# 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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### Fixed point property for the total space and base space of a principal bundle

Is there a principal bundle $P\to X$ such that $P$ has the fixed point property but $X$ does not have? Is there an example of this situation where $P,X$ and the fibers are compact smooth manifolds?(...
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### Are the properties of not having fixed points and being its own inverse enough to characterise the antipodal map?

While trying to solve a problem I ended up with a smooth map $f:S^n \to S^n$ which I know is fixed point free and such that $f \circ f$ is the identity map of the sphere. I would like to conclude ...
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### Finding if a fixed point is attractor or repulsor without differentiation.

Given the function $F(x)=\frac{\pi}{2}\sin(x)$. Find the fixed points and, if they exist, determine if the points are attractors or repulsors without differentiation. I already found the fixed points ...
I’m currently trying to understand the following Proposition from a paper i’m reading: Prop.: Let $X$ and $Y$ be two Hausdorff topological linear spaces. Let $H:X \times Y \rightarrow Y$ be a ...
Say that we have a recurrence of the form $u_{n+1}=f(u_{n})$, where $f:R \to R$, then, what is the conditions on $f$ to have a convergent series $u_{n}$. I have the following questions: Is it enough ...