Fixed point theorems and invariant approximations. (English) Zbl 0673.41037

Let S be a closed, star-shaped subset of normed linear space E, and let T: \(S\to S\) be a nonexpansive mapping. The author first observes that if cl(T(S)) is compact, then T has a fixed point. As a consequence, he proves that if A: \(E\to E\) is a nonexpansive mapping with fixed point a which leaves invariant a subspace M of E, and if A takes bounded subsets of M to relatively compact subsets, then the point a has a best approximation in M which is also a fixed point of A.
Reviewer: R.M.Aron


41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
47H10 Fixed-point theorems
58C30 Fixed-point theorems on manifolds
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