## 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

### MSC:

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

### Keywords:

nonexpansive mapping
Full Text:

### References:

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