# 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 for some continuous function.

Let $f$ be a continuous function on $[a,b]$ ( $f: [a,b]\to \mathbb R$, $a < b$) such that $\int_a^b f(x) \, dx = \frac{b^2-a^2}{2}$. How can we prove that $f$ has a fixed point in $(a,b)$ without ...
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### Proof of Kakutani fixed-point theorem

I'm reading the following proof of the Kakutani fixed-point theorem, written by Prof. McMullen in his functional analysis notes. I don't understand the proof beyond the point where he appeals to ...
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### Converse of Brouwer fixed point theorem

Brouwer fixed point theorem is usually stated in the following way: Let $B^n$ some closed ball of a Euclidean space, and let $f \colon B^{n} \rightarrow B^{n}$ be a continuous map. Then $f$ has a ...
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### Automorphisms of CW complexes and fixed points

Let $X$ be a CW complex, and let $F:X\to X$ be an homeomorphism that sends each cell onto some cell; notice that we could say that $F$ is an automorphism of the CW complex since it preserves its cell ...
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### Confusion on Blackwell's condition for a contraction: $T: B(X)\to B(X)$

Blackwell's condition for a contraction: Why is boundedness neccessary? (Theorem: Blackwell's sufficient condition for a contraction.) Let $X \subset \mathbf{R}^l$ and let $B(X)$ be a space of ...
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### Probabilistic analog of Banach fixed-point theorem

Are there probabilistic analogs of Banach fixed-point theorem? For example, is there a notion of probabilistic contractive mappings that gives rise to a probabilistic fixed-point theorem? Sorry for ...
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### General method for finding invariant subsapces of a nonlinear system

Suppose we are given a system: $$\dot{x_{1}} = f_{1}(x_{1},...,x_{n})$$ $$...$$ $$\dot{x_{n}} = f_{n}(x_{1},...,x_{n})$$ And are interested in finding subspaces of the vector space that are invariant ...
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