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).