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Applications of the proximity map to fixed point theorems in Hilbert space. (English) Zbl 0643.47053

The authors prove that if S is a closed convex subset of a Hilbert space X and f is a l-set-contractive map of S into X, satisfying some closure conditions involving the metric projection of X into S, then there exists a point \(u\in S\) such that \(\| u-f(u)\| =d(f(u),S)\). This result is then applied to prove various fixed point theorems, extending results obtained by Ky Fan, F. Browder, W. Petryshyn, S. Reich and the first author. The basic idea used in the proofs is that the metric projection of a Hilbert space into a closed convex subset is nonexpansive.
Reviewer: S.Cobzaş

MSC:

47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41A50 Best approximation, Chebyshev systems
Full Text: DOI

References:

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