# 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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### 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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### 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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### Reducing Kakutani's fixed-point theorem to Brouwer's using a selection theorem

Kakutani's fixed-point theorem is quite similar to Brouwer's fixed point theorem - the main difference is that Brouwer speaks about single-valued functions and Brouwer about multi-valued functions. ...
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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\...
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### Counterexample to a “modified” Banach Fixed Point Theorem?

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 ...
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### Continuous mapping and fixed points

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?
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### Least square circle via fixed point iteration

You have a collection of 2d points that you want to fit to a circle. Form the sum of the squares of the distances from the points to a generic circle. The variables are the $x,y$ coordinates of the ...
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### Proof about continuity of a function involving the Banach fixed point theorem

Be $X$ and $\Lambda$ metric spaces, with $X$ complete, and $f\in C(X\times\Lambda,X)$. Suppose that exists some $\alpha\in[0,1)$ and, for each $\lambda\in\Lambda$, some $q(\lambda)\in[0,\alpha]$ such ...
Let $p\in\mathbb{N}$. Let $f:I\to\mathbb{R}$ differentiable in the closed interval $I$ (bounded or not), with $f(I) \subset I$, and let $g = f\circ f\circ \cdots \circ f = f^p$, where $\circ$ means ...
I have the following question Show that there is unique real bounded sequence $(a_n: n \in \mathbb{N})$ such that $$a_n = \frac{n+1}{n}+\sum^\infty_{m=1}\frac{\sqrt{3a^2_{m+n}+1}}{4m^2}$$ for all \$n \...