Periodic point theorem for generalized graphic contractions

Let $(X,d)$ be a nonempty metric space and let $n\in \mathbb N^+$. We shall say that $T\colon X\to X$ is a graphic contraction of order $n$ if there exists $\alpha\in (0,1)$ such that the inequality $$d(T^n x,T^{2n}x) \leqslant \alpha d(x,T^nx)$$ holds for all $x\in X$. In the case $n=1$ these mapping are known as graphic contractions and are well studied. In the present paper, we establish a theorem on the existence of periodic points for a graphic contraction of order $n$. Examples of such mappings having different properties are constructed.