Let $(X, d)$ be complete metric space, $q \in [0, 1)$ be a real number, and $f$ be a map that satisfies $$d(f(x), f(y)) \leq q \cdot d(x, y)$$ for all $x, y \in X$. Then, Banach fixed-point theorem says that there is a unique fixed-point of $f$ in $X$.

I am interested in figuring out what happens if we relax the condition on $f$ to require, for all $x, y \in X$:

$$d(f^n(x), f^m(y)) \leq q \cdot d(x, y)\tag{1}$$ for some $n, m \geq 0.$

In general, there need not be a fixed-point when the constraint on $f$ is weakened to the above form. For example, points on a periodic orbit satisfy this weakened constraint, yet they never converge to a fixed-point. This means that, even though we conserve the notion of pairwise convergence, it is not enough to guarantee the existence of a fixed-point.

What can we say about maps that exhibit a fixed point under $(1)$? In other words, what properties do such maps have?