- Schirmer proved that there is a class of smooth self-maps of the unit sphere in Euclidean n-space with the property that any smooth self-map of the unit ball that extends a map of that class must have at least one fixed point in the interior of the ball. We generalize Schirmer’s result by proving that a smooth self-map of Euclidean n-space that extends a self-map of the unit sphere of that class must have at least one fixed point in the interior of the unit ball. Let X be a compact smooth n-manifold, with or without boundary, and let A be an (n − 1)-dimensional smooth submanifold of the interior of X. Let ϕ: A → A be a smooth map and f : (X,A) → (X,A) be a smooth map whose restriction to A is ϕ. If p ∈ A is an isolated fixed point of f that is a transversal fixed point of ϕ, that is, the linear transformation dϕp − IA : TpA → TpA is nonsingular, then the fixed point index of f at p satisfies the inequality |i(X, f, p)| ≤ 1. It follows that if ϕ has k fixed points, all transverse, and the Lefschetz number L(f) > k, then there is at least one fixed point of f in XA. Examples demonstrate that these results do not hold if the maps are not smooth.