Habiniak, Leopold Fixed point theorems and invariant approximations. (English) Zbl 0673.41037 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 Cited in 3 ReviewsCited in 43 Documents 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 PDF BibTeX XML Cite \textit{L. Habiniak}, J. Approx. Theory 56, No. 3, 241--244 (1989; Zbl 0673.41037) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.