Weakly contracting function

A function f:X→Y between 2 metric spaces X,Y is called a weakly contracting function if ∀a,b∈X, d_Y(f(a),f(b))<d_X(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.

..and here's the proof (compare with a contracting function in a complete metric space has one fixed point (proof)).

Uniqueness is trivial: if x and y are distinct fixed points, d(x,y) = d(f(x),f(y)) < d(x,y), a contradiction.

For existence, consider the function g(x) = d(x,f(x)). It is continuous (this is a straightforward check), so by compactness g has a lower bound c, which is attained by some x∈X. If c=0 we are done (it's worth mentioning that in ariel's definition, we require a and b to be different). Suppose it is striclty positive. Then g(f(x)) = d(f(x),f(f(x))) < d(x,f(x)) = g(x) = c, a contradiction. Thus c=0 and x is the unique fixed point of f.

As ariels insinuated, compactness is a necessary assumption here; completeness (as in the "strongly contracting" case) will not suffice. A simple example is the function f defined on the non-negative real numbers via f(x) = x + e^-x.