Hillam's theorem states that if ![f:[a,b]->[a,b]](/images/equations/HillamsTheorem/Inline1.svg)
(where ![[a,b]](/images/equations/HillamsTheorem/Inline2.svg)
denotes the closed interval
from 
to 
on the real line) satisfies a Lipschitz
condition with constant 
, i.e., if
 |
for all ![x,y in [a,b]](/images/equations/HillamsTheorem/Inline6.svg)
,
then the iteration scheme
 |
where 
,
converges to a fixed point of 
.
Hillam's Theorem
See also
Map Fixed PointExplore with Wolfram|Alpha
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.Referenced on Wolfram|Alpha
Hillam's TheoremCite this as:
Weisstein, Eric W. "Hillam's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HillamsTheorem.html