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Fixed points of quasi-nonexpansive mappings and best approximation. (English) Zbl 1183.41030

Summary: Using fixed point, B. Brosowski [Mathematica, Cluj 11(34), 195–220 (1969; Zbl 0207.45502)] proved that if \(T\) is a nonexpansive linear operator on a normed linear space \(X\), \(C\) a \(T\)-invariant subset of \(X\) and \(x\) a \(T\)-invariant point, then the set \(P_C(x)\) of best \(C\)-approximant to \(x\) contains a \(T\)-invariant point if \(P_C(x)\) is non-empty, compact and convex. Subsequently, many generalizations of the Brosowski’s result have appeared. In this paper, we also prove some extensions of the results of Brosowski and others for quasi-nonexpansive mappings when the underlying spaces are metric linear spaces or convex metric spaces.

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)

Citations:

Zbl 0207.45502
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