On the maximum product metric, the displayed map is nonexpansive when is a contraction mapping. If the space has two distinct points, changing only the second input coordinate shows that one step preserves some nonzero distance exactly, so it is not itself a strict contraction. Two iterations apply to both coordinates and therefore contract with the same constant as . This provides a direct example of a map whose iterate is contractive although the map is not.
A contraction constant for a function on a metric space is a single that bounds the ratio for every distinct pair. Pointwise strict inequalities do not imply such a common constant. For a differentiable function on an interval, the mean value theorem shows that supplies a contraction constant.
The same self-map can be a contraction mapping for one metric but not for a bilipschitz equivalent metric. On three points with distances , , the map , has Lipschitz constant . For the discrete metric, its Lipschitz constant is one. The metrics satisfy , so they preserve topology and completeness but not the strict numerical contraction condition. A contraction mapping theorem fixed-point proof therefore needs the specified metric and its contraction constant.
