Contraction mappings and fixed points

Let T map a nonempty complete metric space X to itself. Suppose there is one number q with 0≤q<1 such that every output pair has distance at most q times the corresponding input distance. Then T is a contraction. The word uniform is essential: showing that each particular pair becomes closer, with no common factor below one, does not establish this hypothesis. A derivative bound can supply the factor for a real differentiable map on an interval, but the interval must also be mapped into itself and be complete under its selected metric.

Iterate $x_{n+1}=T(x_n)$. Consecutive distances shrink by powers of q. For q strictly between zero and one, summing these distances through the triangle inequality bounds any long tail by a convergent geometric series. Thus the iterates form a Cauchy sequence, and completeness puts the limit x* in X. A contraction is continuous, so passing to the limit in the iteration equation gives T(x*)=x*. If q=0, the map is constant and reaches its fixed point after one application, provided it is a self-map.

If u and v are both fixed points, their distance is at most q times itself. Since q<1 and distance is nonnegative, that distance must be zero. Thus there is only one fixed point in X. This proof does not require trying different initializations numerically. The iteration converges from every initial point in X because the same geometric estimate applies globally. A contraction on one invariant region does not necessarily describe behavior outside that region, so the theorem's domain should remain explicit when reporting the result.

A useful stopping rule uses the latest observed step rather than the first one. After computing x_n, the remaining distance is at most q/(1−q) times $d(x_n,x_{n-1})$ for n at least one. Equivalently, if $x_{n+1}$ has already been computed, the error at $x_n$ is at most $d(x_{n+1},x_n)/(1-q)$. These are different indexed statements, not contradictory formulas. They are certificates only when q is a proven global contraction factor. Fitting a convenient ratio to a few early steps does not establish a guarantee.

On X=(0,1), the map T(x)=x/2 is a self-map with contraction factor one half, but its only possible fixed point is zero, outside X. Completeness fails. On the complete real line, T(x)=x+1 preserves all distances and has no fixed point, so replacing q<1 by q≤1 is invalid. A formula that sends allowed inputs outside the proposed domain fails the self-map assumption before any convergence argument begins. Checking these examples is a way to audit an application rather than merely memorize the theorem.

The quadratic gradient iteration in the lab is a particularly transparent case. For f(x)=x²/2 and step size eta, its update is T(x)=(1−eta)x. On the real line this is a contraction exactly when |1−eta|<1, or 0<eta<2. For eta between one and two, signs alternate while magnitudes shrink, demonstrating that convergence need not be monotone in position. At eta=2 a nonzero initial value alternates forever, and above two the magnitude grows. These explicit boundaries come from the contraction factor, not from the smoothness of the animation.