# 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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### Literature on the convergence of $x_{n+1} = f(x_n)$ in general

When faced with a recurrence of the form $x_{n+1} = f(x_n)$, my toolbelt for proving convergence is very limited: if $f$ isn't $k$-lipschitzian with $k<1$, and/or if I can't find some complete set ...
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### Brouwer's fixed point theorem in a practical setting

If we assume that a fluid is a continuum then if we have for example a cup of tea and we stir the fluid then there will be a point in the fluid that is on the same location before and after the ...
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### How Gelfond find his limit for $\exp(\pi)$? [duplicate]

$$a_0 = \frac{1}{\sqrt 2}$$ $$a_{n+1} = \frac{( \sqrt {1 - a_n^2} -1)^2}{a_n^2}$$ $$\lim_{n \to \infty} \frac{4^{\frac{1}{2^n}}}{a_{n+1}^{\frac{1}{2^n}}} = \exp(\pi)$$ How did Gelfond find ...
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### Fixed point of a differentiable function

Consider a differentiable fuction $f: \mathbb{R} \rightarrow \mathbb{R}$ for which $f'(t) \neq 1$ for every $t \in \mathbb{R}$. Is it true that $f$ has exactly one fixed point? It is clear to me why ...
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### Show that $f$ has a unique fixed point.

Let $M$ be complete and let $f:M\to M$ be continuous .If $f^k$ is a strict contraction for some integer $k>1$ ,show that $f$ has a unique fixed point. My try: Let us fix $x_0$.Let $g=f^k$. Put ...
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### Find if a fixed-point iteration converges for a certain root

I'm asked to find if the fixed-point iteration $$x_{k+1} = g(x_k)$$ converges for the fixed points of the function $$g(x) = x^2 + \frac{3}{16}$$ which I found to be $\frac{1}{4}$ and $\frac{3}{4}$. ...
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### When to use Newtons's, bisection, fixed-point iteration and the secant methods?

I've been introduced more or less to these methods of finding a root of a function (a point where it intersects the $x$ axis), but I'm not sure when they should be used and what are the advantages of ...
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### Polynomials in two variables over $\mathbb{F}_p$ fixed by $\operatorname{SL}_2(\mathbb{F}_p)$

Let $A=(a_{ij})\in\operatorname{SL}_2(\mathbb{F}_p)$. Consider the ring map $A:\mathbb{F}_p[x,y]\to\mathbb{F}_p[x,y]$ defined by $$A(x)=a_{11}x+a_{21}y$$ $$A(y)=a_{12}x+a_{22}y$$ and extended ...
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### Show that the comb space has fixed-point property

By "comb space", I mean the space $X=([0,1] \times \{0\}) \cup (K \times [0,1])$, where $K=\{ \frac{1}{n} : n \in \mathbb{N}^+ \}$, without the leftmost vertical line segment. How to prove that this ...
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### What is the value of $z$ for any Julia set? Does it influence the graphical result?
I'm trying to understand how Julia sets iterations are done and how those iterations differ from the ones that generate the Mandelbrot set. Both of them use the following function: $f(z) = z^2 + C$ ...