Contraction mapping theorem
General · Mathematics
Study notes
Q: Show T(x) = (x+2)/3 is a contraction on ℝ. Find its fixed point. |T(x)-T(y)| = |x-y|/3: k = 1/3 < 1 ✓ CONTRACTION! Fixed: x = (x+2)/3: 3x = x+2: x = 1. Iterate from 0: 2/3, 8/9, 26/27, ... → 1 ✓! (Unique: can't have two fixed points!)