# 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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### Rate of Convergence of $x_{k+1} = \frac 14 (x_k^2 +sin(x_k) + 1)$

I have shown that the fix point Iteration $$x_{k+1} = \frac 14 (x_k^2 +\sin(x_k) + 1)$$ has exactly one fix point $x^*\in[0,1]$ for every $x_0 \in [0,1]$ with the Banach theorem. Now I want to find ...
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### Proving a function is not a contraction

Let $\psi: C([0,1]; \mathbb{R}) \mapsto C([0,1]; \mathbb{R})$, be such that if $x \in [0,1]$ then $$\psi(f)(x) = \int\limits_{0}^{x} f(t) dt$$ I am asked to show that $\psi$ is not a contraction ...
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### Show that mapping is a contraction?

Show that the mapping $f:\Bbb R \to \Bbb R$, $f(x)=1-xe^x$ is a contraction. I tried everything i could think of but i cant get it to work. Witch is not much since i couldn't really find any ...
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### Algebraic fixed point theorem

I was wondering if there are some "algebraic" fixed point theorems, in group theory. More precisely, given a group $G$ and a group morphism $f : G \to G$, what conditions on $G$ and $f$ should we ...
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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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### Proving a function to be a contraction

I'm studying contractions and I am trying to understand under which conditions a function is actually a contraction. I have understood that a continuous function $f: [a,b]\rightarrow\mathbb{R}$ is a ...
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### Whether a continuous function has fixed point or not when the domain and range are not $[0,1]$

Which of the following is false $?$ $A.$ Any continuous function from $[0,1]$ to $[0,1]$ has a fixed point. $B.$ Any homeomorphism from $[0,1)$ to $[0,1)$ has a fixed point. $C.$ Any bounded ...
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### The conjugate of a complex function

This may be a stupid question. I am trying to explain/prove the following: Let $f$ be a complex rational function of degree $d ≥ 2$ with fix point $z_0 \neq 0$. It is always possible to conjugate $f$...
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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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### Does any continuous function from the open unit interval $(0,1)$ to itself has a fixed point?

I know that the result is true for closed interval $[0,1]$ by using intermediate value property. But in the case where we consider open interval $(0,1)$ does the solution change?
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### Proof there is no fixpoint

I want to proof that a certain function (in this case tan(x) on (0, 1/4)) doesn't have a fixpoint. Is there a general approach to try that? For example an "iff" fixpoint theorem?
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### What is the relationship between eigenvector and computing PageRank?

I read several papers about PageRank and didn't get stuck in understanding the idea of PageRank because it is simple, but I got stuck in computing PageRank, those papers talk about some mathematical ...
### Every finite group of congruences of the $n$-dimensional Euclidean space has a fixed point
Is there an "absolute" proof of the fact that if $G$ is a finite group of congruences of the $n$-dimensional Euclidean space, then $G$ has a fixed point?