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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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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 $f^{-1}(0)$ ...
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4answers
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Iterated functions

A function F(x) when composed in itself number of times is called an iterated function. Let $F(x)=\frac{1}{1+x}$ $F(\frac{1}{2})=\frac{2}{3}$ $F(\frac{2}{3})=\frac{3}{5}$ $F(\frac{3}{5})=\frac{5}{...
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$b)$ what happen if $X$ equals $[0,1) $or $(0,1)$? [closed]

Suppose that $f \colon [a, b] \to [a, b]$ is continuous. (Note that the range of $f$ is a subset of $[a, b]$) $a) $Prove that there exists at least one point $x \in [a, b]$ such that $f(x) = x$. A ...
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Determine $ \ a_{\max} \ $ and $ \ a_{\min} \ $ so that the above difference equation is well-defined.

Consider the Logistic equation $ \ x_{n+1}=a x_n (1-x_n ) \ $ and constrain $ \ x_i \ $ in $ \ [0,1] \ $. Determine $ \ a_{\max} \ $ and $ \ a_{\min} \ $ so that the above difference equation is ...
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Existence of unique fixed point in compact Metric space [duplicate]

Let $(X,d)$ be compact. Show: for a map $f$ that when $\forall x, y \in X$ with $x\neq y$ $d(f(x),f(y))<d(x,y)$ is fulfilled. Then $f$ has a unique fixed point.
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A question about topology [closed]

Assume $(X,d)$ is a compact metric space, $f:X\rightarrow X$, if $x\neq y$,then $d(f(x),f(y))<d(x,y)$ prove that there exists $x$,such that $f(x)=x$ Thanks for your help.
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Fixed point of closed disk [duplicate]

Let $D = \{(x,y)\in \mathbb{R}^2: x^2 + y^2 ≤ 1 \}$. Let $A \subset \mathrm{int}D$. Let $A$ be connected and compact and let $D \setminus A$ be connected. Let $f:A \longrightarrow A$ be a continuous ...
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1answer
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Brouwer's fixed point theorem

Is the statement is true or false, please explain the reason Every continuous map $f \colon S^1 \to S^1$ has a fixed point where $S^1$ is a unit circle in $\mathbb R^2$ follows from Brouwer's fixed ...
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Fixed point of interior of closed disk

Let $D = \{(x,y)\in \mathbb{R}^2: x^2 + y^2 \leq 1 \}$. Let $A \subset \mathrm{int}D$. Let $A$ be connected and compact and let $D \setminus A$ be connected. Let $f:A \longrightarrow A$ be a ...