Abstract
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.
Highlights
1 Introduction For spaces X, Y and subsets V ⊆ X, W ⊆ Y, a map f : X → Y is an extension of a map φ : V → W if f (x) = φ(x) for all x ∈ V
The Brouwer fixed point theorem implies that a map f : B → B must have at least one interior fixed point if it is an extension of a map φ : S → S that has no fixed points
If d ≥ and f : B → B is a smooth extension of φd, f has at least one interior fixed point. It is demonstrated in [ ] that interior fixed points of extensions need not exist if d ≤ or if f is not smooth. Schirmer generalized this interior fixed point result to smooth extensions f : Bn → Bn for n ≥ to show in Example . of [ ] that if f is a smooth extension of a ‘sparse’ map φ : Sn– → Sn, a generalization of φd that is defined below, of degree d such that (– )nd ≥, f must have at least one interior fixed point
Summary
Of [ ] that if f is a smooth extension of a ‘sparse’ map φ : Sn– → Sn– , a generalization of φd that is defined below, of degree d such that (– )nd ≥ , f must have at least one interior fixed point. In Section , we prove that a smooth extension f : R → R of a power map φd : S → S for d ≥ must have at least one interior fixed point.
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