Mathematics Asked by Daniel Huff on January 1, 2022
Consider a map
begin{align}
F:&[0~1]^mathbb{N}rightarrow mathbb{R}^mathbb{N}\
&{x_k}rightarrow {y_k}=F{x_k}
end{align}
with the following properties:
${y_k} rightarrow 0$ for all ${x_k}in[0~1]^mathbb{N}$
$y_k=f(y_{k-1},x_k)$ for all $kinmathbb{N}$, where $f(cdot,cdot)$ is a continuous function and $y_0$ is a given constant.
Can we guarantee all the possible sequences ${y_k}$ converge uniformly to zero?
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