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Fixed points and best approximation in metric spaces. (English) Zbl 1180.41024

Summary: By generalizing a theorem of G. Meinardus [Arch. Ration. Mech. Anal. 14, 301–303 (1963; Zbl 0122.30801)], B. Brosowski [Mathematica, Cluj 11(34), 195–220 (1969; Zbl 0207.45502)] proved a result on invariant approximation using fixed point theory. Subsequently, many generalizations of Brosowski’s result appeared. This paper also deals with some extensions and generalizations of Brosowski’s result by using the notion of contractive jointly continuous family, discussed by W. G. Dotson jun. [Proc. Am. Math. Soc. 38, 155–156 (1973; Zbl 0274.47029)] thereby extending and generalizing various known results on the subject As a consequence, a result ensuring the existence of invariant points for a pair of commuting mappings is also given in the paper.

MSC:

41A50 Best approximation, Chebyshev systems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
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