 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-325/2/i/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

For

0 < α < 1

, an averaged operator has the form

T = (1 - α) I + α R

R

, meaning

∥ R z - Rw ∥_{2} \leq ∥ z - w ∥_{2}

for all

z, w

. The word averaging refers to a convex combination with the identity. It does not mean averaging function values over a neighborhood.

∥ T z - Tw ∥_{2}^{2} \leq ∥ z - w ∥_{2}^{2} - \frac{1 - α}{α} ∥ (I - T) z - (I - T) w ∥_{2}^{2} .

To see it, put

d = z - w

and

r = R z - Rw

and expand

∥ (1 - α) d + α r ∥_{2}^{2} = (1 - α) ∥ d ∥_{2}^{2} + α ∥ r ∥_{2}^{2} - α (1 - α) ∥ d - r ∥_{2}^{2} .

Use

∥ r ∥_{2} \leq ∥ d ∥_{2}

and

(I - T) z - (I - T) w = α (d - r)

. Conversely, substituting

R = [T - (1 - α) I] / α

into the same expansion proves its nonexpansiveness from the displayed inequality. The strict range

0 < α < 1

matters for convergence of unrelaxed iterations: a general nonexpansive mapping such as

T = - I

can have nonconvergent alternating iterates.

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