A function f:X→Y between 2 metric spaces X,Y is called a weakly contracting function if a,b∈X, dY(f(a),f(b))<dX(a,b). That is, any 2 points are brought closer together by f. But unlike a contracting function (which see), there is no bound on the degree of contraction.

For example, the function sin:R→[-1,1] is weakly contracting (because |cos(x)|=|sin'(x)|≤1, so by the mean value theorem we have that for all a<b there exists a<c<b with |sin(a)-sin(b)|=|cos(c)||a-b|≤|a-b|; when |cos(c)|=1 we still don't get |sin(a)-sin(b)|=|a-b|, because sin is not a linear function on any interval), but not contracting (consider points a=0, b close to a; the proof that (sin(x)/x)→1 as x→0 shows sin is not contracting).

Unlike a contracting function, a weakly contracting function f:X→X on a complete metric space X might fail to have a fixed point. It's very easy to draw such a function on R: draw the line y=x, then another increasing curve above it. The curve should approach y=x asymptotically as x→∞, keeping its slope below 1. Then it is not a contracting function (because its slope approaches 1 as x→∞), but it is weakly contracting. It does not have a fixed point, though, as it steadfastly remains above the line y=x.

Nonetheless, we have this:

Theorem. A weakly contracting function f:X→X on a compact metric space X has a unique fixed point.