Proximity maps, best approximations and fixed points (Q2724922)

Firstly, the author stated a condition which is more general than that of \textit{T. C. Lin} and \textit{C. L. Yen} [J. Approximation Theory 52, No. 2, 141-148 (1988; Zbl 0643.47053)], to prove that the Fan's best approximation theorem holds for generalized contractive maps. This condition allows us to prove that the Fan's best approximation theorem holds for generalized contraction maps. Secondly, discussed are iteration methods for the actual construction of such best approximation points for contraction maps and non-expansive maps. Thirdly, introduced are the conditions on when the best approximation of a sequence of contractions will converge to the best approximation of a contraction. Lastly, common best approximation and common fixed point theorems for non-expansive maps are proved.