# 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.

1,413 questions
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### Is there a name for this theorem about the convergence of a function?

Let $f(x)$ be a continuous function over $\mathbb{R}$ such that for all $a < b$, we have $a < f(a) < b$. Then, for any $x < b$, the sequence $\{t_n\}$ defined by $t_0 = x, t_n = f(t_{n-1})$...
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### Let $f: \mathbb R\to \mathbb R$ be a continuous function. Which of the following are sufficient conditions for $f$ to have a fixed point in $[0, 1]$?

(a) $f(0)=f(1)$ (b)$f(1)<0<f(0)$ (c) $0<f(1)<f(0)$ (d) $f(0)<0<1<f(1)$ To obtain a fixed point, we should find $x=f(x)$ but how do I obtain the necessary conditions? What ...
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### logistic map with lamba greater than 4

I was doing some recreational math about the logistic map. (If you're not familiar with what the logistic map is, here are some links you can check out) https://www.youtube.com/watch?v=ETrYE4MdoLQ ...
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### Reconstructing a function from iterates at zero?

Say we have a function $f(x)$ such that $f(0)\neq0$ and construct its iterates at zero e.g. $f^3(0)=f(f(f(0)))$. Let it also be a one-one function such that it has a unique inverse so the $f^{-1}(0)$ ...