Common fixed points and best approximation. (English) Zbl 0858.41022

Al-Thagafi, M. A.

J. Approximation Theory 85, No. 3, 318-323 (1996).

The subject of the fixed points of different systems is an intensively studied domain. It is also the case in the paper under review, which main purposes are to prove a fixed-point theorem generalizing results of Dotson and Habiniak and to extend, to generalize and unify these results on fixed points and common fixed points of best approximation.
The author proposes a theorem which gave the conditions of the existence of a fixed point in a subset of a normed linear space. Through another two theorems it is proved a common fixed-point generalization of Habiniak’s extension of a fixed-point result of Dotson in the second section.
The third part consists of a few applications of best approximations and contains a proposition and two theorems which refer to an extension of the first theorem (from the introduction) and generalize the results of Singh, Hick and Humphries.
The last chapter, “Further applications to best approximations” extends Habiniak’s result and proves the existence of a common fixed-point of best approximation as a generalization of Smoluk’s result and of Habiniak (in a precise particular case).

Reviewer: I.Grosu (Iaşi)

MSC:

41A50	Best approximation, Chebyshev systems
54H25	Fixed-point and coincidence theorems (topological aspects)

Keywords:

fixed-point; best approximation