Let $B = \{\, x \in \Re^m : \|x\| \le 2 \,\}$, and let $f : B \to B$ be a continuous function whose set of fixed points is $S^k = \{\, x \in B : \|x\| = 1, x_{k+2} = \cdots = x_m = 0 \,\}$. Can it happen that for arbitrarily small neighborhoods $U \subset C(B,B)$ of $f$ there is a continuous $G : S^k \to U$ such that $\langle x, y \rangle < 0$ for all $x \in S^k$ and all fixed points $y$ of $G(x)$? (The answer is no when $k = 0$, by the theory of the fixed point index, so assume that $k \ge 1$.)
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