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

Habiniak, Leopold

J. Approximation Theory 56, No. 3, 241-244 (1989).

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

References:

[1]	Dotson, W. J., Fixed point theorems for nonexpansive mappings on star-shaped subsets of Banach spaces, J. London Math. Soc., 4, 408-410 (1972) · Zbl 0229.47047
[2]	Meinardus, G., Invarianz bei linearen Approximationen, Arch. Rational Mech. Anal., 14, 301-303 (1963) · Zbl 0122.30801
[3]	Schauder, J. P., Der Fixpunktsatz in Funktionalraumen, Studia Math., 2, 171-180 (1930) · JFM 56.0355.01
[4]	Smoluk, A., Invariant approximations, Mathematyka Stosowana, 17, 17-22 (1981), [Polish] · Zbl 0539.41038
[5]	Subrahmanyam, P. V., Erratum, J. Math. Phys. Sci., 9, 195 (1975)
[6]	Subrahmanyam, P. V., An application of a fixed point theorem to best approximation, J. Approx. Theory, 20, 165-172 (1977) · Zbl 0349.41013