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

We extend Lin's result (Canad. Math. Bull. 22, No. 4 (1979), 513-515) and Singh and Watsons' result (J. Approx. Theory 39 (1983), 72-76) to more general 1-set-contractive maps. This class of 1-set-contractive maps includes condensing (or densifying) maps and nonexpansive maps; it also includes other important maps such as semicontractive maps and LANE maps. As applications of our theorems, fixed point theorems are proved under various conditions. The main idea we use, is one due to Cheney and Goldstein (Proc. Amer. Math. Soc. 10 (1959), 448-450), that a proximity map in Hilbert space is nonexpansive.