# Tagged Questions

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)...

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### Show that the only intervals having the fixed point property are the closed intervals.

By Fixed Point Theorem, I know that it deals with closed interval, for eg, [0,1]. And this theorem will be false if [0,1] is replaced by (0,1). The counter example will be $f:(0,1)\rightarrow (0,1)$ ...
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### Fixed point, bounded derivative

Let $p\in\mathbb{N}$. Let $f:I\to\mathbb{R}$ differentiable in the closed interval $I$ (bounded or not), with $f(I) \subset I$, and let $g = f\circ f\circ \cdots \circ f = f^p$, where $\circ$ means ...
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### Can there be a metric space where no contraction has a fixed point?

We know that: If $X$ is a metric space, then every contraction has at most one fixed point. (Note: if metric space is complete, then we have existence and uniqueness) I wonder if there can be a ...
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### How many fixed points does this function have?

The function is $f :\overline{\Bbb R}\to \overline{\Bbb R}, x \mapsto x^5$. So does it have $3$ or $5$ fixed points ? Thanks in advance !
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### Why is this easy “proof” of Brouwer's Fixed Point Theorem not correct/common?

Brouwer's Fixed Point Theorem states, essentially, that any continuous function on a closed disc to itself has a fixed point. I am familiar with the proof based on the impossibility of a retraction ...
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### Prove that $(Y,\mathcal T_1)$ also has the fixed point property

Let $(X\mathcal T)$ have the fixed point property and let $(Y,\mathcal T_1)$ be a space homeomorphic to $(X, \mathcal T)$. Prove that $(Y,\mathcal T_1)$ also has the fixed point property. I know ...
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### Uniqueness of solution to integral equation with “endogenous” kernel

Let $x,y\in C$ and consider the functional equation $T:y\mapsto x$ implicitly defined by the following integral equation: $$x(t) = \int_{-\infty}^\infty K(x(t),t,s) y(s) \, ds,$$ with $K$ given. ...
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### How to confirm that if a function $f\circ f$ is a strong contraction, then $f$ has a fixed point or not?
Suppose that $(X,d)$ is a complete metric space and $f:X \rightarrow X$ is such that $f\circ f$ is a strong contraction. Must $f$ have a fixed point? So, it is given that $f\circ f$ is a strong ...