The article deals with some results concerned with periodic points for some single-valued operators in metric spaces. The author presents the definition (due to I. A. Rus) of a fixed point structure and presents five theorems about the existence of
-periodic points. These theorems are based on an abstract lemma given by I. A. Rus; they are analogues of the classical Knaster-Tarski, Krasosel’skiĭ, Nemytskiĭ-Edelstein, Browder-Göhde-Kirk, Perov fixed point theorems. It should be remarked that the definition of a fixed point structure, in the author’s formulation, is not completely clear, and so the same can be said about the theorems of the article.