(Luận văn thạc sĩ) phương pháp hiệu chỉnh lặp giải bài toán chấn nhận tách đa tập trong không gian hilbert

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈

◆●❯❨➍◆ ❚❍➚ ❚❘❆◆●

P❍×❒◆● P❍⑩P ❍■➏❯ ❈❍➓◆❍ ▲➄P ●■❷■ ❇⑨■ ❚❖⑩◆
❈❍❻P ◆❍❾◆ ❚⑩❈❍ ✣❆ ❚❾P ❚❘❖◆●
❑❍➷◆● ●■❆◆ ❍■▲❇❊❘❚

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❍⑩■ ◆●❯❨➊◆ ✕ ✷✵✷✵


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈
◆●❯❨➍◆ ❚❍➚ ❚❘❆◆●

P❍×❒◆● P❍⑩P ❍■➏❯ ❈❍➓◆❍ ▲➄P ●■❷■ ❇⑨■
❚❖⑩◆ ❈❍❻P ◆❍❾◆ ❚⑩❈❍ ✣❆ ❚❾P ❚❘❖◆●
❑❍➷◆● ●■❆◆ ❍■▲❇❊❘❚
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ö♥❣
▼➣ sè✿ ✽ ✹✻ ✵✶ ✶✷

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈
●❙✳❚❙✳ ◆●❯❨➍◆ ❇×❮◆●

❚❍⑩■ ◆●❯❨➊◆ ✕ ✷✵✷✵


▲í✐ ❝↔♠ ì♥

❚ỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s ữớ ữớ
t t ữợ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ♥❣❤✐➯♥ ❝ù✉ ✤➸ tæ✐
❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳
❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❍✐➺✉✱ ❝→❝ t❤➛② ❣✐→♦✱ ❝æ ❣✐→♦ tr♦♥❣
❦❤♦❛ ❚♦→♥ ✕ ❚✐♥✱ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✕✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t➟♥ t➻♥❤
❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ t↕✐ ❚r÷í♥❣✳
❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ỡ tợ rữớ ữ ỡ
tổ ũ ỗ t↕♦ ✤✐➲✉ ❦✐➺♥ ✈➲ ♠å✐ ♠➦t ✤➸ tæ✐
t❤❛♠ ❣✐❛ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➺♥ ❝ù✉✳
◆❤➙♥ ❞à♣ ♥➔②✱ tỉ✐ ❝ơ♥❣ ①✐♥ ❣û✐ ớ ỡ t tợ
trữớ ữ ỗ ữớ t ✤➣ ✤ë♥❣
✈✐➺♥✱ ❦❤➼❝❤ ❧➺✱ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳


ử ử

ớ ỡ
ởt số ỵ t tt
✤➛✉
❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✶✳✶✳

✐✐
✐✈
✶

✷

❈→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ ❣✐↔✐ t➼❝❤ ❤➔♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷

✶✳✶✳✶✳

❑❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳





ởt số t t





ỗ ữợ

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

✶✳✶✳✹✳

❚♦→♥ tû tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✶✳✺✳

✣✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹

✶✳✷✳

P❤→t ❜✐➸✉ ❜➔✐ t♦→♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

✶✳✸✳

P❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼

❈❤÷ì♥❣ ✷✳ P❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❝❤♦ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥
t→❝❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤❛ t➟♣
✶✾
✷✳✶✳

▼ët sè ❜ê ✤➲ ❝➛♥ t❤✐➳t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✾

✷✳✷✳

❚❤✉➟t t♦→♥ ✈➔ sü ❤ë✐ tö

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✷

✷✳✸✳

❱➼ ❞ö sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✾

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✸✶
✸✷


ởt số ỵ t tt

H

ổ rt tỹ

H

ổ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝õ❛

N

t➟♣ sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠

N∗

t➟♣ sè ♥❣✉②➯♥ ữỡ

R

t ủ số tỹ

C

t õ ỗ ừ







t ré♥❣

∀x

✈ỵ✐ ♠å✐

lim sup xn

❣✐ỵ✐ ❤↕♥ tr➯♥ ❝õ❛ ❞➣② sè

{xn }

lim inf xn

ợ ữợ ừ số

{xn }

xn x0



{xn }

❤ë✐ tö ♠↕♥❤ ✈➲

xn

❞➣②

{xn }

❤ë✐ tö ②➳✉ ✈➲

H

H

x

n→∞
n→∞

x0

F ix(T )

❤♦➦❝

F (T )

x0

x0

t t ở ừ

f

ữợ ừ ỗ

PC

tr

C

f

T


▼ð ✤➛✉

❇➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✤â♥❣ ✈❛✐ trá ✤➦❝ ❜✐➺t q✉❛♥ trå♥❣ tr♦♥❣ ✈✐➺❝ ♠æ ❤➻♥❤
❤â❛ ♥❤✐➲✉ ❜➔✐ t♦→♥ ♥❣÷đ❝ ①✉➜t ❤✐➺♥ tr♦♥❣ t❤ü❝ t➳ ♥❤÷ ❜➔✐ t♦→♥ ♥➨♥ ❤➻♥❤ ↔♥❤✱
❝❤ư♣ ❤➻♥❤ ❝ë♥❣ ❤÷ð♥❣ tø✱ ❦❤ỉ✐ ♣❤ư❝ ↔♥❤✳ ▼ët tr♦♥❣ ♥❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✤➣
✈➔ ✤❛♥❣ ✤÷đ❝ ♥❤✐➲✉ t→❝ ❣✐↔ sû ❞ö♥❣ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ❧➔ ♣❤÷ì♥❣
♣❤→♣ ❝❤✐➳✉ tr♦♥❣ ✤â ❝➛♥ ♣❤↔✐ t❤ü❝ ❤✐➺♥ ♣❤➨♣ ❝❤✐➳✉ tr t ỗ
õ ừ ổ rt ❚✉② ♥❤✐➯♥✱ ✈✐➺❝ t➼♥❤ ↔♥❤ ❝õ❛ →♥❤ ①↕ ❝❤✐➳✉ ♠➯tr✐❝
tr➯♥ ởt t ỗ õ t ý ụ ổ tỹ t❤✐✳ ❉♦ ✈➟②✱ ❝➛♥ ①➙② ❞ü♥❣
❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❤✐➺✉ q✉↔ ❤ì♥✳ ✣➲ t➔✐ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉
❝❤➾♥❤ ❧➦♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✤❛ t➟♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ✣â ❧➔
❜➔✐ t♦→♥ t➻♠ ♠ët ✤✐➸♠ tở ừ ởt ồ t õ ỗ tr ❦❤æ♥❣
❣✐❛♥ ❍✐❧❜❡rt ♠➔ ↔♥❤ ❝õ❛ ♥â q✉❛ ♠ët →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❣✐ỵ✐ ♥ë✐ ♥➡♠ ✈➔♦ ❣✐❛♦
❝õ❛ ♠ët ❤å ❝→❝ t õ ỗ tr ởt ổ rt ởt
t ứ õ ỵ t ỵ tt ỗ tớ ứ õ ỵ tỹ t
ở ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ❤❛✐ ❝❤÷ì♥❣ ❝❤➼♥❤✿

❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❧✉➟♥ ✈➠♥ ✤➲ ❝➟♣ ✤➳♥ ♠ët sè ✈➜♥ ✤➲ ❝ì ❜↔♥ ❝õ❛ ❣✐↔✐ t➼❝❤

❤➔♠✱ ♣❤→t ❜✐➸✉ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✤❛ t➟♣✱ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤✱
♣❤÷ì♥❣ ♣❤→♣ ❧➦♣✱ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣✳

❈❤÷ì♥❣ ✷✳ P❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤
tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✤❛ t➟♣
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❧✉➟♥ ✈➠♥ t➟♣ tr✉♥❣ tr➻♥❤ ❜➔② ❧↕✐ ♠ët ❝→❝❤ ❝❤✐ t✐➳t ❝→❝ ❦➳t
q✉↔ ❝õ❛ ◆✳ ❇✉♦♥❣✱ P✳❚✳❚✳ ❍♦➔✐✱ ❑✳❚ ❇➻♥❤ ❬✸❪ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣
❣✐↔✐ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✤❛ t➟♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳


ữỡ

tự
ữỡ ỗ ử ▼ư❝ ✶✳✶ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛
❣✐↔✐ t➼❝❤ ❤➔♠✳ ▼ö❝ ✶✳✷ ♣❤→t ❜✐➸✉ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤✳ ▼ư❝ ✶✳✸ ✤➲ ❝➟♣ ✤➳♥
♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣
❝→❝ t➔✐ ❧✐➺✉ ❬✸✱ ✹❪✳

✶✳✶✳ ❈→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ ❣✐↔✐ t
ổ rt

X
ổ tỡ

ữợ ①→❝ ✤à♥❤ tr♦♥❣

X

tr➯♥ tr÷í♥❣ sè t❤ü❝

R✳

❚➼❝❤ ✈ỉ

❧➔ ♠ët →♥❤ ①↕

·, · : X × X → R
(x, y) → x, y
t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙②✿
✭✐✮
✭✐✐✮

x, x ≥ 0

✈ỵ✐ ♠å✐

y, x = x, y

x ∈ X ✱ x, x = 0 ⇔ x = 0❀

✈ỵ✐ ♠å✐

x, y ∈ X ❀

✭✐✐✐✮

x + x , y = x, y + x , y

✭✐✈✮

λx, y = λ x, y

❙è

x, y

✤÷đ❝ ❣å✐ ❧➔

◆❤➟♥ ①➨t ✶✳✶✳
✭✐✮
✭✐✐✮
✭✐✐✐✮

✈ỵ✐ ♠å✐

✈ỵ✐ ♠å✐

x, x , y ∈ X ❀

x, y X R

t ổ ữợ ừ ❤❛✐ ✈➨❝tì x, y tr♦♥❣ X ✳

❚ø ✤à♥❤ ♥❣❤➽❛ s✉② r❛ ✈ỵ✐ ♠å✐

x, y + z = x, y + x, z
x, λy = λ x, y

x, y, z ∈ X, λ ∈ R✱

t❛ ❝â

❀

❀

x, 0 = 0.

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳
R✱ ·, ·

❈➦♣

(X, ·, · )✱ tr♦♥❣ ✤â X

❧➔ t➼❝❤ ✈æ ữợ tr

X

ữủ ồ

ởt ổ t t tr

ổ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt t❤ü❝✳


✸

▼➺♥❤ ✤➲ ✶✳✶✳ ▼å✐ ❦❤æ♥❣ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤
❝❤✉➞♥✱ ✈ỵ✐ ❝❤✉➞♥ ①→❝ ✤à♥❤ ❜ð✐

x =

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳

X

◆➳✉

❧➔ ❦❤æ♥❣ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt t❤ü❝ ✈➔ ✤➛② ừ ố ợ

s tứ t ổ ữợ t




H



ợ x X.

x, x

X

ữủ ồ

ổ rt t❤ü❝✳

❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ❉➣②

{xn }

✤÷đ❝ ❣å✐ ❧➔

❍ë✐ tư ♠↕♥❤ tợ tỷ x H ỵ xn → x✱ ♥➳✉

xn − x → 0

❦❤✐

n → ∞❀
✭✐✐✮

❍ë✐ tö tợ tỷ x H ỵ xn
n

ợ ồ

x



xn , y x, y



y H

ú ỵ ✶✳✶✳
✭✐✮ ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt

H ✱ ❤ë✐ tư ♠↕♥❤ ❦➨♦ t❤❡♦ ❤ë✐ tư ②➳✉✱ ♥❤÷♥❣ ✤✐➲✉

♥❣÷đ❝ ❧↕✐ ❦❤ỉ♥❣ ✤ó♥❣✳
✭✐✐✮ ▼å✐ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✤➲✉ ❝â t➼♥❤ ❝❤➜t ❑❛❞❡❝✲❑❧❡❡✱ tù❝ ❧➔ ♥➳✉ ❞➣②
tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
t❤➻

xn → x

❦❤✐

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳

H

t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥

xn → x

✈➔

{xn }

xn

x

n → ∞✳

❈❤♦

C

❧➔ t➟♣ ❝♦♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt

H✳

❑❤✐ ✤â

C

✤÷đ❝

❣å✐ ❧➔
✭✐✮

❚➟♣ ✤â♥❣ ♥➳✉ ♠å✐ ❞➣② {xn} ⊂ C t❤ä❛ ♠➣♥ xn → x ❦❤✐ n → ∞✱ t❛ ✤➲✉ ❝â
x ∈ C❀

✭✐✐✮

❚➟♣ ✤â♥❣ ②➳✉
✤➲✉ ❝â

✭✐✐✐✮

♥➳✉ ♠å✐ ❞➣②

{xn } ⊂ C

t❤ä❛ ♠➣♥

xn

x

❦❤✐

n → ∞✱

t❛

x ∈ C❀

❚➟♣ ❝♦♠♣❛❝t

♥➳✉ ♠å✐ ❞➣②

♣❤➛♥ tû t❤✉ë❝

{xn } ⊂ C

✤➲✉ ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tư ✈➲ ♠ët

C❀

❚➟♣ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ ♥➳✉ ♠å✐ ❞➣② {xn} ⊂ C ✤➲✉ ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tö❀

✭✈✮ ❚➟♣ ❝♦♠♣❛❝t ②➳✉ ♥➳✉ ♠å✐ ❞➣② {xn } ⊂ C ✤➲✉ ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tö ②➳✉ ✈➲

✭✐✈✮

♠ët ♣❤➛♥ tû t❤✉ë❝

C❀


✹
✭✈✐✮

❚➟♣ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ ②➳✉ ♥➳✉ ♠å✐ ❞➣② {xn} ⊂ C ✤➲✉ ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐
tö ②➳✉✳

◆❤➟♥ ①➨t ✶✳✷✳
✭✐✮ ▼å✐ t➟♣ ❝♦♠♣❛❝t ✤➲✉ ❧➔ t➟♣ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐✱ ♥❤÷♥❣ ✤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❦❤ỉ♥❣
✤ó♥❣✳
✭✐✐✮ ▼å✐ t➟♣ ✤â♥❣ ②➳✉ ✤➲✉ ❧➔ t➟♣ ✤â♥❣✱ ♥❤÷♥❣ ✤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❦❤ỉ♥❣ ✤ó♥❣✳

▼➺♥❤ ✤➲ ✶✳✷✳ ❈❤♦ H ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ✈➔ C ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ H ✳
❑❤✐ ✤â✱ t❛ ❝â ❝→❝ s
C t ỗ õ t C ❧➔ t➟♣ ✤â♥❣ ②➳✉❀
✭✐✐✮ ◆➳✉ C ❧➔ t➟♣ ❜à ❝❤➦♥ t❤➻ C ❧➔ t➟♣ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ ②➳✉✳

✣à♥❤ ♥❣❤➽❛ ✶✳✻✳
❍✐❧❜❡rt t❤ü❝
tû

H✳

PC (x) ∈ C

❈❤♦

C

❧➔ ♠ët t➟♣ ❝♦♥ ❦❤→❝ ré♥❣✱ ỗ õ ừ ổ

t r ợ ộ

x H

tỗ t t ởt

tọ

x PC (x) = inf x − y .
y∈C

P❤➛♥ tû
→♥❤ ①↕

PC (x)

❤➻♥❤ ❝❤✐➳✉ ❝õ❛ x ❧➯♥ C ✈➔
t❤➔♥❤ PC (x) ✤÷đ❝ ❣å✐ ❧➔ ♣❤➨♣

✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ tr➯♥ ✤÷đ❝ ❣å✐ ❧➔

PC : H → C

❜✐➳♥ ♠é✐ ♣❤➛♥ tû

❝❤✐➳✉ ♠➯tr✐❝ tø H ❧➯♥ C ✳

x∈H

✣➦❝ tr÷♥❣ ❝õ❛ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ ✤÷đ❝ ❝❤♦ ❜ð✐ ữợ

C ởt t ỗ õ rộ ừ ổ

rt tỹ H ✳ ❑❤✐ ✤â✱ →♥❤ ①↕ PC : H → C ❧➔ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tø H ❧➯♥ C
❦❤✐ ✈➔ ❝❤➾ ❦❤✐
x − PC (x), y − PC (x) ≤ 0

◆❤➟♥ ①➨t ✶✳✸✳

y ∈ C✱
π
α≤ ✳
2

❱➲ ♣❤÷ì♥❣ ❞✐➺♥ ❤➻♥❤ ❤å❝✱ ✈ỵ✐ ♠å✐

t↕♦ ❜ð✐ ❝→❝ ✈➨❝tì

❱➼ ❞ư ✶✳✶✳ Rn

✈ỵ✐ ♠å✐ y ∈ C.

x − PC (x)

✈➔

y − PC (x)

t❤➻

♥➳✉ t❛ ❣å✐

❧➔ ổ rt tỹ ợ t ổ ữợ

n

x, y =

k αk
k=1

α

❧➔ ❣â❝


✺
tr♦♥❣ ✤â

x = (λ1 , λ2 , . . . , λn )✱ y = (α1 , α2 , . . . , αn )
n

x

2

n

= x, x =

❑❤æ♥❣ ❣✐❛♥ l2 ✱ ✈ỵ✐

|αk |2 .

αk αk =
k=1

❱➼ ❞ư ✶✳✷✳

✈➔ ❝❤✉➞♥ ❝↔♠ s✐♥❤

k=1

x = {λk }, y = {αk }✱

t❛ ✤à♥❤ ♥❣❤➽❛

∞

λk k

x, y =
k=1
t

Ã, Ã

t ổ ữợ

(l2 , Ã, Ã )

✶✳✶✳✷✳ ▼ët sè t➼♥❤ ❝❤➜t
✣à♥❤ ❧➼ ✶✳✶
X✱

❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳

✭❇➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲❙❝❤✇❛rt③✮

✳ ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt

✈ỵ✐ ♠å✐ x, y ∈ X t❛ ❧✉æ♥ ❝â ❜➜t ✤➥♥❣ t❤ù❝ s❛✉

| x, y |2 ≤ x, x . y, y .

✭✶✳✶✮

❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ y = 0 ❜➜t ✤➥♥❣ t❤ù❝ ❤✐➸♥ ♥❤✐➯♥ ✤ó♥❣✳ ●✐↔ sû y = 0 ❦❤✐ ✤â
✈ỵ✐ ♠å✐ sè

λ∈R

t❛ ✤➲✉ ❝â

x + λy, x + λy ≥ 0
tù❝ ❧➔

x, x + λ y, x + λ x, y + |λ|2 y, y ≥ 0.
❈❤å♥

λ=−

x, y
y, y

t❛ ✤÷đ❝

x, x −

| x, y |2
≥ 0 ⇔ | x, y |2 ≤ x, x . y, y .
y, y

ỵ ữủ ự ♠✐♥❤✳

✣à♥❤ ❧➼ ✶✳✷✳ ●✐↔ sû {xn}n, {yn}n ❧➔ ❤❛✐ ❞➣② ❤ë✐ tư ②➳✉ ✤➳♥ a, b tr♦♥❣ ❦❤ỉ♥❣
❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt t❤ü❝ X ✳ ❑❤✐ ✤â

lim xn , yn = a, b .

n→∞

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû n→∞
lim xn = a✱ lim yn = b tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ X ✳ ❚❛ s➩ ❝❤ù♥❣
n→∞
♠✐♥❤

lim xn , yn = a, b

n→∞

tr♦♥❣

R✳

❚❤➟t ✈➟②✱ t❛ ❝â

| xn , yn − a, b | = | xn , yn + xn , b − xn , b − a, b |


✻

≤ | xn , yn − b + xn − a, b |
≤ xn . yn − b + xn − a . b .
❱➻ ❞➣②

{xn }n

n ∈ N✳

❑❤✐ ✤â t õ t tự

ở tử tr

X

tỗ t

M >0

xn ≤ M

s❛♦ ❝❤♦

✈ỵ✐ ♠å✐

| xn , yn − a, b | ≤ M xn . yn − b + xn − a . b .
❈❤♦

n → ∞✱

s✉② r❛

lim xn , yn = a, b .

n
ỵ ữủ ự

✶✳✸✳ ❱ỵ✐ ♠å✐ x, y t❤✉ë❝ ❦❤ỉ♥❣ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt X t❛ ❧✉æ♥ ❝â ✤➥♥❣ t❤ù❝
❤➻♥❤ ❜➻♥❤ ❤➔♥❤ s❛✉

x+y

2

+ x−y

2

= 2( x

2

+ y 2 ).

✭✶✳✷✮

❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠å✐ x, y ∈ X ✱ t❛ ❝â
x+y

2

= x + y, x + y = x

2

+ y

2

+ x, y + y, x ,

x−y

2

= x − y, x − y = x

2

+ y

2

− x, y − y, x .

❈ë♥❣ ❤❛✐ ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ✤÷đ❝ ✤➥♥❣ t❤ù❝ ✭✶✳✷✮
⑩♣ ❞ư♥❣ ✤➥♥❣ t❤ù❝ ❤➻♥❤ ❜➻♥❤ ❤➔♥❤ ❝❤♦ ❤❛✐ ✈➨❝tì

x−y

✈➔

x−z

t❛ ❝â ❤➺

q✉↔ s❛✉✳

❍➺ q✉↔ ✶✳✶✳ ●✐↔ sû X ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt ✈➔ x, y, z ∈ X ✳ ❑❤✐ ✤â✱ t❛ ❝â
✤➥♥❣ t❤ù❝ ❆♣♦❧❧♦♥✐✉s

2( x y

2

2

+ xz )=4

y+z
x
2

2

+ yz

ỗ ữợ
H


ởt ổ rt

t ỗ ♥➳✉ ∀x, y ∈ C ✱ ∀λ ∈ [0; 1] t❛ ❝â λx + (1 − λ)y ∈ C ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✽✳
✭✐✮ ▼ët t➟♣

λx ∈ C ✳

C ⊆ H

✤÷đ❝ ❣å✐ ❧➔

♥â♥

❝â ✤➾♥❤ t↕✐

0

♥➳✉

2

C⊆H

.

✤÷đ❝ ❣å✐ ❧➔

∀x ∈ C ✱ ∀λ ≥ 0

t❤➻


✼
✭✐✐✮

C

✭✐✐✐✮ ◆â♥

♠å✐

x0

✤÷đ❝ ❣å✐ ❧➔ ♥â♥ ❝â ✤➾♥❤ t↕✐

C

❝â ✤➾♥❤ t↕✐

x, y C

ợ ồ



0

ữủ ồ

, à > 0



C =

t

C x0

õ õ t

0

õ ỗ C ởt t ỗ ợ

x + ày C

t ỗ tr

H



x C

õ t✉②➳♥

♥❣♦➔✐ ❝õ❛ C t↕✐ x ∈ C ✱ ♥â♥ ✤è✐ ❝ü❝ ✈➔ ♥â♥ ✤è✐ ♥❣➝✉ ❝õ❛ C ❧➔ ❝→❝ t➟♣ ữủt
ữủ ỵ

NC (x) := {w ∈ H : w, y − x ≤ 0, ∀y ∈ C},
C0 := {w ∈ H : w, x ≤ 0, ∀x ∈ C},
C+ := {w ∈ H : w, x 0, x C}.

r ỗ t ừ f ỵ epif ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ❝ỉ♥❣ t❤ù❝
epif := {(x, r) ∈ C × R : f (x) ≤ r}.

✭✐✐✮

▼✐➲♥ ❤ú✉ ❤✐➺✉ ừ f ỵ domf ữủ ♥❣❤➽❛ ❜ð✐ ❝æ♥❣ t❤ù❝
domf := {x ∈ C : f (x) < +}.




ợ ồ

f

ữủ ồ



f

ữủ ồ

t❤÷í♥❣ ♥➳✉ domf = ∅ ✈➔ f (x) >

x ∈ C

ỗ tr C
f (x + (1 λ)y) ≤ λf (x) + (1 − λ)f (y), ∀x, y C, [0; 1];



ỗ t tr C ♥➳✉
f (λx + (1 − λ)y) < λf (x) + (1 − λ)f (y), ∀x, y ∈ C, x = y, (0; 1);



ỗ

tr

C

ợ số

>0

ợ

x, y ∈ C, ∀λ ∈ (0; 1)

t❛ ❝â

1
f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) − λ(1 − λ)α x − y 2 ;

2
✭✐✈✮

▲ã♠ tr➯♥ C f ỗ tr C




t


f

ỗ t ỗ tr



f

ỗ tr



f

ỗ s r


P

C



domf

sỷ

x H

epif

f

C

t

x



ỗ tr ổ rt

ữủ ồ

f (
x)

f

ữủ ồ

H

ữợ ❝õ❛ ❤➔♠ f t↕✐ x¯ ∈ H ♥➳✉
f

t↕✐

x¯

∀x ∈ H.

✤÷đ❝ ồ

ữợ ừ

ởt tữỡ ữỡ t ❝â

∂f (¯
x) := {x∗ ∈ H : x∗ , x − x¯ ≤ f (x) − f (¯
x),
✭✐✐✐✮ ❍➔♠

∀x ∈ H}.

ữợ t x f (x) = ∅✳

✶✳✶✳✹✳ ❚♦→♥ tû tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt

✣à♥❤ ♥❣❤➽❛ ✶✳✶✹✳ H
❈❤♦

H

✤÷đ❝ ❣å✐ ❧➔

❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ❚♦→♥ tû ✤ì♥ trà

t♦→♥ tû ✤ì♥ ✤✐➺✉ ♥➳✉

T (x) − T (y), x − y ≥ 0,

❱➼ ❞ö ✶✳✸✳

❈❤♦ t♦→♥ tû

T

①→❝ ✤à♥❤ tr➯♥

T (x) = x,
❑❤✐ ✤â

T

❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ✈➻ ✈ỵ✐ ♠å✐

R

❚♦→♥ tû ✤❛ trà

T :H→

∀x, y ∈ H.

❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝

∀x ∈ R.
x, y ∈ R✱

t❛ ❝â

T (x) − T (y), x − y = x − y, x − y = x y



C

t ỗ

tt ữợ ừ
t

ỗ tr

H ì R

t ỗ tr

x , x x f (x) − f (¯
x),

f

f

T : H → 2H

2

≥ 0,

∀x, y ∈ R.

✤÷đ❝ ❣å✐ ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉

♥➳✉

u − v, x − y ≥ 0, ∀x, y ∈ domT, ∀u ∈ T (x), ∀v ∈ T (y)
tr♦♥❣ ✤â ❞♦♠T

❱➼ ❞ö

= {z(z)}

ỗ

T = f : H H

ừ

f

f : H → [−∞, +∞]✱

❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉✳

❦❤✐ ✤â ữợ


✾

❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠å✐ x, y

∈ domT ✱ u ∈ T (x), v ∈ T (y)✱

t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤

r➡♥❣✿

u − v, x − y ≥ 0.
❚❤ü❝ ✈➟②✱ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ữợ ừ ỗ t õ

u T (x) = ∂f (x)

❦❤✐ ✈➔ ❝❤➾ ❦❤✐✿

f (z) − f (x) ≥ u, z − x , ∀z ∈ H.
❚❤❛②

z=y

t❛ ❝â✿

f (y) − f (x) ≥ u, y − x ⇔ f (y) − f (x) ≥ − u, x − y .
❚÷ì♥❣ tü✱

v ∈ T (y) = ∂f (y)

✭✶✳✸✮

❦❤✐ ✈➔ ❝❤➾ ❦❤✐✿

f (z) − f (y) ≥ v, z − y , ∀z ∈ H.
❚❤❛②

z=x

t❛ ❝â✿

f (y) − f (x) ≥ v, x − y .

✭✶✳✹✮

❈ë♥❣ ❤❛✐ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✸✮ ✈➔ ✭✶✳✸✮✱ t❛ ✤÷đ❝✿

v, x − y − u, x − y ≤ 0 ⇔ v − u, x − y ≤ 0
❤❛②

u − v, x − y ≥ 0.
❱➟②

T = ∂f

❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✻✳

❚♦→♥ tû ✤❛ trà

u − v, x − y > 0

✣à♥❤
ợ số

ợ

T : H 2H

ữủ ồ ✤ì♥ ✤✐➺✉ ❝❤➦t ♥➳✉✿

∀x, y ∈ domT, x = y, ∀u ∈ T (x), ∀v ∈ T (y).

❚♦→♥ tû ✤❛ trà

T : H → 2H

✤÷đ❝ ❣å✐ ❧➔ ✤ì♥ ✤✐➺✉ ♠↕♥❤ ♥➳✉

α ∈ R, α > 0✱ ∀x, y ∈ domT, ∀u ∈ T (x)✱ ∀v ∈ T (y)✱

t❛ ❝â

x − y, u − v ≥ α x − y 2 .

▼➺♥❤ ✤➲ ✶✳✹✳ ❚♦→♥ tû t✉②➳♥ t➼♥❤ A : H → H ❧➔ ✤ì♥ ✤✐➺✉ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
Az, z ≥ 0,

∀z ∈ H.


✶✵

❈❤ù♥❣ ♠✐♥❤✳ ❍✐➸♥ ♥❤✐➯♥ domA = H ✈➔ A ❧➔ t♦→♥ tû ✤ì♥ trà✳ ❚❍❡♦ ✤à♥❤ ♥❣❤➽❛✱
A

❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐✿

Ax − Ay, x − y ≥ 0,

∀x, y ∈ H,

A(x − y), x − y ≥ 0,

∀x, y ∈ H.

❤❛②

✣➦t

z = x − y✱

t❛ ❝â✿

Az, z ≥ 0,

∀z ∈ H.

▼➺♥❤ ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳

▼➺♥❤ ✤➲ ✶✳✺✳ ❈→❝ t➼♥❤ ❝❤➜t s❛✉ ❧➔ ❧✉ỉ♥ ✤ó♥❣✳
✭✐✮ T : H → 2H ✤ì♥ ✤✐➺✉ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ T −1 : H → 2H ❧➔ ✤ì♥ ✤✐➺✉✳
✭✐✐✮ ◆➳✉ Ti : H → 2H (i = 1, 2)✱ ❧➔ ❝→❝ t♦→♥ tû ✤ì♥ ✤✐➺✉ ✈➔ ♥➳✉ λi ≥ 0 ✭i = 1, 2✮✱
t❤➻
λ1 T1 + λ2 T2

❝ô♥❣ ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉✳
✭✐✐✐✮ ◆➳✉ A : T → T ❧➔ t♦→♥ tû t✉②➳♥ t➼♥❤✱ b ∈ H ✈➔ ♥➳✉ T : H → H ❧➔ t♦→♥ tû
✤ì♥ ✤✐➺✉ t❤➻
S(x) = A∗ T (Ax + b)

❝ơ♥❣ ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉✳ ◆❣♦➔✐ r❛✱ ♥➳✉ A ❧➔ ✤ì♥ →♥❤ ✈➔ T ❧➔ t♦→♥ tû ✤ì♥
✤✐➺✉ ❝❤➦t t❤➻ S ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝❤➦t✳ Ð ✤➙②✱ A∗ ❧➔ t♦→♥ tû ❧✐➯♥ ❤ñ♣ ❝õ❛
A✳
❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛✱ t♦→♥ tû T ❧➔ ✤ì♥ ✤✐➺✉ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐✿
u − v, x − y ≥ 0,

∀x, y ∈ domT, ∀u ∈ T (x), ∀v ∈ T (y),

❤❛②

x − y, u − v ≥ 0,
✣✐➲✉ ♥➔② ❝❤♦ t❤➜②

T −1

∀u, v ∈ domT −1 , ∀x ∈ T −1 (u), ∀y ∈ T −1 (v).

❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉✳

✭✐✐✮ ❍✐➸♥ ♥❤✐➯♥ t❛ ❝â✿

dom(λ1 T1 + λ2 T2 ) = {z ∈ H : λ1 T1 (z) + λ2 T2 (z) = ∅} = domT1 ∩ domT2 .


✶✶
●✐↔ sû

x, y ∈ domT1 ∩ domT2
✈➔

u ∈ (λ1 T1 + λ2 T2 ) = λ1 T1 (x) + λ2 T2 (x),
v ∈ (λ1 T1 + λ2 T2 ) = λ1 T1 (y) + λ2 T2 (y).
▲➜②

ui ∈ Ti (x), vi (y) ∈ Ti (y) ✭i = 1, 2✮

s❛♦ ❝❤♦✿

u = λ1 u1 + λ2 u2 ,
❉♦

T1 , T2

v = λ1 v1 + λ2 v2 .

❧➔ ❝→❝ t♦→♥ tû ✤ì♥ ✤✐➺✉ ♥➯♥ t❛ ❝â

u1 − v1 , x − y ≥ 0,

✭✶✳✺✮

u2 − v2 , x − y ≥ 0.

✭✶✳✻✮

λ1

◆❤➙♥ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✺✮ ✈ỵ✐

✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✻✮ ✈ỵ✐

λ2

rỗ ở t

ữủ

u v, x y 0.

✣✐➲✉ ✤â ❝❤ù♥❣ tä

λ1 T1 + λ2 T2

❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉✳

✭✐✐✐✮ ▲➜②

x, y ∈ domT, u ∈ S(x) = A∗ T (Ax + b), v ∈ S(y) = A∗ T (Ay + b).
❈❤å♥

u1 ∈ T (Ax + b)

✈➔

v1 ∈ T (Ay + b)

s❛♦ ❝❤♦

u = A∗ u1 , v = A∗ v1 .
❉♦ t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛

T✱

t❛ ❝â

v − u, y − x = A∗ v1 − A∗ u1 , y − x = v1 − u1 , (Ay + b) − (Ax + b) ≥ 0.
❚ø ✤â ❝❤ù♥❣ tä
●✐↔ sû

Ay = Ax✱
T

A

S

❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉✳

❧➔ ✤ì♥ →♥❤ ✈➔

❦➨♦ t❤❡♦

T

❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝❤➦t✳ ❑❤✐ ✤â✱ ♥➳✉

Ay + b = Ax + b✳

●✐↔ sû

u, v, u1 , v1

x=y

t❤➻

✤÷đ❝ ❧➜② ♥❤÷ ð tr➯♥✱ ✈➻

❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝❤➦t ♥➯♥

v1 − u1 , (Ay + b) = (Ax + b) > 0.


✶✷
❙✉② r❛

v − u, y − x > 0.
❚ø ✤â ❝❤ù♥❣ tä

S

❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝❤➦t✳ ▼➺♥❤ ✤➲ ✤÷đ❝ ự


ỗ t

Gr(T )

T : H 2H

❚♦→♥ tû ✤ì♥ ✤✐➺✉

❝õ❛

T

✤÷đ❝ ❣å✐ ❧➔

t♦→♥ tû ❝ü❝ ✤↕✐

❦❤ỉ♥❣ ❧➔ t tỹ sỹ ừ ỗ t ừ t ♠ët

t♦→♥ tû ✤ì♥ ✤✐➺✉ ♥➔♦ ❦❤→❝✳

❱➼ ❞ư ✶✳✺✳

T : R → 2R ✤÷đ❝ ❝❤♦ ❜ð✐


1
♥➳✉ x > 0



T (x) = [0; 1] ♥➳✉ x = 0



−x2 ♥➳✉ x < 0

❚♦→♥ tû ✤❛ trà

❝ỉ♥❣ t❤ù❝

❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳

M (x, y) ∈
/ Gr(T ) t❛ ❧✉ỉ♥ t➻♠ ✤÷đ❝ ✤✐➸♠ M0 (x0 , y0 ) ∈
−−→

−−−→
❤❛✐ ✈➨❝tì OM ✈➔ OM0 ❧➔ ❣â❝ tị✱ ✤✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔

❚❤➟t ✈➟②✱ ✈ỵ✐ ♠å✐ ✤✐➸♠

Gr(T )

s❛♦ ❝❤♦ ❣â❝ ❣✐ú❛

−−→ −−−→
(x, y), (x0 , y0 ) = OM .OM0 < 0.
❉♦ ✈➟②

T

❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✾✳
H1 → H2
✭✐✮
✭✐✐✮

❣å✐ ❧➔ ♠ët

❈❤♦

H1 ✱ H2

→♥❤ ①↕ t✉②➳♥ t➼♥❤ ❤❛② t♦→♥ tû t✉②➳♥ t➼♥❤ ♥➳✉✿

A(x1 + x2 ) = Ax1 + Ax2
A(αx) = αAx

✈ỵ✐ ♠å✐

✣à♥❤ ♥❣❤➽❛ ✶✳✷✵✳

❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ▼ët →♥❤ ①↕

❈❤♦

✈ỵ✐ ♠å✐

x ∈ H1

H1 ✱ H2

A :

x1 , x2 ∈ H1 ❀

✈➔ ✈ỵ✐ ♠å✐ sè

α✳

❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ▼ët t♦→♥ tû

A :

t♦→♥ tû ❧✐➯♥ tư❝ ♥➳✉ xn → x0 ❧✉ỉ♥ ❧✉æ♥ ❦➨♦ t❤❡♦ Axn → Ax0✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✶✳ ❚♦→♥ tû t✉②➳♥ t➼♥❤ A : H1 → H2 ❣å✐ ❧➔ t♦→♥ tû ❜à ❝❤➦♥

H1 → H2

❣å✐ ❧➔

✭❣✐ỵ✐ ♥ë✐✮ ♥➳✉ ❝â ♠ët số

r>0



(x H1 )

Ax r x

ỵ r➡♥❣ ❝❤✉➞♥ ❜➯♥ tr→✐ ❜➜t ✤➥♥❣ t❤ù❝ ❧➔ ❝❤✉➞♥ tr♦♥❣
❧➔ ❝❤✉➞♥ tr♦♥❣

✭✶✳✼✮

H2 ✱ ❝á♥ ❝❤✉➞♥ ❜➯♥ ♣❤↔✐

H1 ✮✳

✣à♥❤ ❧➼ ✶✳✹✳ ▼ët t♦→♥ tû t✉②➳♥ t➼♥❤ A : H1 → H2 ❧➔ ❧✐➯♥ tö❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥â

❜à ❝❤➦♥✳

ự sỷ t tỷ A tử rữợ ❤➳t t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♣❤↔✐ ❝â
♠ët ❤➡♥❣ sè

r

Ax ≤ r

✤➸ ❝❤♦

x

✈ỵ✐ ♠å✐

❝â

x = 1✳

❚❤➟t ✈➟②✱ ♥➳✉ tr→✐ ❧↕✐

tù❝ ❧➔

(∀x) (∃xn ) : xn = 1, Axn > n,
t❤➻ ❧➜②

xn
n

xn =

t❛ ❝â

xn → 0

✈➔

Axn = A
tr→✐ ✈ỵ✐ ❣✐↔ t❤✐➳t
❱ỵ✐ ♠å✐

x=0

A

xn
Axn
=
>1
n
n

❧✐➯♥ tư❝✳ ❱➟② ♣❤↔✐ ❝â

x
x

t❛ ❝â

= 1✱

◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû ❝â ❤➡♥❣ sè

r

r

✈ỵ✐ t➼♥❤ ❝❤➜t tr➯♥✳

❝❤♦ ♥➯♥

Ax
≤ r✱
x

❞♦ ✤â

t❤ä❛ ♠➣♥ ❝ỉ♥❣ t❤ù❝ ✭✶✳✼✮ ✈➔

Ax ≤ r x .
xn → x0 ✳

❚❛ ❝â

Axn − Ax0 = A(xn − x0 ) ≤ r xn − x0 → 0.
❱➟②

A

❙è

❧✐➯♥ tư❝ t↕✐

r>0

A

✈➔♦

C

s❛♦ ❝❤♦

✤÷đ❝ ❣å✐ ❧➔

PC

tø

❀

(∀x ∈ H1 ) Ax ≤ r x

✣à♥❤ ♥❣❤➽❛ ✶✳✷✷✳
H

✈➔

✳ ◆❤÷ ✈➟②✿

(∀x ∈ H1 ) Ax ≤ A . x

✭✐✐✮ ◆➳✉

A

♥❤ä ♥❤➜t t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✶✳✼✮ ❣å✐ ❧➔ ❝❤✉➞♥ ❝õ❛ t tỷ

ữủ ỵ


x0

A r

t

ợ ởt t õ ỗ

C

ừ

x PC (x) = inf yC x y

H

tỗ t↕✐ ♠ët →♥❤ ①↕

✈ỵ✐ ♠å✐

x ∈ H✳

⑩♥❤ ①↕

PC

♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ ❧➯♥ C ✳ ❇✐➳t r➡♥❣ PC ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t

✭✈➻ ✈➟② ❦❤æ♥❣ ❣✐➣♥✮ ✈➔

x − PC x
❚❛ ❦➼ ❤✐➺✉ ❋✐①(T )

2

+ PC x − p

2

= {x ∈ C : T x = x}

≤ x − p 2 , x ∈ H, p ∈ C.
T✳

❧➔ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕

❇ê ✤➲ ✶✳✶✳ ❈❤♦ {ak }✱ {bk } ✈➔ {ck } ❧➔ ❞➣② ❝→❝ sè t❤ü❝ s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ k ≥ 1✱
ak+1 ≤ (1 − bk )ak + bk ck ✱ ak ≥ 0✱ bk ∈ [0, 1]

t❤ä❛ ♠➣♥ bk → 0 ❦❤✐ k → ∞ ✈➔
= ∞ ✈➔ lim supk→∞ ck ≤ 0 t❤➻ limk→∞ ak = 0✳
❇ê ✤➲ ✶✳✷✳ ❈❤♦ {ak } ❧➔ ♠ët ❞➣② số tỹ s tỗ t {ak } ❝õ❛
{ak } s❛♦ ❝❤♦ ak < ak +1 ✈ỵ✐ ồ l N+ õ tỗ t ởt ❞➣② ❦❤æ♥❣ ❣✐↔♠
{mk } ⊆ N+ s❛♦ ❝❤♦ mk → ∞, am ≤ am +1 ✈➔ ak ≤ am +1 ✈ỵ✐ ♠å✐ sè k ∈ N+
✤õ ❧ỵ♥✳ ❚❤➟t ✈➟②✱ mk = max{j ≤ k : aj ≤ aj+1}✳
∞
k=1 bk

l

l

l

k

k

k


✶✹

✶✳✶✳✺✳ ✣✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥
❈❤♦

C

❧➔ t➟♣ ỗ õ rộ tr ổ rt tỹ

T :C→H

⑩♥❤ ①↕

T

✤÷đ❝ ❣å✐ ❧➔

❑❤ỉ♥❣ ❣✐➣♥ ♥➳✉

Tx − Ty ≤ x − y ,
✭✐✐✮

✈➔

❧➔ ♠ët →♥❤ ①↕✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✸✳
✭✐✮

H

∀x, y ∈ C;

❑❤æ♥❣ ❣✐➣♥ ❝❤➦t ♥➳✉
Tx − Ty

2

≤ T x − T y, x − y ,

∀x, y ∈ C.

❱➜♥ ✤➲ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❧ỵ♣ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ❧➔ ✤➲ t➔✐ ❝â t➼♥❤
t❤í✐ sü ✈➔ t❤✉ ❤ót sü q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ tr♦♥❣
ữợ ữợ t ởt số ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ t➻♠ ✤✐➸♠ ❜➜t
✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳

❇➔✐ t♦→♥ ✶✳✶✳
❍✐❧❜❡rt

❈❤♦

C

H✱ T : C → C

❧➔ ♠ët t ỗ õ rộ tr ổ
ởt →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳
❍➣② t➻♠

P❤➛♥ tû

x∗ ∈ C

x∗ ∈ C : T (x∗ ) = x∗ .

✭✶✳✽✮

t❤ä❛ ♠➣♥ ✭✶✳✽✮ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕

❚➟♣ ✤✐➸♠ t ở ừ

T

ỵ

T

F ix(T )

ỹ tỗ t ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ rt
ữủ ỵ ữợ

C t ỗ õ ừ ổ ❣✐❛♥ ❍✐❧❜❡rt H ✈➔
T :C→C

❧➔ ♠ët →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ❑❤✐ ✤â✱ T ❝â ➼t ♥❤➜t ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣✳

◆❤➟♥ t

ứ t ỗ t ừ ổ rt

❦❤æ♥❣ ❣✐➣♥

T✱

t❛ t❤➜② ♥➳✉ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣

H

✈➔ t➼♥❤ ❧✐➯♥ tử ừ

F ix(T )

rộ t õ

t ỗ ✤â♥❣✳
❚❛ t❤➜② r➡♥❣ ♥➳✉

T

❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝❤➦t t❤➻

T

❧➔ ♠ët t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥✳ ❱➻

✈➟② t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t ❧➔ ởt t tỷ ổ ú ỵ r
ổ t❤➻ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛

T ✱ F ix(T )

❧➔ õ ỗ

T

ú ỵ



0 < < 1

t Pr ①→❝ ✤à♥❤ ❜ð✐

T :C →C

❧➔ →♥❤ ①↕ ❝♦ ✭tù❝ ❧➔

♠↕♥❤ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❝õ❛

T✳

Tx − Ty ≤ ρ x − y

x0 ∈ C

✈➔

xn+1 = T (xn )

✈ỵ✐

❤ë✐ tư

❚✉② ♥❤✐➯♥ ✤✐➲✉ ♥➔② ❦❤ỉ♥❣ ❝á♥ ✤ó♥❣

✤è✐ ✈ỵ✐ ❧ỵ♣ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳

❇ê ✤➲ ✶✳✸✳ ❈❤♦ C ❧➔ ♠ët t➟♣ ỗ õ ừ ổ rt tỹ H

♠ët →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ✈ỵ✐ ❋✐①(T ) = ∅✳ ◆➳✉ {xk } ❧➔ ♠ët ❞➣②
tr♦♥❣ C ❤ë✐ tö ②➳✉ ✤➳♥ x ✈➔ ♥➳✉ {(I −T )xk } ❤ë✐ tö ♠↕♥❤ ✤➳♥ y t❤➻ (I −T )x = y✳
✣➦❝ ❜✐➺t✱ ♥➳✉ y = 0 t❤➻ x ∈ ❋✐①(T )✳
❇ê ✤➲ ✶✳✹✳ ◆➳✉ ❝→❝ →♥❤ ①↕ {Ti}ki=1 ❧➔ tr✉♥❣ ❜➻♥❤ ✈➔ ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣
t❤➻
k
❋✐①(T1T2 · · · Tk ) = ❋✐①(Ti).

T :C →C

i=1

❇ê ✤➲ ✶✳✺✳ ❈❤♦ C ❧➔ ♠ët t ỗ õ ừ ổ rt H ✈➔
❝❤♦ T

❧➔ ♠ët →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tø C ✈➔♦ H ✳ ◆➳✉ F ix(T ) = ∅ t❤➻
F ix(T ) = F ix(PC T )✳
❇ê ✤➲ ✶✳✻✳ ❈❤♦ T ❧➔ ♠ët →♥❤ ①↕ L✲❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③ ✈➔ η✲✤ì♥ ✤✐➺✉ tr
ổ rt H õ ợ à ∈ (0, 2η/L2)✱ λ ∈ (0, 1) t❛ ❧✉æ♥ ❝â
:C →H

T λ x − T λ y ≤ (1 − λτ ) x − y ,

tr♦♥❣ ✤â τ
x ∈ H✳

= 1−

1 − µ(2η − µL2 ) ∈ (0, 1)

✈➔ T x = (I àT )x ợ ồ

ờ ❈❤♦ ❞➣② {xn} ✈➔ {zn} ❧➔ ❝→❝ ❞➣② ❜à ❝❤➦♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
H

s❛♦ ❝❤♦

xn+1 = (1 − βn )xn + βn zn ,

n ≥ 1,

tr♦♥❣ ✤â {βn} ⊂ [0, 1] t❤ä❛ ♠➣♥
0 < lim inf βn ≤ lim sup βn < 1.
n→∞

◆➳✉ lim sup(

n→∞

zn+1 − zn − xn+1 − xn ) ≥ 0

n→∞

t❤➻ n→∞
lim

xn − zn = 0✳

✶✳✷✳ P❤→t ❜✐➸✉ ❜➔✐ t♦→♥
❈❤♦

·

H1 ✈➔ H2 ❧➔ ❤❛✐ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt tỹ ợ t ổ ữợ Ã, Ã



J1



J2

t ❝❤➾ sè✱

{Ci }i∈J1

✈➔

{Qj }j∈J2

✈➔ ❝❤✉➞♥

❧➔ ❤❛✐ ❤å ❝→❝ t➟♣ ❝♦♥



ỗ õ tr ổ
ợ ở tứ

H1



H1



H2

tữỡ ự ❝❤♦

A

❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤

H2 ✳

❇➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✤❛ t➟♣ ❧➔ t➻♠ ♠ët ✤✐➸♠

x ∈ C := ∩i∈J1 Ci
Γ

❑➼ ❤✐➺✉

s❛♦ ❝❤♦

Ax ∈ Q := ∩j∈J2 Qj .

✭✶✳✾✮

❧➔ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✾✮✳

❇➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✤❛ t➟♣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✤➣ ✤÷đ❝ ✤➲
①✉➜t ❜ð✐ ❈❡♥s♦r ✈➔ ❊❧❢✈✐♥❣ ❬✶✵❪ ✤➸ ♠æ ❤➻♥❤ ❝→❝ ❜➔✐ t ữủ t tứ
ử ỗ ỷ ❤➻♥❤ ↔♥❤ ❬✽❪✳ ▼ỵ✐ ✤➙②✱ ❜➔✐ t♦→♥ ♥➔② ❝ơ♥❣ ❝â t❤➸ ❞ị♥❣ ✤➸
♠ỉ ❤➻♥❤ ❤â❛ ❝÷í♥❣ ✤ë ✤✐➲✉ ❜✐➳♥ ①↕ trà✳
❚r♦♥❣ tr÷í♥❣ ❤đ♣ ✤ì♥ ❣✐↔♥✱ ❦❤✐

J1 = J2 = {1}✱

t❛ ❝â ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥

t→❝❤ ✭❙❋P✮✱ ✤â ❧➔ t➻♠ ♠ët ✤✐➸♠

x∈C

s❛♦ ❝❤♦

Ax ∈ Q.

✭✶✳✶✵✮

●✐↔ t❤✐➳t ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ❝â ♥❣❤✐➺♠ ✭❝â ♥❣❤➽❛ ✭✶✳✶✵✮ ❝â ♥❣❤✐➺♠✮✱ ❇②r♥❡
❬✽❪ ✤➣ ✤➲ ①✉➜t ♣❤÷ì♥❣ ♣❤→♣

CQ✱

ð ✤➙② ❞➣② ❧➦♣

{xk }

①→❝ ✤à♥❤ ❜ð✐

xk+1 = PC (I − γA∗ (I − PQ )A)xk , k ≥ 1,
tr♦♥❣ ✤â

I

❧➔ →♥❤ ①↕ ✤ì♥ ✈à tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥

❝❤✐➳✉ ♠➯tr✐❝ tø
❤➡♥❣ sè ❞÷ì♥❣✱

H1

✈➔

H2

t÷ì♥❣ ù♥❣ ①✉è♥❣

H1

C

✈➔

✈➔

H 2 ✱ PC

✭✶✳✶✶✮
✈➔

PQ

❧➔ ❝→❝ ♣❤➨♣

Q✱ γ ∈ (0, 2/ A 2 )

❧➔ ♠ët

A∗ ❧➔ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝õ❛ A✳ P❤÷ì♥❣ ♣❤→♣ ♥➔② ❤ë✐ tư ②➳✉ tr♦♥❣

❦❤ỉ♥❣ ❣✐❛♥ ✈ỉ ❤↕♥ ❝❤✐➲✉ ✤➳♥ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤✱ ✈ỵ✐ ♠ët ✤✐➸♠
①✉➜t ♣❤→t
♣❤→♣

CQ

tr➯♥ t➟♣

x∗

x1

❜➜t ❦➻✳ ❑❤✐ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ❦❤æ♥❣ ❝â ♥❣❤✐➺♠✱ ♣❤÷ì♥❣

❤ë✐ tư ②➳✉ ✤➳♥ ✤✐➸♠ ❝ü❝ t✐➸✉ ❝õ❛ ❤➔♠ ✤♦

C✱

q(x) = (I − PQ )Ax 2 /2

✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❧➔ ❜➔✐ t♦→♥ ❝ü❝ t✐➸✉ ❝â r➡♥❣ ❜✉ë❝ ♥➔② ❝â ♥❣❤✐➺♠✳ ✣➦t

❧➔ ✤✐➸♠ ❝ü❝ t✐➸✉ ❝õ❛

q(x) tr➯♥ C

✈➔

F := A∗ (I − PQ )A✳ ❑❤✐ ✤â✱ x∗

❧➔ ♥❣❤✐➺♠

❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ s❛✉✿

x∗ ∈ C : F x∗ , x − x∗ ≥ 0 ∀x ∈ C.

✭✶✳✶✷✮

❱➻ ✈➟②✱ ♣❤÷ì♥❣ ♣❤→♣ ✭✶✳✶✶✮ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣r❛❞✐❡♥t✳ ❉➵ ❞➔♥❣ ♥❤➟♥ t❤➜②

✤÷đ❝

F

❧➔ ♠ët

tư❝ ▲✐♣s❝❤✐t③✳

(1/ A 2 )✲→♥❤

①↕ ✤ì♥ ✤✐➺✉ ♥❣÷đ❝ ♠↕♥❤ tr♦♥❣

H1 ✱

✈➻ ✈➟② ♥â ❧✐➯♥


✶✼

✶✳✸✳ P❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣
❇➔✐ t♦→♥ ✤➦t ❦❤ỉ♥❣ ❝❤➾♥❤ ♣❤✐ t✉②➳♥ ✭✶✳✶✷✮ ✈ỵ✐ t♦→♥ tû ✤ì♥ ✤✐➺✉ ✈➔ ❧✐➯♥ tư❝
▲✐♣s❝❤✐t③ ❝â t❤➸ ❣✐↔✐ ❜➡♥❣ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ tr♦♥❣ ❬✺✱ ✼❪✱ ♠ët
tr♦♥❣ sè ✤â ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣✱

xk+1 = PC (I − γk (F + αk I))xk , k ≥ 1,

✭✶✳✶✸✮

✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐ ❇❛❦✉s❤✐♥s❦② ❬✻❪ ✈➔ ❇r✉❝❦ ❬✾❪✳ ❚✐➳♣ t❤❡♦✱ ❳✉ ❬✶✷❪ ①➨t ✭✶✳✶✸✮
tr♦♥❣ tr÷í♥❣ ❤đ♣

F = A∗ (I − PQ )A✳

❙ü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ✭✶✳✶✸✮ tr♦♥❣ ❝↔ ❤❛✐

tr÷í♥❣ ❤đ♣ ✤÷đ❝ ✤↔♠ ❜↔♦ ❜ð✐ ♠ët sè ✤✐➲✉ ❦✐➺♥ ❝õ❛

γk → 0

❦❤✐

k → ∞✳

γk

✈➔

αk ✱

tø ✤â s✉② r❛

❨❛♦ ✈➔ ♠ët sè t→❝ ❣✐↔ ❦❤→❝ ❬✶✸❪ ✤➣ t❤❛② ✤ê✐ ✤✐➲✉ ❦✐➺♥

❜➡♥❣ ✤✐➲✉ ❦✐➺♥ ②➳✉ ❤ì♥ ❝❤ù❛

limk→∞ (γk+1 − γk ) = 0

γk

✈➔ ✤✐➲✉ ❦✐➺♥ ♥➔② ✤➣ ✤÷đ❝

❧♦↕✐ ❜ä tr♦♥❣ ❬✶✶❪ ❜ð✐ ❈❤✉❛♥❣✳
❑❤✐

J1 = 1, ..., N

✈➔

J2 = 1, ..., M

ð ✤➙②

N✱ M < 1

t❤➻ ♥❣÷í✐ t❛ ✤➣ ✤➲ ①✉➜t

♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ✭✶✳✺✮ ♥❤÷✿

xn+1 = [PCN (I − γ∇q )]...[PC1 (I − γ∇q )]xn ,
M

N

λi PCi

yn+1 =

n ≥ 0,

βj A∗ (I − PQj )Ayn ,

yn − γ

n ≥ 0,

✭✶✳✶✹✮

j=1

i=1
M

βj A∗ (I − PQj )Azn ,

zn − γ

zn+1 = PC[n+1]

n ≥ 0,

j=1
tr♦♥❣ ✤â

x∈C

M
j=1 βj

q(x) = (1/2)

✈➔

C[n] = Cn

PQj Ax−Ax

2

✱

∇q(x) =

M
∗
j=1 βj A (I −PQj )Ax✱

mod N ✈➔ ❤➔♠ ♠♦❞ ❧➜② ❝→❝ ❣✐→ trà tr♦♥❣

{1, 2, ..., N }✳

✣➙②

❧➔ ♠ët ✤ë♥❣ ❧ü❝ ✤➸ ❝❤ó♥❣ t❛ ♥❣❤✐➯♥ ❝ù✉ t❤✉➟t t♦→♥ tê♥❣ q✉→t s❛✉ ✤➸ t↕♦ r❛ ❝→❝
❞➣②

{xn }, {yn }

✈➔

{zn }

t÷ì♥❣ ù♥❣✱ t❤ỉ♥❣ q✉❛ ❝→❝ ❝ỉ♥❣ t❤ù❝

xn+1 = (1 − αn )xn + αn TN ...T2 T1 xn ,

✭✶✳✶✺✮

N

yn+1 = (1 − βn )yn + βn

λi Ti y n ,

✭✶✳✶✻✮

zn+1 = (1 − γn+1 )zn + γn+1 T[n+1] zn ,

✭✶✳✶✼✮

i=1

tr♦♥❣ ✤â

T[n] = Tn

{T1 , T2 , ..., Tn }

mod N , {αn }, {βn } ✈➔ ❞➣②

{γn }

❧➔ ❝→❝ ❞➣② tr♦♥❣

(0, 1)

✈➔

❧➔ ❝→❝ ❞➣② ❝õ❛ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳ ❈❤ó♥❣ t❛ s➩ ❝❤➾ r❛ r➡♥❣✱

tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ tê♥❣ q✉→t

X ✱ ❝→❝ ❞➣② {xn }, {yn } ✈➔ {zn } ✤÷đ❝ s✐♥❤ ❜ð✐


✶✽
✭✶✳✶✺✮✱ ✭✶✳✶✻✮ ✈➔ ✭✶✳✶✼✮ ❤ë✐ tö ②➳✉ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛

{T1 , T2 , ..., Tn }✱

t÷ì♥❣ ù♥❣✳
▲÷✉ þ r➡♥❣✱ ✤➸
❧➔ tü →♥❤ ①↕ ❝õ❛

C

❧➔ ♠ët t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤

X

✈➔

A, B

C ✱ t❛ sû ❞ö♥❣ Dρ (A, B) ✤➸ ❜✐➸✉ t❤à sup{ Ax−Bx : x ≤ ρ}✱

♥❣❤➽❛ ❧➔

Dρ (A, B) := sup{ Ax − Bx : x ≤ ρ}.
❱ỵ✐ ♠ët sè ✤✐➲✉ ❦✐➺♥ ✤➦t ❧➯♥ t❤❛♠ sè

α✱ β ✱ γ

t❤➻ t❤✉➟t t♦→♥ ❧➔ ❤ë✐ tư ②➳✉✳

❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ tỉ✐ ✤➣ ✤➲ ❝➟♣ ✤➳♥ ❝→❝ ỡ ừ t ỗ
t t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✈➔ tr➻♥❤ ❜➔② tê♥❣ q✉→t ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉
❝❤➾♥❤ ❧➦♣✳ Ð ❝❤÷ì♥❣ s❛✉✱ tỉ✐ ①➨t t❤✉➟t t♦→♥ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❦❤✐
❝❤å♥ ♠ët ❝→❝❤ tê♥❣ q✉→t✱ tù❝ ❧➔
✳

J1

✈➔

J2

J1

✈➔

❝❤ù❛ ❜➜t ❦➻ ♣❤➛♥ tû ♥➔♦ tø

1

J2

✤÷đ❝

✤➳♥

∞✳


❈❤÷ì♥❣ ✷

P❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❝❤♦
❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ tr♦♥❣
❦❤ỉ♥❣ ❣✐❛♥ ✤❛ t➟♣
◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❝❤♦
❜➔✐ t♦→♥ ❝❤➜♣ t tr ổ t ử t ỗ ✸ ♠ö❝✿ ♠ö❝ ✷✳✶
tr➻♥❤ ❜➔② ♠ët sè ❜ê ✤➲ ❝➛♥ t❤✐➳t✱ ♠ö❝ ✷✳✷ tr➻♥❤ ❜➔② t❤✉➟t t♦→♥ ✈➔ sü ❤ë✐ tö✱
♠ö❝ ✷✳✸ tr➻♥❤ ❜➔② ♠ët sè ✈➼ ❞ö✳

✷✳✶✳ ▼ët sè ❜ê ✤➲ ❝➛♥ t❤✐➳t
❇ê ✤➲ ✷✳✶✳ ❈❤♦ H1 ✈➔ H2 ❧➔ ❤❛✐ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✱ ❝❤♦ Tj ✈ỵ✐ ♠å✐ j ∈ J2

❧➔ ♠ët →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ H2 s❛♦ ❝❤♦ ∩j∈J ❋✐①(Tj ) = ∅ ✈➔ ❝❤♦ A ❧➔ ♠ët
→♥❤ ①↕ t✉②➳♥ t➼♥❤ ❣✐ỵ✐ ♥ë✐ tø H1 ✈➔♦ H2✳ ❑❤✐ ✤â✱
2

∩j∈J2 A−1

❋✐①(Tj ) = ∩j∈J ❋✐①(I − γA∗(I − Tj )A) = A−1(∩j∈J ❋✐①(Tj )),
2

2

tr♦♥❣ ✤â γ ❧➔ ♠ët sè ❞÷ì♥❣✳
❈❤ù♥❣ ♠✐♥❤✳ ❘ã r➔♥❣✱ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ✤➥♥❣ t❤ù❝ t❤ù ♥❤➜t ❝❤➾ ❝➛♥ ❝❤➾ r❛ r➡♥❣
A−1
✈ỵ✐ ♠å✐

j ∈ J2 ✳

❚❤➟t ✈➟②✱ ♥➳✉

z − γA∗ (I − Tj )Az = z ✱
❋✐①(I − γA
❦❤✐

γ > 0✳

∗

❋✐①(Tj )

❦❤✐

z∈

= ❋✐①(I − γA∗ (I − Tj )A),
z ∈ A−1
❋✐①(I

❋✐①(Tj )✱ ❝â ♥❣❤➽❛ ❧➔

− γA∗ (I − Tj )A)✳

Az = Tj Az ✱

t❤➻

❚ø ❜❛♦ ❤➔♠ t❤ù❝

z∈

(I − Tj )A)✱ ❝â ♥❣❤➽❛ ❧➔ γA∗ (I − Tj )Az = 0 s✉② r❛ A∗ (I − Tj )Az = 0✱

❉♦ ✤â✱

Tj Az = Az + wj , A∗ wj = 0.
▲➜② ♠ët ♣❤➛♥ tû

Tj Ap = Ap✳

p ∈ A−1

❋✐①(Tj )✳ ❑❤✐ ✤â✱ t❛ ❝â

Ap ∈

❋✐①(Tj )✱ ❝â ♥❣❤➽❛ ❧➔

❉♦ ✤â✱

Az − Ap

2

≥ Tj Az − Tj Ap
= Az − Ap

2

2

= Az − Ap + wj

+ wj

2

2

+ 2 wj , A(z − p)


✷✵

❱➻ ✈➟②✱

2

+ 2 A ∗ wj , z − p

= Az − Ap

2

+ wj

= Az − Ap

2

+ wj 2 .

wj = 0✳ ❈â ♥❣❤➽❛ ❧➔ Tj Az = Az ✱ ❦❤✐ z ∈ A−1 ❋✐①(Tj )✳ ✣➥♥❣ t❤ù❝ t❤ù ❤❛✐

✤÷đ❝ s✉② r❛ tø

∩j∈J2 A−1

❋✐①(Tj )

= A−1 (∩j∈J2 ❋✐①(Tj )).

❙✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣

♠✐♥❤✳

❇ê ✤➲ ✷✳✷✳ ❈❤♦ H1, H2, A ✈➔ γ ❣✐è♥❣ ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✷✳✶ ✈➔ ❝❤♦ Tj ✈ỵ✐ ♠å✐
❧➔ ♠ët →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr♦♥❣ H2 s❛♦ ❝❤♦ ∩∞j=1❋✐①(Tj ) = ∅✳ ❑❤✐ ✤â✱

j ∈ N+

C˜ := ∩j∈N+ ❋✐①(I − γA∗ (I − Tj )A) = ❋✐①(T∞ ),

tr♦♥❣ ✤â T∞ = I − γA∗(I − V∞)A✱ V∞ = ∞j=1 ηj Tj ✈➔ ηj t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥
✭η✮✳
❈❤ù♥❣ ♠✐♥❤✳ ◆❤÷ t❛ ✤➣ ❜✐➳t →♥❤ ①↕ V∞ ❧➔ ❦❤ỉ♥❣ ❣✐➣♥ ✈ỵ✐ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣
❋✐①(V∞ )
❱ỵ✐ ♠é✐

˜
= ∩∞
j=1 ❋✐①(Tj ) rữợ t t ự tự C ⊂ ❋✐①(T∞ )✳
˜ ✱ t❛ ❝â (I − γA∗ (I − Tj )A)z = z ✈ỵ✐ ♠å✐ j ∈ N+ ✳ ❚ø ✤â✱ t❛
✤✐➸♠ z ∈ C

❝â

∞

ηj (I − γA∗ (I − Tj )A)z = z,
j=1
❞♦ ✤â

I − γA∗ (I − V∞ )A)z = z,
✈➻

I

✈➔

A∗

❧➔ ❤❛✐ →♥❤ ①↕ t✉②➳♥ t➼♥❤✳ ✣✐➲✉ ✤â ❝â ♥❣❤➽❛ ❧➔

❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ ❋✐①(T∞ )

p∈

❋✐①(V∞ )✳ ❉♦

z∈

⊂ C˜ ✳

❋✐①(T∞ ) ♥➯♥ t❛ ❝â

▲➜② ❤❛✐ ✤✐➸♠ ❜➜t ❦➻

A∗ (I − V∞ )Az = 0✳

❧➟♣ ❧✉➟♥ tữỡ tỹ ữ tr ự ờ ợ

✤â

V∞

t❛ ♥❤➟♥ ✤÷đ❝ ✤➥♥❣ t❤ù❝

z ∈ ❋✐①(T∞ )✳

z ∈ ❋✐①(T∞ )✳

V∞ Az = Az ✱

♥❣❤➽❛ ❧➔

z∈

❋✐①(T∞ ) ✈➔

❚✐➳♣ t❤❡♦✱ ❜➡♥❣

J2 = {1}

✈➔

T1

t❤❛②

γA∗ (I − V∞ )Az = 0✱

❙✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳

❇ê ✤➲ ✷✳✸✳ ❈❤♦ H ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ✈➔ ❝❤♦ Si✱ ✈ỵ✐ ♠å✐ i ∈ N+ ❧➔ ♠ët

→♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t tr♦♥❣ H ✳ ●✐↔ sû ✤✐➲✉ ❦✐➺♥ ✭β ✮ ✤÷đ❝ t❤ä❛ ♠➣♥✳ ❑❤✐ ✤â✱
❝→❝ →♥❤ ①↕ S∞ := ∞i=1 βiSi ✈➔ I − S∞ ụ ổ t
ự rữợ t t r❛ r➡♥❣ S∞ ❧➔ ❦❤ỉ♥❣ ❣✐➣♥ ❝❤➦t✳ ❱ỵ✐ ♠ư❝ ✤➼❝❤

♥➔②✱ t❛ ①➨t →♥❤ ①↕

Sk :=

k
˜
i=1 (βi /βk )Si ✳ ❱➻

Si

❧➔ ❦❤ỉ♥❣ ❣✐➣♥ ❝❤➦t ✈ỵ✐ ♠å✐