= Invariant ball from sublinear growth
{title2=$|f(x)|\le A|x|^\alpha+B,\quad0<\alpha<1$}
A <continuous> Euclidean self-map with this growth has a <fixed point>. Choose $R\ge(2A)^{1/(1-\alpha)}$ and $R\ge2B$. Then $|f(x)|\le R$ whenever $|x|\le R$, so the closed radius-$R$ ball is invariant and <Brouwer fixed-point theorem>[Brouwer] applies. Existence does not imply convergence of iterating the original map: $x\mapsto1-x$ on the unit <interval> has a <fixed point> but a two-cycle from either endpoint.
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