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Past exam of the mathematics course of the University of Cambridge
/
2022
/
iii
/
Paper 339
/
3
/
a
/
Solution
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Past exam of the mathematics course of the University of Cambridge
2022
iii
Paper 339
3
a
2026-09-28
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A
map
T
is
a
firmly nonexpansive mapping
when
∥
T
x
−
T
y
∥
2
≤
(
T
x
−
T
y
)
T
(
x
−
y
)
(1)
for all
x
,
y
. Put
p
=
prox
f
(
x
)
and
q
=
prox
f
(
y
)
. Then
x
−
p
∈
∂
f
(
p
)
and
y
−
q
∈
∂
f
(
q
)
. Monotonicity of the
subdifferential
gives
[(
x
−
p
)
−
(
y
−
q
)
]
T
(
p
−
q
)
≥
0
,
(2)
which rearranges to
∥
p
−
q
∥
2
≤
(
p
−
q
)
T
(
x
−
y
)
.
(3)
Ancestors
(11)
a
3
Paper 339
iii
2022
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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