Let $(X,d)$ be a complete metric space and $T$ a mapping of $X$ into itself. Suppose that for each $x\in X$ there exists a positive integer $n=n(x)$ such that for all $y\in X$, $$ d(T^nx,T^ny)\leq \alpha\max\{d(x,y), d(x,Ty), d(x,T^2y),\dots, d(x,T^ny), d(x,T^nx)\}, $$ holds tor some $\alpha<1$. With these assumptions our main result states that $T$ has a unique fixed point. This generalizes an earlier result of V. M. Sehgal and a recent result of the author.