Hillam's Theorem

If $f:[a,b]\to[a,b]$ (where $[a,b]$ denotes the Closed Interval from $a$ to $b$ on the Real Line) satisfies a Lipschitz Condition with constant $K$, i.e., if

$$\vert f(x)-f(y)\vert\leq K\vert x-y\vert$$

for all $x,y\in[a,b]$, then the iteration scheme

$$x_{n+1}=(1-\lambda)x_n+\lambda f(x_n),$$

where $\lambda=1/(K+1)$, converges to a Fixed Point of $f$.

References

Falkowski, B.-J. ``On the Convergence of Hillam's Iteration Scheme.'' Math. Mag. 69, 299-303, 1996.

Geist, R.; Reynolds, R.; and Suggs, D. ``A Markovian Framework for Digital Halftoning.'' ACM Trans. Graphics 12, 136-159, 1993.

Hillam, B. P. ``A Generalization of Krasnoselski's Theorem on the Real Line.'' Math. Mag. 48, 167-168, 1975.

Krasnoselski, M. A. ``Two Remarks on the Method of Successive Approximations.'' Uspehi Math. Nauk (N. S.) 10, 123-127, 1955.