# 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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### algebraic or homotopical proof for Kakutani fixed point theorem

As Kakutani fixed point theorem is a genral case of Brouwer fixed point theorem, and one can read the proof from homotopy theory books. I wonder if there is any proof for the Kakutani using homotopy ...
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### Equivalence of Brouwers fixed point theorem and Sperner's lemma

I'm looking for a proof that Brouwer's fixed point theorem implies Sperner's lemma. All proof I've found just prove that there must be at least one completely colored n-simplex, not that there must be ...
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### Point-set topological proof of Brouwer's fixed point theorem

I have tried to understand the point-set topological proof of Brouwer's fixed point theorem presented in Cou11. But I couldn't clarify some parts. Here are the theorem and its proof. Theorem: There ...
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### How many fixed points can a function have?

For dimension one, it is easy to think in samples of continuous functions $f:[a,b]\rightarrow [a,b]$ with one, two, three,... fixed points. Or even, infinitely fixed points (take the idendity map). ...
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### System of equations and the Brouwer's Fixed-Point Theorem.

Let's consider the following system of equations: \begin{eqnarray}{ e^{xyz} = \frac{x}{\sqrt{e^{2xyz}+1}}\\ \cos(x+y+z) = \frac{y}{\sqrt{e^{2xyz}+1}}\\ \sin(x+y+z) = \frac{z}{\sqrt{e^{2xyz}+1}} }...
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### Fixed point method for non-homogeneous ode and pde

During my course in mathematical methods for physics, my professor introduced us to the "method of fixed point" for studying non-homogeneous systems of odes, odes and pdes. Although I think I ...
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### Existence and Uniqueness of Equilibrium Points in Non-Linear Dynamical Systems

Let $\dot{X} = F(X)$ be a non-linear dynamical system. I'm interested in knowing if there are any existence theorems for an equilibrium point, that is, an $X^*$ that satisfies: $$F(X^*) = 0$$ I ...
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### Equivalence between Brouwer fixed-point theorem and Borsuk-Ulam theorem. Is there a simple proof of equivalence between them?

I wonder if Brouwer's fixed-point theorem and Borsuk-Ulam's theorem are equivalent. Brouwer's fixed-point theorem (simple form). Let $B_{\mathbb{R}^{n}}[0,1]=\{x\in \mathbb{R}^n: \|x-0\|\leq 1\}$ ...
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### Fredholm alternative for elliptic PDE: Is $L_\mu^{-1}$ contractive?

In Evans' PDE, §6.2, in Theorem 4 he proves a Fredholm alternative based dichotomy for second order uniformly elliptic pde. The general idea: For $\mu\geq\gamma\geq0$ with $\gamma\geq0$ from Thm. 2, ...
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### differentiable contractive mappings

Some time ago, I managed to show that $sin(x)$ and $cos(x)$ were both contractive almost everywhere and then I noted that $cos$ was the derivative of $sin$ so there might be a deeper connection. Let'...
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### What is this numerical method for solving Ax=b called?

Following is the pseudo-code of a simple iterative method of solving $Ax=b$ where $A$ is an $n\times n$ matrix. ...
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### On the Proof of the Perron-Frobenius Theorem.

The Perron-Frobenius theorem states that a square matrix with nonnegative entries has a real nonnegative eigenvalue. One possible proof uses the Brouwer fixed point theorem, and every proof I've seen ...
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### Existence of fixed points for this Markov operator.

Perhaps math overflow is a better place to put this but I'm looking for some mathematical results that I might be able to apply to see if an operator I'm considering has a fixed point. In particular ...
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### Fixed Point of an infinite-dimensional map

I have the following fixed point equation: $$F(t)= \displaystyle\sum_{r=1}^M \int_{\mathbb{R}} F ((t-c)r) g(r) h(c) \: \mathrm{d}c$$ where: $F$ is a cumulative density function of a certain random ...
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### Is it hard to find out for which $r$ the infinite power tower $r\uparrow r \uparrow r\uparrow \cdots$ converges?
It is well known that the infinite power tower $$r\uparrow r\uparrow r\uparrow\cdots$$ with $r>0$ converges if and only if $e^{-e}\le r\le e^{1/e}$. I tried to prove it and I got stuck in the ...