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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 $f(x)=1-x$ with starting point $x_1=0$ gives the nonconvergent alternating sequence $0,1,0,1,\dots$, although $1/2$ is a fixed point.

For an invariant ball from sublinear growth, chooseThen $A{R}^{\alpha }\le R/2$ 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 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.