Applications of the proximity map to fixed point theorems in Hilbert space (Q1101940)

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.