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Results tagged with fixed-point-theorems
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user 5903
A fixed-point theorem is a result saying that a function $F$ will have at least one fixed point (a point $x$ for which $F(x) = x$), under some conditions on $F$ that can be stated in general terms.
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Fixed point theorem for the uncountable power of an interval
Let $f:[0,1]^\kappa\to[0,1]^\kappa$ be given. For a finite subset $F$ of $\kappa$ let $A_F=\{x\in[0,1]^\kappa: \pi_F(x)=\pi_F(f(x))\}$, where $\pi_F$ is the projection onto $[0,1]^F$. The set $A_F$ is …
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