x)=\frac t{1-x^2}$ ...
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109 views

If $q(x)=x^2+1$, does $q^{\circ 1/2}$ exist? [duplicate]

I've been doing a lot of research about functional half-iteration, and I posed the following question to myself: Consider the function $q:\mathbb R\mapsto\mathbb R$ defined as $$q(x)=x^2+1$$ ...
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1answer
95 views

Quotient Spaces Defined By Bijection

I was working with a question in topology and came to the following statement that I can't seem to figure out: Let $f:\mathbb R^2\rightarrow\mathbb R^2$ be a homeomorphism with no fixed points. ...
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1answer
569 views

What is the role of fixed point theorems in modern mathematics?

About Fixed Point Theorems, Wikipedia says: Results of this kind are amongst the most generally useful in mathematics. This seems an accurate statement: indeed, there are many journals ...
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1answer
256 views

Fixed point theorem involving nonexpansive mapping on uniformly convex Banach space

This is an exercise from the Dirk Werner's book about fixed point theorem: Let $X$ be a uniformly convex Banach space. Let $F:B_X\to X$ be a nonexpansive mapping, i.e. $$ \forall x,y\in B_X: \|F(x)-F(...
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1answer
66 views

Transforming Nested Fixed-Point Formulas into Infinitary Logic Formulas with Finitely many Variables

There is a definition (actually a description of how it could be defined) of a fixed-point logic formula. The formula is in inflationary fixed point logic (IFP) in this case but it could also be ...
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1answer
1k views

Show that sequence approaches fixed point of a function

Problem Let $f(x)$ be a differentiable function on $\Bbb R$ with $\left|\,f ' (x)\right| \leq r < 1$, where $r$ is constant. Then consider the sequence $\{x_n\}$ such that $x_1 = 0$, $x_{n+1} = f(...
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1answer
353 views

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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136 views

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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425 views

Godel's Diagonalization Lemma As Application Of Lawvere's Fixed Point Theorem

I've read through this paper with applications of Lawvere's fixed point theorem. On the diagonalization lemma, they say the following: For one thing, how can $f$ and $\Phi_{\cal E}$ be well ...
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125 views

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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265 views

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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215 views

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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401 views

Vector valued contraction

I am familiar with the Banach Fixed Point Theorem and I have used it to prove existence and uniqueness of functional equations in Banach spaces like $C(X)$, the space of bounded continuous function $f:...
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591 views

To prove : If $f^n$ has a unique fixed point $b$ then $f(b)=b$

If $f: \mathbb R \to \mathbb R$ be a function such that for some $n_o \in \mathbb N$ , the $n_o$th iterate of $f$ has a unique fixed point $b$ , then how to prove that $f(b)=b$ ? I cant think of ...
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Picard's existence theorem, successive approximations and the global solution

Picard's existence theorem states that if $U$ is an open subset of $\mathbb{R}^2$ and $f$ is a continuous function on $U$ that is Lipschitz continuous with respect to the second variable then there is ...
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349 views

Where is my mistake using the Banach theorem for $x^2 - 2 = 0$?

Consider example $x^2 - 2 = 0$. I can rewrite so I get $x^2 + x - 2 = x$. If I define $\phi(x) = x^2 + x - 2$, I need to solve $\phi(x) = x$. $\phi$ is Lipschitz-continuous, since it's differentiable. ...
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3answers
431 views

Can't find all roots to function with fixed point method

I have a function $f(x) = x^2-8x-10\cos(2x)+15$ and I'm supposed to via Matlab find all roots for this function. I can see graphically that it has $4$ roots and I've written a matlab function to find ...
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4answers
829 views

Newton's method with no real roots

So as the title would suggest I'm currently reading about Newton's method for finding roots. I'm having trouble understanding the reasoning for a function without a root. It reads as following: "...
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3answers
106 views

How many fixed points are there for $f:[0,4]\to [1,3]$

Let , $f:[0,4]\to [1,3]$ be a differentiable function such that $f'(x)\not=1$ for all $x\in [0,4]$. Then which is correct ? (A) $f$ has at most one fixed point. (B) $f$ has unique fixed point. (C) $...
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37 views

Do the closed unit disk $D$ and $f(D)$ intersect, if $||f(x)-x||\le2$ for all $x\in D$?

In $\Bbb R^n$ let $D=\{x:||x||\le1\}$, and let $f:D\to\Bbb R^n$ be continuous with the property that $||f(x)-x||\le2$ for all $x\in D$. Is it true that $D\cap f(D)\neq\varnothing$ (where $f(D)=\{f(...
4
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1answer
103 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 ...
4
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3answers
134 views

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 ...
4
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1answer
71 views

Brouwer's fixed point theorem and continuous functional dependence on the fixed point.

The famous Brower's fixed-point theorem states that any $ f $ function that maps a compact and convex set itself has a fixed point. I would like to know if minor disturbances in the function $ f $ ...
4
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2answers
222 views

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 ...