Common Fixed Points And Related Problems In Best Approximation

It is well known that the theory on fixed points enters many fields in mathematics and has been applied to researches for various problems. As extension and generalization of it, the common fixed points for pair or family of mappings have also been paid much attention to, and become an active field. In particular, many results have been obtained for mappings satisfying commutativity. In order to study the associated problems to non-commutative mappings, the concepts of weak commutativity, R—weak commutativity, compatibility and others, and the associated classes of non-commutative mappings were introduced, and some advances have been achieved. But, due to tools and techniques, there are a lot of problems to be studied for non-commutative mappings, in comparison with commutative mappings. Particularly, it is an urgent need to introduce much wider classof non-commutative mappings and to determine the existence of common fixed points, which has great significances both in theory and applications.One of the important applications of the theorems for fixed points and common fixed points is to verify, for a given fixed point or common fixed point, the existence of the best approximating elements with the property of fixed point in an approximation subset. In 1964, using the fixed point theorem of Schauder, Meinardus proved that some properties of a function could be transferred to the best approximating elements of it, which is named as "invariant approxiamtion". After that, best approximation of fixed points and common fixed points became a topic with much attention. This topic has special significances in non-strictly convex spaces.