= Solution
<Brouwer fixed-point theorem>[Brouwer]'s fixed-point theorem says that every <continuous> self-map of a nonempty closed <Euclidean ball> has a <fixed point>. It does not promise convergence of repeated application. For example \b[$f(x)=1-x$ with starting point $x_1=0$] gives the nonconvergent alternating <sequence> $0,1,0,1,\ldots$, although $1/2$ is a <fixed point>.
For an <invariant ball from sublinear growth>, choose
$$
R\ge\max\{(2A)^{1/(1-\alpha)},\,2B\}.
$$
Then $AR^\alpha\le R/2$ and $B\le R/2$. For every $|x|\le R$ the given estimate yields $|f(x)|\le A R^\alpha+B\le R$. Thus $f$ restricts to a <continuous> self-map of the <closed ball> of this radius, and <Brouwer fixed-point theorem>[Brouwer] gives a <fixed point>. The exponent $\alpha<1$ ensures that the linear radius eventually dominates its sublinear growth; no <differentiability> or contraction estimate is needed.
Back to article page
