Luận án Tiến sĩ Một số phương pháp giải bài toán tìm không điểm của toán tử đơn điệu cực đại và bài toán chấp nhận tách nhiều tập

❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖

❱■➏◆ ❍⑨◆ ▲❹▼ ❑❍❖❆ ❍➴❈
❱⑨ ❈➷◆● ◆●❍➏ ❱■➏❚ ◆❆▼
❍➴❈ ❱■➏◆ ❑❍❖❆ ❍➴❈ ❱⑨ ❈➷◆● ◆●❍➏
✖✖✖✖✖✖✖✖✖✖

P❍❸▼ ❚❍➚ ❚❍❯ ❍❖⑨■

▼❐❚ ❙➮ P❍×❒◆● P❍⑩P ●■❷■ ❇⑨■ ❚❖⑩◆ ❚➐▼
❑❍➷◆● ✣■➎▼ ❈Õ❆ ❚❖⑩◆ ❚Û
✣❒◆ ✣■➏❯ ❈Ü❈ ✣❸■ ❱⑨ ❇⑨■ ❚❖⑩◆ ❈❍❻P ◆❍❾◆
❚⑩❈❍ ◆❍■➋❯ ❚❾P

▲❯❾◆ ⑩◆ ❚■➌◆ ❙ß ❚❖⑩◆ ❍➴❈

❍⑨ ◆❐■ ✲ ✷✵✷✷


❱■➏◆ ❍⑨◆ ▲❹▼ ❑❍❖❆ ❍➴❈ ❱⑨ ❈➷◆● ◆●❍➏ ❱■➏❚ ◆❆▼
❍➴❈ ❱■➏◆ ❑❍❖❆ ❍➴❈ ❱⑨ ❈➷◆● ◆●❍➏
✳✳✳✳✳✳✳✳✳✳✳✳✳✳✯✯✯✳✳✳✳✳✳✳✳✳✳✳✳✳✳

P❍❸▼ ❚❍➚ ❚❍❯ ❍❖⑨■

▼❐❚ ❙➮ P❍×❒◆● P❍⑩P ●■❷■ ❇⑨■ ❚❖⑩◆ ❚➐▼
❑❍➷◆● ✣■➎▼ ❈Õ❆ ❚❖⑩◆ ❚Û
✣❒◆ ✣■➏❯ ❈Ü❈ ✣❸■ ❱⑨ ❇⑨■ ❚❖⑩◆ ❈❍❻P ◆❍❾◆
❚⑩❈❍ ◆❍■➋❯ ❚❾P
▲❯❾◆ ⑩◆ ❚■➌◆ ❙ß ❚❖⑩◆ ❍➴❈
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ư♥❣

▼➣ sè✿
✾
ữớ ữợ ồ
ữớ

◆ë✐ ✲ ✷✵✷✷


✐✐

▲❮■ ❈❆▼ ✣❖❆◆
❈→❝ ❦➳t q✉↔ ✤↕t ✤÷đ❝ tr♦♥❣ ❧✉➟♥ →♥ ổ tr ự ừ tổ
ữủ t ữợ sỹ ữợ ừ ữớ t
q ợ ữ ữủ tr tr ❝ỉ♥❣ tr➻♥❤ ❝õ❛ ♥❣÷í✐
❦❤→❝✳
❚ỉ✐ ①✐♥ ❝❤à✉ tr→❝❤ ♥❤✐➺♠ ✈➲ ♥❤ú♥❣ ❧í✐ ❝❛♠ ✤♦❛♥ ❝õ❛ ♠➻♥❤✳


✐✐✐

▲❮■ ❈❷▼ ❒◆
▲✉➟♥ →♥ ♥➔② ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❍å❝ ✈✐➺♥ ❑❤♦❛ ❤å❝ ✈➔ ❈æ♥❣ ♥❣❤➺✱
❱✐➺♥ ❍➔♥ ❧➙♠ ❑❤♦❛ ❤å❝ ổ t ữợ sỹ ữợ t
t ❝õ❛ ●❙✳ ❚❙✳ ◆❣✉②➵♥ ❇÷í♥❣✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝
tỵ✐ ❚❤➛②✳
❚r♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✱ t❤æ♥❣ q✉❛ ❝→❝ ❜➔✐ ❣✐↔♥❣ ✈➔
s❡♠✐♥❛r t→❝ ❣✐↔ ❧✉ỉ♥ ♥❤➟♥ ✤÷đ❝ sü q✉❛♥ t➙♠ ❣✐ó♣ ✤ï ✈➔ ỳ ỵ õ
õ qỵ ừ ộ ▲÷✉✱ ❚❙✳ ◆❣✉②➵♥ ❈ỉ♥❣ ✣✐➲✉✱ P●❙✳❚❙✳
◆❣✉②➵♥ ❚❤à ❚❤✉ ❚❤õ②✱ ❚❙✳ ◆❣✉②➵♥ ❚❤à ◗✉ý♥❤ ❆♥❤✱ ❚❙✳ ◆❣✉②➵♥ ❚❤à
❚❤ó② ❍♦❛✱ ❚❙✳ ◆❣✉②➵♥ ✣➻♥❤ ❉÷ì♥❣✱ ❚❙✳ ◆❣✉②➵♥ ❉÷ì♥❣ ◆❣✉②➵♥✳ ❚ø ✤→②

❧á♥❣ ♠➻♥❤ t→❝ ❣✐↔ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❝→❝ t❤➛② ❝æ
❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ tỵ✐ ❇❛♥ ❧➣♥❤ ✤↕♦✱ ❝→❝ t❤➛② ❝ỉ ❝ị♥❣ t♦➔♥
t❤➸ ❝→♥ ❜ë✱ ❝æ♥❣ ♥❤➙♥ ✈✐➯♥ t❤✉ë❝ ❱✐➺♥ ❈æ♥❣ ♥❣❤➺ t❤æ♥❣ t✐♥✱ ❍å❝ ✈✐➺♥
❑❤♦❛ ❤å❝ ✈➔ ❈æ♥❣ ♥❣❤➺✱ ❱✐➺♥ ❍➔♥ ❧➙♠ ❑❤♦❛ ❤å❝ ✈➔ ❈æ♥❣ ♥❣❤➺ ❱✐➺t ◆❛♠
✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t✱ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔
♥❣❤✐➯♥ ❝ù✉✳
❚→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉✱ ❝→❝ t❤➛② ❝æ tr♦♥❣ ❇ë
♠æ♥ ❚♦→♥ ✲ ❑❤♦❛ ❈ì sð ❝ì ❜↔♥ ✲ ✣↕✐ ❤å❝ ❍➔♥❣ ❤↔✐ ❱✐➺t ◆❛♠✱ ❝ò♥❣ t♦➔♥
t❤➸ ❛♥❤ ❝❤à ❡♠ ♥❣❤✐➯♥ ❝ù✉ s✐♥❤✱ ỗ ổ q t
ở tr ờ õ õ ỳ ỵ qỵ t→❝ ❣✐↔ tr♦♥❣
s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱ s❡♠✐♥❛r✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ →♥✳
❚→❝ ❣✐↔ ①✐♥ ❦➼♥❤ t➦♥❣ ♥❤ú♥❣ ♥❣÷í✐ t❤➙♥ ②➯✉ tr♦♥❣ ❣✐❛ ✤➻♥❤ ❝õ❛ ♠➻♥❤✱
♥❤ú♥❣ ♥❣÷í✐ ✤➣ ❧✉æ♥ ✤ë♥❣ ✈✐➯♥✱ ❝❤✐❛ s➫ ✈➔ ❦❤➼❝❤ ❧➺ ✤➸ t→❝ ❣✐↔ ❝â t❤➸ ❤♦➔♥
t❤➔♥❤ ❝æ♥❣ ✈✐➺❝ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ♠➻♥❤✱ ♥✐➲♠ ✈✐♥❤ ❤↕♥❤ t♦ ❧ỵ♥
♥➔②✳

❚→❝ ❣✐↔


▼ư❝ ❧ư❝

❚r❛♥❣ ❜➻❛ ♣❤ư
✐
▲í✐ ❝❛♠ ✤♦❛♥
✐✐
▲í✐ ❝↔♠ ì♥
✐✐✐
▼ư❝ ❧ư❝
✐✈
▼ët sè ỵ t tt

ữỡ ởt số ♥✐➺♠ ❜➔✐ t♦→♥ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❝ì ❜↔♥ ✽
✶✳✶✳ ▼ët sè ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✷✳ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ t♦→♥ tû ✤ì♥ ✤✐➺✉ ✳ ✶✹
✶✳✷✳✶✳ P❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ✈➔ ♠ët sè ❝↔✐ ❜✐➯♥ ✳ ✳ ✳ ✳ ✳ ✶✹
✶✳✷✳✷✳ P❤÷ì♥❣ ♣❤→♣ t→❝❤ t✐➳♥✲❧ị✐ ✈➔ ♠ët sè ❝↔✐ ❜✐➯♥ ✳ ✳ ✳ ✳ ✳ ✶✾
✶✳✸✳ ❇➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ♥❤✐➲✉ t➟♣ ✈➔ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐

✷✸

✶✳✸✳✶✳ P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✭❙❋P✮ ✳ ✳ ✷✹
✶✳✸✳✷✳ P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ♥❤✐➲✉ t➟♣
✭▼❙❙❋P✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✶✳✹✳ ▼ët sè ❜ê ✤➲ ❜ê trñ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

❈❤÷ì♥❣ ✷✳ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ t♦→♥ tû ✤ì♥
✤✐➺✉ ❝ü❝ ✤↕✐ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
✸✺
✷✳✶✳ P❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ✈ỵ✐ ❞➣② t❤❛♠ sè ❜➜t ❦ý ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✷✳✷✳

❱➼ ❞ö sè ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

❈❤÷ì♥❣ ✸✳ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ tê♥❣ ❤❛✐
t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt

✺✵


✈

✸✳✶✳ P❤÷ì♥❣ ♣❤→♣ ❞↕♥❣ t→❝❤ t✐➳♥ ❧ị✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵
✸✳✷✳ ❱➼ ❞ö sè ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽

❈❤÷ì♥❣ ✹✳ P❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❝❤♦ ❜➔✐ t♦→♥ ❝❤➜♣
♥❤➟♥ t→❝❤ ♥❤✐➲✉ t➟♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
✼✹
✹✳✶✳ P❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ✈➔ ♥❣❤✐➺♠ ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t ✳ ✳ ✳ ✼✹
✹✳✷✳ P❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❝❤♦ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤
♥❤✐➲✉ t➟♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻
✹✳✸✳

❱➼ ❞ö sè ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼

❑➳t ❧✉➟♥
✽✾
❉❛♥❤ ♠ư❝ ❝→❝ ❝ỉ♥❣ tr➻♥❤ ✤➣ ❝ỉ♥❣ ❜è ❧✐➯♥ q✉❛♥ ✤➳♥ ❧✉➟♥ →♥ ✾✵
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦
✾✶


ởt số ỵ t tt
t ủ số t❤ü❝

R
En

❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ n✲❝❤✐➲✉

H

❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt

2H

t➟♣ t➜t ❝↔ ❝→❝ t➟♣ ừ ổ H

x, y

t ổ ữợ ừ ✈➨❝ tì x ✈➔ y

∥x∥

❝❤✉➞♥ ❝õ❛ ✈➨❝ tì x

inf M

❝➟♥ ữợ ú ừ t ủ số M

sup M

tr ú ❝õ❛ t➟♣ ❤đ♣ sè M

max M
min M

sè ❧ỵ♥ ♥❤➜t tr♦♥❣ t➟♣ ❤ñ♣ sè M
sè ♥❤ä ♥❤➜t tr♦♥❣ t➟♣ ❤ñ♣ sè M

D(A)

♠✐➲♥ ①→❝ ✤à♥❤ ❝õ❛ t♦→♥ tû A

R(A)

♠✐➲♥ ❣✐→ trà ❝õ❛ t♦→♥ tû A

A−1

→♥❤ ①↕ ♥❣÷đ❝ ❝õ❛ t♦→♥ tû A

A∗

→♥❤ ①↕ ủ ừ t tỷ A

I

ỗ t

f (x)

ữợ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ f t↕✐ ✤✐➸♠ x

lim inf xn

❣✐ỵ✐ ữợ ừ số {xn }

lim sup xn

ợ tr➯♥ ❝õ❛ ❞➣② sè {xn }

n→∞

n→∞

xn → x

❞➣② {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ x

xn ⇀ x

❞➣② {xn } ❤ë✐ tö ②➳✉ ✈➲ x

F ix(T )

t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T

ZerA

t➟♣ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ t♦→♥ tû A

SF P

❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤

M SSF P

❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ♥❤✐➲✉ t➟♣


▼ð ✤➛✉

◆❤✐➲✉ ❜➔✐ t♦→♥ tr♦♥❣ ❦❤♦❛ ❤å❝ ❦ÿ t❤✉➟t ✭❜➔✐ t♦→♥ ❜✐➳♥ ♣❤➙♥✱ ❜➔✐ t♦→♥
❝ü❝ trà✱ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✱ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✱ ✳✳✳✮ ✈➔ tr♦♥❣
✤í✐ sè♥❣ ✭❜➔✐ t♦→♥ ❦➳ ❤♦↕❝❤ s↔♥ ①✉➜t✱ ❜➔✐ t♦→♥ ✈➟♥ t↔✐✱ ❜➔✐ t♦→♥ ❦❤➞✉ ♣❤➛♥
t❤ù❝ ➠♥✱ ✳✳✳✮ ✤➲✉ ❞➝♥ ✤➳♥ ❜➔✐ t♦→♥ tê♥❣ q✉→t ❧➔ t➻♠ ❝ü❝ t✐➸✉ ❝õ❛ ♠ët
♣❤✐➳♠ ❤➔♠ f tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❤♦➦❝ ✈æ ❤↕♥ ❝❤✐➲✉✳ ❈❤♦ ✤➳♥ ♥❛②✱
❝â ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ✤÷đ❝ ✤➲ ①✉➜t ✤➸ t➻♠ ❝ü❝ t✐➸✉ ❝õ❛ ♠ët ♣❤✐➳♠ ❤➔♠
♥❤÷✿ ♣❤÷ì♥❣ ♣❤→♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ✭♣❤÷ì♥❣ ♣❤→♣ ❣r❛❞✐❡♥t✮✱ ♣❤÷ì♥❣ ♣❤→♣
❣r❛❞✐❡♥t ❧✐➯♥ ❤đ♣✱ ♣❤÷ì♥❣ ♣❤→♣ ❉❛♥t③✐❣ ❝❤♦ ❜➔✐ t♦→♥ q✉② ❤♦↕❝❤ t✉②➳♥ t➼♥❤
✈➔ ❝→❝ ❝↔✐ ❜✐➯♥ ❝õ❛ ❝❤ó♥❣✳ ▼ët ♣❤÷ì♥❣ ♣❤→♣ ✤➦❝ ❜✐➺t q✉❛♥ trồ t
ỹ t ừ ỗ ✤➳♥ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ✤÷đ❝
✤➲ ①✉➜t ❜ð✐ ▼❛rt✐♥❡t ❬✶❪ ✈➔♦ ♥➠♠ ✶✾✼✵✳ ❱➻ ✤✐➸♠ ❝ü❝ t✐➸✉ ❝õ❛ ởt
ỗ ổ ừ ữợ ❝õ❛ ♣❤✐➳♠ ❤➔♠ ✤â✱ ♥➠♠ ✶✾✼✻✱
❘♦❝❦❛❢❡❧❧❛r ❬✷❪ ✤➣ ✤➲ ①✉➜t ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛
♠ët t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ T tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✱ tù❝ ❧➔✿
❚➻♠ ♣❤➛♥ tû p∗ ∈ H

s❛♦ ❝❤♦

0 ∈ T p∗ .

✭✵✳✶✮

❚→❝ ❣✐↔ ✤➣ ①➙② ❞ü♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣

xk+1 = Jk xk + ek

❤♦➦❝

xk+1 = Jk (xk + ek ), k ≥ 1,

✭✵✳✷✮

tr♦♥❣ ✤â Jk = (I + rk T )−1 ❧➔ t♦→♥ tû ❣✐↔✐ ❝õ❛ T ✈ỵ✐ t❤❛♠ sè rk > 0✱ ek ❧➔
✈➨❝ tì s❛✐ sè ✈➔ I ❧➔ →♥❤ ①↕ ✤ì♥ ✈à tr➯♥ H ✳ ➷♥❣ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣
♣❤÷ì♥❣ ♣❤→♣ ✭✵✳✷✮ ❤ë✐ tư ②➳✉ tỵ✐ ♠ët ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ T ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ t➟♣
❦❤æ♥❣ ✤✐➸♠ ❝õ❛ T ❦❤→❝ ré♥❣✱

∞

∥ek ∥ < ∞ ✈➔ rk ≥ ε > 0 ✈ỵ✐ ♠å✐ k ≥ 1✳

k=1

◆➠♠ u
ăr r r ữỡ ❣➛♥ ❦➲ ❝❤➾ ✤↕t ✤÷đ❝
sü ❤ë✐ tư ②➳✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈æ ❤↕♥ ❝❤✐➲✉✳ ◆➠♠ ✶✾✾✷✱ ❊❝❦st❡✐♥
✈➔ ❇❡rts❡❦❛s ❬✹❪ ✤➲ ①✉➜t ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ tê♥❣ q✉→t ❧➔ ♠ð rë♥❣
❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❝❤♦ ❜➔✐ t♦→♥ ✭✵✳✶✮✳ ❚✉② ♥❤✐➯♥✱ ❝→❝ t→❝ ❣✐↔
❝ơ♥❣ ❝❤➾ t❤✉ ✤÷đ❝ sü ❤ë✐ tư ②➳✉ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣✳ ✣➸ t❤✉ ✤÷đ❝ sü ❤ë✐ tö


✷

♠↕♥❤✱ ♠ët sè ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ✤➣ ✤÷đ❝ ✤÷❛ r❛ ♥❤÷✿

♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❤✐➺✉ ❝❤➾♥❤ ❚✐❦❤♦♥♦✈ ❝õ❛ ▲❡❤❞✐❤✐ ✈➔ ▼♦✉❞❛❢✐
✭✶✾✾✻✮ ❬✺❪ ✈➔ ✤÷đ❝ ♠ð rë♥❣ ❜ð✐ ❳✉ ✭✷✵✵✻✮ ❬✻❪✱ ❇♦✐❦❛♥②♦ ✈➔ ▼♦r♦s❛♥✉ ✭✷✵✶✷✮
❬✼❪❀ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❝♦ ❝õ❛ ❑❛♠✐♠✉r❛ ✈➔ ❲✳❚❛❦❛❤❛s❤✐ ✭✷✵✵✵✮ ❬✽❪
✈➔ ✤÷đ❝ tê♥❣ q✉→t ❜ð✐ ❨❛♦ ✈➔ ◆♦♦r ✭✷✵✵✽✮ ❬✾❪❀ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ♠➲♠
❝õ❛ ❲✳❚❛❦❛❤❛s❤✐ ✭✷✵✵✼✮ ❬✶✵❪✳ ❚r♦♥❣ ❤➛✉ ❤➳t ❝→❝ ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣
✤✐➸♠ ❣➛♥ ❦➲ ❝ơ♥❣ ♥❤÷ ❜↔♥ t❤➙♥ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ t❤❛♠ số rk ừ
t tỷ ữợ ♠ët ❤➡♥❣ sè ❧ỵ♥ ❤ì♥ ✵✳ ●➛♥ ✤➙②✱ ♥➠♠
✷✵✶✼✱ tr♦♥❣ ❬✶✶❪✱ ◆✳ ❇÷í♥❣✱ P✳❚✳❚✳ ❍♦➔✐ ✈➔ ◆✳❉✳ ◆❣✉②➵♥ ✤➣ tr➻♥❤ ởt
số ợ ừ ữỡ trữớ ủ rk tợ
ử t rk t❤♦↔ ♠➣♥

∞

rk < +∞✳ ▼ët ❝➙✉ ❤ä✐ ✤÷đ❝ ✤➦t r

k=1

ự õ tỗ t ởt ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❤ë✐ tư ♠➔
sü ở tử t ữủ ợ {rk } ♠ët ❞➣② sè ❜➜t ❦ý tr♦♥❣ (0, ∞)
❦❤æ♥❣❄
❑❤✐ ♣❤✐➳♠ ❤➔♠ ỹ t tờ ừ ỗ t♦→♥ ♥➔②
❞➝♥ ✤➳♥ ❜➔✐ t♦→♥ t➻♠ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ tê♥❣ ❤❛✐ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐

A, B, ✤â ❧➔ ❜➔✐ t♦→♥✿
❚➻♠ ♣❤➛♥ tû p∗ ∈ H

s❛♦ ❝❤♦

0 ∈ (A + B)p∗ .

✭✵✳✸✮

❇➔✐ t♦→♥ ✭✵✳✸✮ t❤✉ ❤ót ✤÷đ❝ sü ❝❤ó þ ❝õ❛ ♥❤✐➲✉ ♥❤➔ ♥❣❤✐➯♥ ❝ù✉ ✈➻ ♥â
❧➔ ❝èt ❧ã✐ ❝õ❛ ♥❤✐➲✉ ❜➔✐ t♦→♥ ♥❤÷✿ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✱ ❜➔✐ t♦→♥ ❝❤➜♣
♥❤➟♥ t→❝❤✱ ❜➔✐ t♦→♥ ❝ü❝ t✐➸✉ ❤â❛ ✭①❡♠ ❬✶✷✱ ✶✸✱ ✶✹❪✮ ✈ỵ✐ ❝→❝ ù♥❣ ❞ư♥❣ tr♦♥❣
❤å❝ ♠→②✱ ①û ỵ t ữủ t t t q trồ
tr ỵ tt t ồ ụ ữ tr ự ❞ư♥❣ t❤ü❝ t➳ ♥➯♥ ❝→❝ ♣❤÷ì♥❣
♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ ✭✵✳✸✮ ữủ t tr ữợ q
t ❝ù✉✱ ✤✐➸♥ ❤➻♥❤ ❧➔ P❡❛❝❡♠❛♥✲❘❛❝❤❢♦r❞ ✭✶✾✺✺✮ ❬✶✺❪✱ ❉♦✉❣❧❛s✲
❘❛❝❤❢♦r❞ ✭✶✾✺✻✮ ❬✶✻❪✱ ▲✐♦♥s ✈➔ ▼❡r❝✐❡r ✭✶✾✼✾✮ ❬✶✼❪✱ P❛sst② ✭✶✾✼✾✮ ❬✶✽❪✱
❈♦♠❜❡tt❡s ✭✷✵✵✹✮ ❬✶✾❪✱ ❚❛❦❛❤❛s❤✐✱ ❲♦♥❣ ✈➔ ❨❛♦ ✭✷✵✶✵✮ ❬✷✵❪✱ ❚s❡♥❣ ✭✷✵✵✵✮
❬✷✶❪✱ ▼❛❧✐ts❦② ✭✷✵✶✽✮ ❬✷✷❪✱ ❙❡♠❡♥♦✈ ✭✷✵✶✽✮ ❬✷✸❪✱✳✳✳ Ð ❱✐➺t ◆❛♠✱ tr♦♥❣ ♠ët
sè ♥➠♠ trð ❧↕✐ ✤➙②✱ ❜➔✐ t♦→♥ ✭✵✳✸✮ ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ ♥❣❤✐➯♥ ❝ù✉ t♦→♥ ❣✐↔✐ t➼❝❤
✈➔ t♦→♥ ù♥❣ ❞ö♥❣ t➻♠ ❤✐➸✉ ✈➔ ợ t ởt số t tr ữợ õ
ổ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❜➔✐ t♦→♥ ♥➔② ❝â t❤➸ ❦➸ ✤➳♥ ♥❤÷✿ ✣✳❱✳ ❚❤ỉ♥❣ ✈➔
●✐❜❛❧✐ ✭✷✵✶✽✮ ❬✷✹❪✱ ✣✳❱✳ ❚❤ỉ♥❣ ✈➔ ❈❤♦❧❛♠❥✐❛❦ ✭✷✵✶✾✮ ❬✷✺❪✱ ✣✳❱✳ ❚❤æ♥❣ ✈➔
◆✳❚✳ ❱✐♥❤ ✭✷✵✶✾✮ ❬✷✻❪✱ ▲✳❉✳ ▼÷✉✱ P✳❑✳ ❆♥❤✱ ✣✳❱✳ ❍✐➺✉ ✭✷✵✷✵✮ ❬✷✼❪✱✳✳✳


✸

❚❛ ❜✐➳t r➡♥❣✱ ♥➳✉ tê♥❣ ❆✰❇ ❝ô♥❣ ❧➔ ♠ët t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✱ t❤➻
❝â t❤➸ →♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✵✳✷✮ ✈ỵ✐ ❚❂❆✰❇ ✤➸ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛
tê♥❣✳ ❚✉② ♥❤✐➯♥✱ ♥❤✐➲✉ ❦❤✐ ❚ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ❝❤♦ ❞ị ❆ ✈➔
❇ ❧➔ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳ ❉♦ ✤â✱ ❝❤➾ ❝â t❤➸ ①➙② ❞ü♥❣ ♠ët ♣❤➨♣ ❧➦♣ ❞ü❛ ✈➔♦
t♦→♥ tû ❣✐↔✐ ❝õ❛ tø♥❣ t♦→♥ tû ❆ ✈➔ ❇✳ ✣✐➲✉ ♥➔② ❝ơ♥❣ ❧đ✐ t❤➳✱ ♥❣❛② ❝↔ ❦❤✐
❚ ❧➔ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✱ ♥❤÷♥❣ ✈✐➺❝ t➼♥❤ ❣✐→ trà ❝õ❛ t♦→♥ tû ❣✐↔✐ ❝õ❛ ❚ ❦❤â
❤ì♥ ✈✐➺❝ t➼♥❤ ♥â ❝❤♦ tø♥❣ ❆ ✈➔ ❇✳ ❇ð✐ ✈➟②✱ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ ❝❤♦ ❣✐↔✐ ❜➔✐
t♦→♥ ✭✵✳✸✮ ❝❤➼♥❤ ❧➔ sû ❞ö♥❣ t♦→♥ tû ❣✐↔✐ JrA , JrB ❝õ❛ A ✈➔ B t❤❛② ❝❤♦ ❞ò♥❣
t♦→♥ tû ❣✐↔✐ JrA+B ❝õ❛ A + B ✳

P❤÷ì♥❣ ♣❤→♣ t→❝❤ ❝ê ✤✐➸♥ ❝õ❛ P❡❛❝❡♠❛♥✲❘❛❝❤❢♦r❞ ❬✶✺❪✱ ❉♦✉❣❧❛s✲❘❛❝❤❢♦r❞
❬✶✻❪ ✤÷đ❝ ✤➲ ①✉➜t ✈➔♦ ♥❤ú♥❣ ♥➠♠ ✶✾✺✵ ❝❤♦ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❦❤✐ ❝↔ A
✈➔ B ✤➲✉ ❧➔ t♦→♥ tû t✉②➳♥ t➼♥❤ ✤ì♥ trà✳ ◆➠♠ ✶✾✼✾✱ tr♦♥❣ ❬✶✼❪✱ ▲✐♦♥s ✈➔
▼❡r❝✐❡r ✤➣ ♠ð rë♥❣ sì ỗ t sr trữớ ủ
ợ A B ❧➔ ❝→❝ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ✤❛ trà✳ ▼ët ♣❤÷ì♥❣ ♣❤→♣ t→❝❤
t❤ỉ♥❣ ❞ư♥❣ ❦❤→❝ ✤÷đ❝ ✤÷❛ r❛ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ✭✵✳✸✮ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤
t✐➳♥✲❧ị✐✳ P❤÷ì♥❣ ♣❤→♣ ♥➔② ✤÷đ❝ ✤➲ ①✉➜t ❜ð✐ ▲✐♦♥s ✈➔ ▼❡r❝✐❡r ❬✶✼❪✱ P❛sst②
❬✶✽❪ ợ xk ữủ ❜ð✐✿

xk+1 = Jk (I − rk A)xk , k ≥ 1,

✭✵✳✹✮

tr♦♥❣ ✤â A, B ❧➔ ❝→❝ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr➯♥ H ✱ Jk = (I + rk B)−1
❧➔ t♦→♥ tû ❣✐↔✐ ❝õ❛ B ✱ {rk } ❧➔ ❞➣② sè ❞÷ì♥❣✳ ❚✉② ♥❤✐➯♥✱ ❞➣② ❧➦♣ xk ①→❝
✤✐♥❤ ❜ð✐ ✭✵✳✹✮ ❝ơ♥❣ ❝❤➾ ❤ë✐ tư ②➳✉ tỵ✐ ♠ët ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ A + B ✳ ✣➸
t❤✉ ✤÷đ❝ sü ❤ë✐ tư ♠↕♥❤✱ ♠ët sè ❝↔✐ t✐➳♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ t✐➳♥✲❧ị✐
✤➣ ữủ ữ r ợ ữủ ỹ t ❤đ♣ ✈ỵ✐ ❝õ❛ ▼❛♥♥ ✭①❡♠
❬✶✾✱ ✷✽❪✮✱ ❍❛❧♣❡r♥ ✭①❡♠ ❬✷✾❪✮✱ ▼❛♥♥✲❍❛❧♣❡r♥ ✭①❡♠ ❬✷✵✱ ✸✵❪✮✳ ◆➠♠ ✷✵✵✵✱
❚s❡♥❣ ❬✷✶❪ ✤➣ ✤➲ ①✉➜t ♠ët ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ t✐➳♥✲❧ị✐ ❝❤➾ ❝➛♥
✤✐➲✉ ❦✐➺♥ A ❧➔ ✤ì♥ ✤✐➺✉ ✈➔ ❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③ tr➯♥ t➟♣ ỗ õ ừ
tr ừ õ ♠ët sè t→❝ ❣✐↔ ❝ơ♥❣ ♥❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ ♣❤→♣
♥➔② ♠➔ ❦❤ỉ♥❣ ❝➛♥ ❣✐↔ t❤✐➳t t♦→♥ tû A ❧➔ α✲ ♥❣÷đ❝ ✤ì♥ ✤✐➺✉ ♠↕♥❤ ✭①❡♠
❬✷✷✱ ✷✹✱ ✷✺✱ ✷✻✱ ✷✼❪✮✳ ❙ü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ t❤✉ ✤÷đ❝ ✤➲✉
❝➛♥ ✤✐➲✉ ❦✐➺♥ t❤❛♠ sè rk ❝õ❛ t♦→♥ tû ❣✐↔✐ ♣❤↔✐ ❜à ữợ ởt
số ợ ỡ ởt ✤➲ ♥↔② s✐♥❤ tø ✤➙② ❧➔ ❧✐➺✉ ❝â t❤➸ ①➙② ❞ü♥❣ ✤÷đ❝
♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ ✭✵✳✸✮ ♠➔ sü ❤ë✐ tử t ữủ ợ

rk tợ ❤♦➦❝ ✤✐➲✉ ❦✐➺♥ tê♥❣ q✉→t ❤ì♥ ❝❤♦ ❞➣② t❤❛♠ sè ❝õ❛ t♦→♥ tû

✹

❣✐↔✐ ✤â ❧➔ {rk } ❧➔ ♠ët ❞➣② sè ❜➜t ❦ý tr♦♥❣ (0, α) ✈ỵ✐ α > 0 ❦❤ỉ♥❣❄ ✣➸ tr↔
❧í✐ ❝❤♦ ❝➙✉ ❤ä✐ ♥➔②✱ tr♦♥❣ ❈❤÷ì♥❣ ✸ ❝õ❛ ❧✉➟♥ →♥✱ ❝❤ó♥❣ tỉ✐ ✤➲ ①✉➜t ♣❤÷ì♥❣
♣❤→♣ ❞↕♥❣ t→❝❤ t✐➳♥✲❧ị✐ ❝❤♦ t sỹ ở tử t ữủ
ợ ✤✐➲✉ ❦✐➺♥ ♥❤÷ ✤➣ ♥➯✉ tr➯♥ ❝❤♦ ❞➣② t❤❛♠ sè {rk } ❝õ❛ t♦→♥ tû ❣✐↔✐✳
❑❤✐ ❤➔♠ f ❧➔ tê♥❣ ừ ỗ ởt ọ t t ✤÷đ❝
✤➦t r❛ ❧➔ ❧✐➺✉ ❝â t❤➸ ♠ð rë♥❣ ❝→❝ ❦➳t q✉↔ tr➯♥ ❝❤♦ tr÷í♥❣ ❤đ♣ ♥➔② ✤÷đ❝
❦❤ỉ♥❣❄ ❈➙✉ ❤ä✐ ♥➔② ❝➛♥ ♥❤✐➲✉ ❝æ♥❣ sù❝ ♥❣❤✐➯♥ ❝ù✉ t✐➳♣ t❤❡♦✳ Ð ✤➙② ú
tổ ợ t ởt trữớ ủ ử t ổ t↔ ❜➔✐ t♦→♥ tr♦♥❣ t❤ü❝ t➳✳ ✣â ❧➔
❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ♥❤✐➲✉ t➟♣ ✭▼❙❙❋P✮✿
❚➻♠ x ∈ C :=

Ci , s❛♦ ❝❤♦ Ax ∈ Q :=
i∈J1

Qj ,

✭✵✳✺✮

j∈J2

tr♦♥❣ ✤â {Ci }i∈J1 ✈➔ {Qj }j∈J2 t÷ì♥❣ ù♥❣ ❧➔ ❤❛✐ ❤å ❝→❝ t➟♣ ỗ õ
tr ổ rt tỹ H1 H2 ✱ J1 , J2 ❧➔ ❝→❝ t➟♣ ❝❤➾ sè ❜➜t ❦➻✱

A : H1 → H2 ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥✳
❚r♦♥❣ tr÷í♥❣ ❤đ♣ ✤ì♥ ❣✐↔♥ J1 = J2 = {1} t❤➻ ❜➔✐ t♦→♥ ✭✵✳✺✮ trð t❤➔♥❤
❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✭❙❋P✮✿
❚➻♠ x ∈ C s❛♦ ❝❤♦ Ax ∈ Q,

✭✵✳✻✮

ð õ C, Q tữỡ ự t ỗ ✤â♥❣ tr♦♥❣ H1 ✈➔ H2 .
❇➔✐ t♦→♥ ✭✵✳✻✮ ✤÷đ❝ ✤÷❛ r❛ ✤➛✉ t✐➯♥ ❜ð✐ ❈❡♥s♦r ✈➔ ❊❧❢✈✐♥❣ ❬✸✶❪ ✈➔♦ ♥➠♠
✶✾✾✹ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✤➸ ♠æ ❤➻♥❤ ❤â❛ ❜➔✐ t♦→♥
♥❣÷đ❝✱ ❜➔✐ t♦→♥ ♥↔② s✐♥❤ tø ✈✐➺❝ ♣❤ư❝ ỗ tr t t
t ✤➙②✱ ♥❣÷í✐ t❛ ❝á♥ ♣❤→t ❤✐➺♥ r❛ r➡♥❣ ❙❋P ❝ơ♥❣ ❝â t❤➸ ✤÷đ❝ →♣
❞ư♥❣ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ✤✐➲✉ ❝❤➳ ❝÷í♥❣ ✤ë ①↕ trà✳ ❚❤✉➟t t♦→♥ ✤÷đ❝ ❝→❝ t→❝ ❣✐↔
✤➲ ①✉➜t ❝❤♦ ❙❋P ❧✐➯♥ q✉❛♥ ✤➳♥ ✈✐➺❝ t➼♥❤ t♦→♥ ♠❛ tr➟♥ ừ
tr A tr ộ ữợ ♥➔② ❞➝♥ ✤➳♥ ✈✐➺❝ t➼♥❤ t♦→♥ trð ♥➯♥ ❦❤â
❦❤➠♥✱ ✤➦❝ t tr õ tữợ ợ ❦❤➢❝ ♣❤ư❝ ✈➜♥ ✤➲
♥➔②✱ ♥➠♠ ✷✵✵✷✱ ❇②r♥❡ ❬✸✷❪ ✤➣ ✤÷❛ r ởt ữỡ ợ ữủ ồ
tt t ❈◗ ✈ỵ✐ ❞➣② ❧➦♣ {xk } ❝â ❞↕♥❣✿

xk+1 = PC (I − γAt (I − PQ )A)xk , k ≥ 1,



C Q tữỡ ự t ỗ ✤â♥❣ ✈➔ ❦❤æ♥❣ ré♥❣ tr♦♥❣ Rn , Rm ✱ A
❧➔ ♠❛ tr➟♥ t❤ü❝ ❝ï ♠①♥✱ I ❧➔ ♠❛ tr➟♥ ✤ì♥ ✈à✱ At ❧➔ ♠❛ tr➟♥ ❝❤✉②➸♥ ✈à ❝õ❛

A✳ ❚❤✉➟t t♦→♥ ❈◗ ❝❤➾ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝ ♣❤➨♣ ❝❤✐➳✉ trü❝ ❣✐❛♦ PC , PQ ❧➯♥
C ✈➔ Q ♥➯♥ ❝â t❤➸ t❤ü❝ ❤✐➺♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ PC , PQ ❞➵ ❞➔♥❣ t➼♥❤ t♦→♥✳


✺

◆➠♠ ✷✵✵✻✱ ❳✉ ❬✸✸❪ ①➨t ❙❋P tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ổ
r t tữỡ ữỡ ợ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣✳ ❉♦ ✤â✱ t→❝ ❣✐↔

→♣ ❞ö♥❣ t❤✉➟t t♦→♥ ❑r❛s♥♦s❡❧✬s❦✐✐✲▼❛♥♥ ✭❑▼✮ ❝❤♦ ❙❋P ✭✵✳✻✮✳ ❚r♦♥❣ ❬✸✹❪✱
♥➠♠ ✷✵✶✵✱ ❳✉ ❝❤➾ r❛ r➡♥❣ t❤✉➟t t♦→♥ ❈◗ ✈➔ ❑▼ ❝❤➾ ❤ë✐ tư ②➳✉ tr♦♥❣ ❦❤ỉ♥❣
❣✐❛♥ ❍✐❧❜❡rt ✈ỉ ❤↕♥ ❝❤✐➲✉✳ ✣➸ t❤✉ ✤÷đ❝ sü ❤ë✐ tư ♠↕♥❤✱ t→❝ ❣✐↔ ✤➲ ①✉➜t
♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❝õ❛ ❇❛❦✉s❤✐♥s❦② ❬✸✺❪ ✈➔ ❇r✉❝❦ ❬✸✻❪ ❝â ❞↕♥❣✿

xk+1 = PC [I − γk (A∗ (I − PQ )A + αk I)]xk , k ≥ 1.

✭✵✳✽✮

❈→❝ t→❝ ❣✐↔ ❨❛♦ ✈➔ ❝ë♥❣ sü ❬✸✼❪✱ ❈❤✉❛♥❣ ❬✸✽❪ ❝ô♥❣ ❝❤ù♥❣ ♠✐♥❤ sỹ ở
tử ừ ữỡ ợ ✤➦t ❧➯♥ ❝→❝ ❞➣② t❤❛♠ sè ②➳✉
❤ì♥✳ ▼ët ♣❤÷ì♥❣ ♣❤→♣ ♥ú❛ ✤÷đ❝ ✤÷❛ r❛ ❝❤♦ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✤â
❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ✭①❡♠ ❏✉♥❣ ✭✷✵✶✻✮ ❬✸✾❪✱ ◆✳ ❇÷í♥❣
✭✷✵✶✼✮ ❬✹✵❪✮✳ ✣➙② ❧➔ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ữủ t ỹ ỵ tữ
ừ ữỡ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✳
❚r♦♥❣ tr÷í♥❣ ❤đ♣ sè t➟♣ J1 , J2 ❤ú✉ ❤↕♥✱ tù❝ ❧➔ J1 = {1, 2, ..., N }; J2 =

{1, 2, ..., M } ✈ỵ✐ N, M > 1✱ ♥➠♠ ✷✵✵✺✱ ❈❡♥s♦r ❝ị♥❣ ❝ë♥❣ sü ❬✹✶❪ ①➨t ▼❙❙❋P
✭✵✳✺✮ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✈➔ ✤➲ ①✉➜t ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉
❣r❛❞✐❡♥t ❝❤♦ ❜➔✐ t♦→♥✳ ❚r♦♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝õ❛ ❈❡♥s♦r ❝ị♥❣ ❝→❝ ❝ë♥❣ sü
t❤❛♠ sè ❧➦♣ ❧➔ ❝è ✤à♥❤ ✈➔ ♣❤ö t❤✉ë❝ ❤➡♥❣ sè ▲✐♣s❝❤✐t③✱ ❤➡♥❣ sè ♥➔② ♣❤ö
t❤✉ë❝ ✈➔♦ ❝❤✉➞♥ ❝õ❛ A ♥➯♥ ✈✐➺❝ t➼♥❤ t♦→♥ ❧➔ ❦❤ỉ♥❣ ✤ì♥ ❣✐↔♥✳ ✣➸ ❦❤➢❝ ♣❤ư❝
✈➜♥ ✤➲ ♥➔②✱ ♥➠♠ ✷✵✵✽✱ ❩❤❛♥❣ ❬✹✷❪ ✤➲ ①✉➜t ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ tü t❤➼❝❤ ù♥❣
❝❤♦ ▼❙❙❋P✱ ð ✤â t❤❛♠ sè ❧➦♣ ❝è ữủ t t số t ờ
ộ ữợ ❧➦♣✳ ▼ët sè ❝↔✐ ❜✐➯♥ ✈➔ ♠ð rë♥❣ ❝õ❛ ♣❤÷ì♥❣ ữủ ợ
t ✹✹✱ ✹✺❪✮ ✈➔ ❍❡ ❝ò♥❣ ❝ë♥❣ sü ✭❬✹✻❪✮✳ ◆➠♠
✷✵✵✻✱ ❳✉ ❬✸✸❪ ✤÷❛ r❛ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤♦ ▼❙❙❋P✳ r
r P tữỡ ữỡ ợ t♦→♥ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣
❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ →♥❤ ①↕ tr✉♥❣ ❜➻♥❤✳ P❤÷ì♥❣ ♣❤→♣ ♥➔② ❝ơ♥❣ ✤÷đ❝ ❝↔✐
t✐➳♥ ✈➔ ♠ð rë♥❣ ❜ð✐ ❲❡♥ ✭✷✵✶✺✮ ❬✹✼❪✱ ◆✳ ❇÷í♥❣ ✭✷✵✶✼✮ ❬✹✵❪✳

❚r♦♥❣ tr÷í♥❣ ❤đ♣ J1 ✈➔ J2 ❧➔ ❝→❝ ❤å ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝✱ tù❝ ❧➔ J1 = J2 =

N+ ✱ ❧➔ t➟♣ t➜t ❝↔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✱ ◆✳ ❇÷í♥❣ ữ r tt
t ở tử tợ ừ P ỗ tớ ụ ừ
t tự ❜✐➳♥ ♣❤➙♥✳ ❚→❝ ❣✐↔ ❝❤➾ r❛ r➡♥❣ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛ t❤✉➟t
t♦→♥ ❧➔ ❝↔✐ t✐➳♥ ❝õ❛ t❤✉➟t t♦→♥ ✤÷đ❝ ✤➲ ①✉➜t ❜ð✐ ❳✉ ❬✸✸❪ ♥➠♠ ✷✵✵✻✳ ◆❤÷
✈➟② ❝❤♦ t❤➜②✱ ❝❤÷❛ ❝â ♥❤✐➲✉ t❤✉➟t t♦→♥ ✤÷đ❝ ✤➲ ①✉➜t ❝❤♦ ▼❙❙❋P ✭✵✳✺✮


✻

tr♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ ❝→❝ t➟♣ ❝❤➾ sè J1 , J2 ❧➔ ✈æ ❤↕♥ ❤♦➦❝ ♠ët tr♦♥❣ ❤❛✐
t➟♣ J1 , J2 ❤ú✉ ❤↕♥✱ t➟♣ ❝á♥ ❧↕✐ ✈ỉ ❤↕♥✳ ❚r♦♥❣ ❈❤÷ì♥❣ ✹ ❝õ❛ ❧✉➟♥ →♥ ❧➔ ♠ët
sè ❦➳t q✉↔ ❝❤ó♥❣ tỉ✐ t ữủ ợ trữớ ủ tr
õ t ❦❤➥♥❣ ✤à♥❤ r➡♥❣✱ ✈✐➺❝ ①➙② ❞ü♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥
t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ✈➔ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤
♥❤✐➲✉ t➟♣ ❧➔ r➜t ❝➛♥ t❤✐➳t ✤➸ ❧➔♠ ♣❤♦♥❣ ♣❤ó ✈➔ ❤♦➔♥ t❤✐➺♥ t❤➯♠ ỵ
tt t q trồ ỳ ỵ ữủ t
tr ú tổ ồ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ❝❤♦ ❧✉➟♥ →♥ ❧➔ ✧▼ët sè ♣❤÷ì♥❣
♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ✈➔ ❜➔✐
t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ♥❤✐➲✉ t➟♣✧✳
▲✉➟♥ →♥ t➟♣ tr✉♥❣ ♥❣❤✐➯♥ ❝ù✉ ✤➲ ①✉➜t ♠ët sè t❤✉➟t t♦→♥ ❧➦♣ ♠ỵ✐ ❝❤♦
❜➔✐ t♦→♥ ✭✵✳✶✮ ✈➔ ✭✵✳✸✮ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❞➣② t❤❛♠ sè ❝õ❛ t♦→♥ tû ❣✐↔✐ ❞➛♥ tỵ✐ ✵
❤♦➦❝ ❧➔ ❞➣② sè ❜➜t ❦ý tr♦♥❣ ♠ët ❦❤♦↔♥❣ ♥➔♦ ✤â❀ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤
❧➦♣ ❝❤♦ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ♥❤✐➲✉ t➟♣ tr♦♥❣ tr÷í♥❣ ❤đ♣ J1 ✈➔ J2 ❧➔
❝→❝ ❤å ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ →♥✱ ♥❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t
❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ✹ ❝❤÷ì♥❣✿
❚r♦♥❣ ❈❤÷ì♥❣ ✶✱ ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à q✉❛♥
trå♥❣ ❝❤♦ ✈✐➺❝ tr➻♥❤ ❜➔② ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ ð ❝❤÷ì♥❣ s ỗ ởt số
ỡ ừ t ỗ ởt số ữỡ t ổ ừ

t tỷ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✱ tê♥❣ ❤❛✐ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✱ ❜➔✐ t♦→♥ ❝❤➜♣
♥❤➟♥ t→❝❤ ♥❤✐➲✉ t➟♣ ✈➔ ❝→❝ ữỡ
r ữỡ ợ t ởt ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲
✤➸ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱
sü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❦❤æ♥❣ ❝➛♥ t❤➯♠
✤✐➲✉ ❦✐➺♥ ♥➔♦ ❦❤→❝ ❧➯♥ t❤❛♠ sè ❝õ❛ t♦→♥ tû ❣✐↔✐ ❝õ❛ t♦→♥ tû ✤➣ ❝❤♦✳
❚r♦♥❣ ❈❤÷ì♥❣ ✸✱ ❝❤ó♥❣ tỉ✐ ✤➣ ♥❤➟♥ ✤÷đ❝ ♠ët ❦➳t q✉↔ t÷ì♥❣ tü ❝❤♦ ❜➔✐
t♦→♥ ❜❛♦ ❤➔♠ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✤ì♥ ✤✐➺✉✳
❚r♦♥❣ ❈❤÷ì♥❣ ✹✱ ✤÷đ❝ ❞➔♥❤ ✤➸ ✤➲ ①✉➜t ♠ët ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤
❧➦♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✈ỵ✐ ❤❛✐ ❤å ổ t õ ỗ
q trồ ừ ữỡ ộ ữợ ũ ỳ ❤↕♥ ❝→❝
t➟♣ ❝õ❛ ❤❛✐ ❤å tr➯♥✳
✣➸ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❣✐↔✐ q✉②➳t ❝→❝ ♠ư❝ t✐➯✉ ✤➦t r❛✱ ❝❤ó♥❣ tỉ✐ ✤➣ sû ❞ư♥❣
❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✈➔ ❝ỉ♥❣ ❝ư ❤✐➺♥ ✤↕✐ ❝õ❛ ❣✐↔✐ t t ỗ ỵ
tt tố ữ ❦➳t q✉↔ ✤➣ ❝â ✈➲ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♥❤ú♥❣ ❜➔✐ t♦→♥


✼

♥➯✉ tr➯♥✳
◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ →♥ ✤➣ ✤÷đ❝ ❜→♦ ❝→♦ t

ã ở t ố ữ t ồ ❧➛♥ t❤ù ✶✺✱ ❇❛ ❱➻✱ ❍➔ ◆ë✐✱
✷✵✲✷✷✴✹✴✷✵✶✼✳

• ❍ë✐ t❤↔♦ ❚è✐ ÷✉ ✈➔ ❚➼♥❤ t♦→♥ ❦❤♦❛ ❤å❝ ❧➛♥ t❤ù ✶✼✱ ❇❛ ❱➻✱ ❍➔ ◆ë✐✱
✶✽✲✷✵✴✹✴✷✵✶✾✳
• ❍ë✐ t❤↔♦ ◗✉è❝ ❣✐❛ ❧➛♥ t❤ù ❳❳■■■ ✈➲ ♠ët sè ✈➜♥ ✤➲ ❝❤å♥ ❧å❝ ❝õ❛ ❈æ♥❣
♥❣❤➺ ❚❤æ♥❣ t✐♥ ✈➔ ❚r✉②➲♥ t❤ỉ♥❣✱ ◗✉↔♥❣ ◆✐♥❤✱ ✺✲✻✴✶✶✴✷✵✷✵✳
• ❙❡♠✐♥❛r ❤➔♥❣ t✉➛♥ ð ♥❤â♠ ❚♦→♥ ù♥❣ ❞ư♥❣ ❝õ❛ ❱✐➺♥ ❈ỉ♥❣ ♥❣❤➺ t❤ỉ♥❣

t✐♥✱ ❱✐➺♥ ❍➔♥ ❧➙♠ ❑❤♦❛ ❤å❝ ✈➔ ❈æ♥❣ ♥❣❤➺ ❱✐➺t ◆❛♠✳


❈❤÷ì♥❣ ✶

▼ët sè ❦❤→✐ ♥✐➺♠ ❜➔✐ t♦→♥ ✈➔
♣❤÷ì♥❣ ♣❤→♣ ❝ì ❜↔♥
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ ✤➲ ❝➟♣ ✤➳♥ ♥❤ú♥❣ ✈➜♥ s ử ợ
t sỡ ữủ ỵ tt ừ ❣✐↔✐ t➼❝❤ ❤➔♠✳ ▼ư❝ ✶✳✷ tr➻♥❤ ❜➔② tê♥❣ q✉❛♥ ❝→❝
♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ t➻♠ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳
▼ư❝ ✶✳✸ ❧➔ ✈➲ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ♥❤✐➲✉ t➟♣ ✈➔ ❝→❝ ❝→❝❤ ❣✐↔✐✳ ▼ư❝
❝✉è✐ ❝ị♥❣ ❝õ❛ ❝❤÷ì♥❣ ❧➔ ♠ët sè ❜ê ✤➲ ❜ê trđ ❝❤♦ ❝→❝ ❝❤÷ì♥❣ s❛✉ ❝õ❛ ❧✉➟♥
→♥✳

✶✳✶✳ ▼ët sè ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥
✣à♥❤ ♥❣❤➽❛ ✶✳✶ ❉➣② xk ⊂ H ✤÷đ❝ ❣å✐ ❧➔ ở tử tợ tỷ

x H ỵ ❤✐➺✉ xk → x✱ ♥➳✉ ∥xk − x∥ → 0 ❦❤✐ k → ∞✳
❉➣② xk ⊂ H ✤÷đ❝ ❣å✐ ❧➔ ❤ë✐ tư ②➳✉ tỵ✐ ♣❤➛♥ tû x ∈ H ✱ ỵ xk x
xk , y x, y⟩ ❦❤✐ n → ∞ ✈ỵ✐ ♠å✐ y ∈ H ✳

◆❤➟♥ ①➨t ✶✳✶ • ❍ë✐ tư ♠↕♥❤ ❦➨♦ t❤❡♦ ❤ë✐ tử ữ ữủ

ổ ú

ã xk ⊂ H t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ∥xk ∥ → ∥x∥ ✈➔ xk ⇀ x t❤➻
xk → x ❦❤✐ k → ∞✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷ ❈❤♦ C ❧➔ ♠ët t➟♣ ❝♦♥ ừ H C ữủ ồ t ỗ

ợ ∀x, y ∈ C ✈➔ ♠å✐ sè t❤ü❝ λ ∈ [0, 1] t❛ ❝â (1 − λ)x + λy ∈ C
r ỏ õ ởt số t s

ã ợ ♠é✐ a ∈ R, z ∈ H ✈➔ z ̸= 0✱ t➟♣ Wa = {x ∈ H : ⟨z, x⟩ = a} ữủ ồ
s tr H

ã ợ ♠é✐ a ∈ R, z ∈ H ✈➔ z ̸= 0✱ ❝→❝ t➟♣ {x ∈ H : ⟨z, x⟩ ≤ a} ✈➔
{x ∈ H : ⟨z, x⟩ ≥ a} ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ♥û❛ ❦❤æ♥❣ ❣✐❛♥ ❝õ❛ H ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸ P❤✐➳♠ ❤➔♠ f : H → (−∞, +∞] ✤÷đ❝ ❣å✐ ❧➔✿




ã ỗ tr H ợ ồ số tỹ ∈ [0, 1]✱ ∀x, y ∈ H t❛ ❝â
f (λx + (1 − λ)y) ≤ λf (x) + (1 − )f (y);
ã ỗ t tr H ợ ồ số t❤ü❝ λ ∈ (0, 1)✱ ∀x, y ∈ H ✱ x ̸= y t❛ ❝â
f (λx + (1 − λ)y) < f (x) + (1 )f (y);
ã tữớ ♥➳✉ ♠✐➲♥ ❤ú✉ ❤✐➺✉ ❝õ❛ ♥â
❞♦♠f = {x ∈ H : f (x) < +∞} =
̸ Ø;

• ♥û❛ ❧✐➯♥ tư❝ ữợ t x0 H ợ {xn } ⊂ H ✱ xn → x0 t❤➻
lim inf f (xn ) ≥ f (x0 );
n→∞

• ♥û❛ ❧✐➯♥ tư❝ tr➯♥ t↕✐ ✤✐➸♠ x0 ∈ H ♥➳✉ ✈ỵ✐ ❞➣② {xn } ⊂ H ✱ xn → x0 t❤➻
lim sup f (xn ) ≤ f (x0 ).
n→∞

✣à♥❤ ♥❣❤➽❛ ✶✳✹ ❈❤♦ f : H (, +] ỗ tữớ ỷ

tử ữợ ữợ f ừ f ❜ð✐

∂f (x) = {z ∈ H : f (y) ≥ f (x) + ⟨y − x, z⟩} , ∀y H.
ữ ữợ ừ ỗ tữớ ỷ tử ữợ f
f : H → 2H ①→❝ ✤à♥❤ ❜ð✐ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥✳

✣à♥❤ ♥❣❤➽❛ ✶✳✺ ❈❤♦ C ❧➔ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ H ✱ I ❧➔ →♥❤ ①↕ ✤ì♥ ✈à
tr➯♥ H ✳ T : C H ữủ ồ

ã Lst tỗ t số L > 0 s ❝❤♦ ✈ỵ✐ ∀x, y ∈ C,
∥T x − T y∥ Lx y;
ã T Lst ợ ❤➡♥❣ sè L < 1❀
• ❦❤ỉ♥❣ ❣✐➣♥ ♥➳✉ T ❧➔ L✲▲✐♣s❝❤✐t③ ✈ỵ✐ ❤➡♥❣ sè L = 1✱ tù❝ ❧➔ ✈ỵ✐ ∀x, y ∈ C,
∥T x − T y∥ ≤ ∥x y;
ã ổ t ợ x, y C,
∥T x − T y∥2 ≤ ⟨T x − T y, x y;
ã ỡ ợ ∀x, y ∈ C,
⟨T x − T y, x − y⟩ ≥ η∥x − y∥2 , η > 0;




ã t ợ x, y C,
⟨T x − T y, x − y⟩ ≤ ∥x − y∥2 − γ∥(I − T )x − (I − T )y2 , [0, 1);
ã ữủ ỡ ♠↕♥❤ ♥➳✉ ✈ỵ✐ ∀x, y ∈ C,
⟨T x − T y, x − y⟩ ≥ α∥T x − T y∥2 , α > 0;
• tr✉♥❣ ❜➻♥❤✱ ♥➳✉ T = (1 − α)I + αN ✈ỵ✐ sè ❝è ✤à♥❤ α ∈ (0, 1) ✈➔ N ❧➔
→♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✱ ❦❤✐ ✤â õ T tr

ỵ t t ở ❝õ❛ T ❧➔ ❋✐①✭❚✮✱ tù❝ ❧➔✿
❋✐①✭❚✮ = {x ∈ C : T x = x} .

✣à♥❤ ♥❣❤➽❛ ✶✳✻ ❈❤♦ C H t ỗ õ x C. ❚➟♣
NC (x) = {w ∈ H : ⟨w, y − x⟩ ≤ 0, ∀y ∈ C} ,
✤÷đ❝ ❣å✐ ❧➔ ♥â♥ ♣❤→♣ t✉②➳♥ ❝õ❛ C t↕✐ x✳

✣à♥❤ ♥❣❤➽❛ ✶✳✼ ❈❤♦ C ởt t rộ ỗ õ ừ H. ợ
ộ x H tỗ t ởt tỷ PC x ∈ C t❤ä❛ ♠➣♥

∥x − PC x∥ = inf ∥x − y∥.
y∈C

P❤➛♥ tû PC x ①→❝ ✤à♥❤ ♥❤÷ tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ x ❧➯♥ C ✈➔ →♥❤
①↕ PC : H → C ❜✐➳♥ ♠é✐ ♣❤➛♥ tû x ∈ H t❤➔♥❤ PC x ✤÷đ❝ ❣å✐ ❧➔ ♣❤➨♣ ❝❤✐➳✉
♠➯tr✐❝ tø H ❧➯♥ C ✳
✣➦❝ tr÷♥❣ ❝õ❛ ♣❤➨♣ tr ữủ ữợ

C t ỗ õ ré♥❣ ❝õ❛ H ✳ ❑❤✐
✤â✱ →♥❤ ①↕ PC : H → C ❧➔ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tø H ❧➯♥ C ❦❤✐ ✈➔ ❝❤➾ ❦❤✐

⟨x − PC x, y − PC x⟩ ≤ 0, ∀y ∈ C.
❚ø ✤â t❛ ❝â ❝→❝ ❤➺ q✉↔

(i) ∥PC x − PC y∥2 ≤ ⟨PC x − PC y, x − y⟩, ∀x, y ∈ H ✱ tù❝ ♣❤➨♣ ❝❤✐➳✉ ❧➔ →♥❤
①↕ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t❀

(ii) ∥x − PC x∥2 ≤ ∥x − y∥2 − ∥y − PC x∥2 , ∀x ∈ H, ∀y ∈ C.

◆❤➟♥ ①➨t ữỡ ồ ợ ồ y C ✱ ♥➳✉ t❛ ❣å✐ α ❧➔

❣â❝ t↕♦ ❜ð✐ ❝→❝ ✈➨❝ tì x − PC x ✈➔ y − PC x t❤➻ α ≥ π/2✳


✶✶

❚♦→♥ tû ✤ì♥ ✤✐➺✉ ✈➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐

❈❤♦ T : H → 2H ❧➔ t♦→♥ tû ✤❛ trà ❝â ♠✐➲♥ ①→❝ ✤à♥❤ ✈➔ ♠✐➲♥ ❣✐→ trà ❧➛♥
❧÷đt ❧➔ D(T ) := {x ∈ H : T x ̸= ∅} ✈➔ R(T ) = {y ∈ T x : x D(T )} .
ỗ t ừ T ỵ ❤✐➺✉ ❣r❛T ✈➔ ①→❝ ✤à♥❤ ❜ð✐
❣r❛T = {(x, u) ∈ H × H : u ∈ T x} .
❚♦→♥ tû ♥❣÷đ❝ T −1 : H → 2H ①→❝ ✤à♥❤ ❜ð✐

T −1 u = {x ∈ H : u ∈ T x}
tù❝ ❧➔ (u, x) ∈ ❣r❛T −1 ⇔ (x, u) rT.
ỵ t ổ ừ T ZerT ✱ tù❝ ❧➔✿

ZerT = {x ∈ D(T ) : 0 ∈ T x}

✣à♥❤ ♥❣❤➽❛ ✶✳✽ ❚♦→♥ tû T ✤÷đ❝ ❣å✐ ❧➔
• ✤ì♥ ✤✐➺✉ ♥➳✉

⟨u − v, x − y⟩ ≥ 0, ∀(x, u), (y, v) ∈ ❣r❛T ;
• ✤ì♥ ✤✐➺✉ ỹ T ỡ ỗ t ừ T ổ tỹ sỹ
tr ỗ t ừ ởt t♦→♥ tû ✤ì♥ ✤✐➺✉ ♥➔♦ ❦❤→❝✳

◆❤➟♥ ①➨t ✶✳✸ ❱ỵ✐ λ > 0✱ ♥➳✉ T ✤ì♥ ✤✐➺✉ t❤➻ T −1 ✈➔ λT ❝ơ♥❣ ✤ì♥ ✤✐➺✉✱
♥➳✉ T ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ t❤➻ T −1 ✈➔ λT ❝ơ♥❣ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳

❱➼ ❞ư ✶✳✶ ▼ët sè ✈➼ ❞ư ✈➲ t♦→♥ tû ✤ì♥ ✤✐➺✉✿

(1) ❚♦→♥ tû ✤ì♥ trà T : R → R ①→❝ ✤à♥❤ ❜ð✐ T (x) = x, ∀x ∈ R✳
(2) ❚♦→♥ tû t✉②➳♥ t➼♥❤ T : H → H t❤ä❛ ♠➣♥ ⟨T x, x⟩ ≥ 0, ∀x ∈ H ✳
(3) ❱ỵ✐ C t õ ỗ ừ H t PC t♦→♥ tû ✤ì♥ ✤✐➺✉✳

❱➼ ❞ư ✶✳✷ ▼ët sè ✈➼ ❞ư ✈➲ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✿

(1) ❈❤♦ f : H (, +] ỗ tữớ ỷ tử ữợ
õ ữợ f : H → 2H ①→❝ ✤à♥❤ ❜ð✐

∂f (x) = {z ∈ H : f (y) ≥ f (x) + ⟨y − x, z⟩} , ∀y ∈ H.
❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳

(2) ❈❤♦ T : H → H ❧➔ α✲tr✉♥❣ ❜➻♥❤✱ α ∈ [0, 1/2]. ❑❤✐ ✤â✱ T ❧➔ t♦→♥ tû
✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳


✶✷

(3) ❈❤♦ T : H → H ❧➔ α✲♥❣÷đ❝ ✤ì♥ ✤✐➺✉ ♠↕♥❤ t❤➻ T ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉
❝ü❝ ✤↕✐✳

✣à♥❤ ♥❣❤➽❛ ✶✳✾ ❈❤♦ t♦→♥ tû ✤❛ trà T : H → 2H ✈➔ sè t❤ü❝ r > 0✳
• ❚♦→♥ tû ❣✐↔✐ ❝õ❛ T ❧➔ Jr = (I + rT )−1 ✳
I − Jr
✳
• ❚♦→♥ tû ①➜♣ ①➾ ❨♦s✐❞❛ ❝õ❛ T ❧➔ Tr =
r

❍➺ q✉↔ ✶✳✶ ✭❬✹✾❪✮ ❈❤♦ t♦→♥ tû T : H → H ❧➔ ✤ì♥ ✤✐➺✉ ✈➔ ❧✐➯♥ tư❝✳ ❑❤✐
✤â✱ T ❧➔ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳

▼➺♥❤ ✤➲ ✶✳✷ ✭❬✹✾❪✮ ❈❤♦ t♦→♥ tû ✤ì♥ ✤✐➺✉ T : H → 2H ✳ ❑❤✐ ✤â✱ T ❧➔ t♦→♥

tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ∀(u, v) ∈ H × H ✱ ♥➳✉✿

⟨y − v, x − u⟩ ≥ 0, ∀(x, y) ∈ ❣r❛T.
t❤➻ v ∈ T (u)✳

▼➺♥❤ ✤➲ ✶✳✸ ✭❬✺✵❪✮ ❈❤♦ t♦→♥ tû ✤ì♥ ✤✐➺✉ T

tr➯♥ H ✱ T ❧➔ ✤ì♥ ✤✐➺✉ ❝ü❝

✤↕✐ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ R(I + T ) = H.

❍➺ q✉↔ ✶✳✷ ✭❬✹✾❪✮ ●✐↔ sû T : H → 2H ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ✈➔ r ❧➔

sè t❤ü❝ ❞÷ì♥❣✳ ❑❤✐ ✤â✿

(i) Jr : H → H ✈➔ I − Jr : H → H ❧➔ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ❝❤➦t ✈➔ ✤ì♥ ✤✐➺✉
❝ü❝ ✤↕✐✳

(ii) Tr ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ✈➔ r✲ ữủ ỡ
1
(iii) Tr tử st ợ ❤➺ sè ✳
r

▼➺♥❤ ✤➲ ✶✳✹ ✭❬✺✵❪✮ ❈❤♦ T ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ✈➔ sè t❤ü❝ r > 0✳

❑❤✐ ✤â✱ t♦→♥ tû ❣✐↔✐ Jr = (I + rT )−1 ❧➔ ❦❤ỉ♥❣ ❣✐➣♥ ❝❤➦t ✭❞♦ ✤â ❧➔ ✤ì♥ trà✮
✈➔ ❝â ♠✐➲♥ ①→❝ ✤à♥❤ ✤➛② ✤õ✳

▼➺♥❤ ✤➲ ✶✳✺ ✭❬✹✾❪✮ ❈❤♦ T : H → 2H ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ỹ õ

ZerT t õ ỗ

✶✳✻ ✭❬✺✶❪✮ ❈❤♦ T

❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr➯♥ H ✱ sè t❤ü❝

r > 0 ✈➔ x ∈ H ✳ ❑❤✐ ✤â 0 ∈ T x ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ Jr (x) = {x}✳
◆❤÷ ✈➟②✱ t❤❡♦ ❦➳t q✉↔ ❝õ❛ ♠➺♥❤ ✤➲ tr➯♥ ❝❤♦ t❤➜② ZerT = ❋✐①(Jr )✳ ❚➟♣
❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ✤â♥❣ trỏ q trồ tr ỵ




tt tố ữ t ở ử t

ã ◆➳✉ T = I − A✱ ✈ỵ✐ A ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✱ t❤➻ ZerT ❝❤➼♥❤ ❧➔ t➟♣ ✤✐➸♠
❜➜t ✤ë♥❣ ừ A
ã T = f ợ f ỗ tữớ ỷ tử ữợ t
ZerT ❧➔ t➟♣ ✤✐➸♠ ❝ü❝ t✐➸✉ ❝õ❛ f ✳

❚ê♥❣ ❝õ❛ ❤❛✐ t♦→♥ tû ✤ì♥ ✤✐➺✉
✣à♥❤ ♥❣❤➽❛ ✶✳✶✵ ❈❤♦ ❤❛✐ t♦→♥ tû A : H → H ✈➔ B : H → H ✳ ❑❤✐ ✤â

A + B : H → H ❧➔ t♦→♥ tû ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐

D(A + B) = D(A) + D(B);
(A + B)(x) = A(x) + B(x).
◆➳✉ A, B ❧➔ ❝→❝ t♦→♥ tû ✤❛ trà tr➯♥ H t❛ ❝â ✤à♥❤ ♥❣❤➽❛ s❛✉✿

A + B = {(x, y + z) | (x, y) ∈ ❣r❛A, (x, z) ∈ ❣r❛B} .
ỵ t ổ ừ A + B Zer(A + B)✱ tù❝ ❧➔✿

Zer(A + B) = {x ∈ H : 0 ∈ (A + B)x} .
❚❛ ❜✐➳t r➡♥❣✱ ♥➳✉ A, B ❧➔ ❝→❝ t♦→♥ tû ✤ì♥ ✤✐➺✉ t❤➻ A + B ❧➔ t♦→♥ tû ✤ì♥
✤✐➺✉✳ ❚✉② ♥❤✐➯♥✱ ♥➳✉ A, B ❧➔ ❝→❝ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ t❤➻ ❝❤÷❛ ❝❤➢❝

A + B ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳

❇ê ✤➲ ✶✳✶ ✭❬✺✷❪✮ ❈❤♦ A : H → H ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ✈➔ ❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③✱

B : H → 2H ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳ ❑❤✐ ✤â✱ A + B : H → 2H ❧➔ t♦→♥
tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳

❇ê ✤➲ ✶✳✷ ✭❬✺✸❪✮ ❈❤♦ A : H → H ❧➔ →♥❤ ①↕ tr➯♥ H ✱ B : H → 2H ❧➔ t♦→♥
tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ✈➔ r ❧➔ sè t❤ü❝ ❞÷ì♥❣✳ ✣➦t Tr = JrB (I − rA)✳ ❑❤✐ ✤â✿

(i) F ix(Tr ) = Zer(A + B)✳
(ii) ❱ỵ✐ 0 < s ≤ r, x ∈ H t❤➻ ∥x − Ts x∥ ≤ 2∥x − Tr x∥✳

❇ê ✤➲ ✶✳✸ ✭❬✺✹❪✮ ❈❤♦ T ❧➔ →♥❤ ①↕ α✲ ♥❣÷đ❝ ✤ì♥ ✤✐➺✉ ♥❣÷đ❝ ♠↕♥❤ ✈➔ r ❧➔
sè t❤ü❝ ❞÷ì♥❣ t❤ä❛ ♠➣♥ r ≤ α✳ ❑❤✐ ✤â✱ A := I − rT ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥
❝❤➦t✳

❇ê ✤➲ ✶✳✹ ✭❬✺✹❪✮

❈❤♦ T ✈➔ S ❧➔ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t t❤➻

1
∥T Sx − T Sy∥2 ≤ ∥x − y∥2 − ∥(I − T S)x − (I − T S)y∥2
2


✶✹

❉♦ ✤â✱ T S ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳

◆❤➟♥ ①➨t ✶✳✹ ❱ỵ✐ B ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ✈➔ A ❧➔ →♥❤ ①↕ α−

♥❣÷đ❝ ✤ì♥ ✤✐➺✉ ♠↕♥❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ α > 0✳ ✣➦t U = JrB ✱

V = (I − rA) ✈ỵ✐ r ≤ α✱ ❦❤✐ ✤â U, V ✤➲✉ ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t✳ ◆❣♦➔✐
r❛✱ ♥➳✉ S := ❋✐①(JrB (I − rA)) ̸= Ø t❤➻ ✈ỵ✐ ❜➜t ❦ý z ∈ S ✈➔ x ∈ D(A)✱ t❤❡♦
❇ê ✤➲ 1.4 ♥❤➟♥ ✤÷đ❝

1
∥JrB (I − rA)x − z∥2 ≤ ∥x − z∥2 − ∥x − JrB (I − rA)x∥2 .
2

✶✳✷✳ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ t♦→♥ tû ✤ì♥ ✤✐➺✉
✶✳✷✳✶✳ P❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ✈➔ ♠ët sè ❝↔✐ ❜✐➯♥
❈❤♦ t♦→♥ tû T : H → 2H ❧➔ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳ ❚❛ ①➨t ❜➔✐ t♦→♥✿
❚➻♠ ♣❤➛♥ tû p∗ ∈ H

s❛♦ ❝❤♦

0 ∈ T p∗ .



ỵ ZerT t ổ ừ T ✳
P❤÷ì♥❣ ♣❤→♣ ❝ê ✤✐➸♥ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ✭✶✳✶✮ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥
❦➲✳ P❤÷ì♥❣ ♣❤→♣ ♥➔② ✤÷đ❝ ✤➲ ①✉➜t ✤➛✉ t✐➯♥ ❜ð✐ ▼❛rt✐♥❡t ❬✶❪ ✈➔♦ ♥➠♠ ✶✾✼✵
✤â ❧➔ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❝ü❝ t✐➸✉✿

min f (x),
x∈H

✭✶✳✷✮

✈ỵ✐ f : H (, +] ỗ tữớ ỷ tử ữợ tọ
ồ t ự {x H : f (x) ≤ α} , α ∈ R ❧➔ ợ ở
rtt ữ r xk ❣✐↔✐ ❜➔✐ t♦→♥ ✭✶✳✷✮ ❧➔✿

xk+1 = arg min Φk (x),
x

✭✶✳✸✮

1
∥x − xk ∥2 ✈ỵ✐ {ck } ❧➔ ❞➣② sè t❤ü❝ ữỡ
2ck
ở tử tợ ởt ừ t♦→♥ ✭✶✳✷✮✳

ð ✤â Φk (x) = f (x) +
♥❤✐➯♥ ❞➣② xk

◆➠♠ ✶✾✼✻✱ ❘♦❝❦❛❢❡❧❧❛r ❬✷❪ ✤➣ tê♥❣ q✉→t ❤â❛ ♣❤÷ì♥❣ ♣❤→♣ ❝õ❛ ▼❛rt✐♥❡t
❬✶❪ ❝❤♦ tr÷í♥❣ ❤đ♣ t♦→♥ tû T ❧➔ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳ ❚→❝ ❣✐↔ ✤➣ ✤➲ ①✉➜t
♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ✈ỵ✐ ❞➣② xk ❝❤♦ ❜ð✐✿

xk+1 = Jk xk + ek

❤♦➦❝

xk+1 = Jk (xk + ek ), k ≥ 1,

✭✶✳✹✮

ð ✤â x1 ∈ H ✱ Jk = (I + rk T )−1 ❧➔ t♦→♥ tû ❣✐↔✐ ❝õ❛ T ✱ {rk } ❧➔ ❞➣② sè t❤ü❝
❞÷ì♥❣✱ ek ❧➔ ✈➨❝ tì s❛✐ sè✳ ❚→❝ ❣✐↔ ✤➣ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ ❞➣② {rk } ❝õ❛


✶✺

t♦→♥ tû ❣✐↔✐ t❤ä❛ ♠➣♥ rk ≥ ε > 0, ∀k ≥ 1✱ ✈➔ ❞➣② s❛✐ sè ek t❤♦↔ ♠➣♥
∞

∥x

k+1

k

− Jk x ∥ ≤ εk

✈ỵ✐

εk < ∞,
k=1

t❤➻ ❞➣② xk ①→❝ ✤à♥❤ ❜ð✐ ✭✶✳✹✮ ❤ë✐ tư ②➳✉ tỵ✐ ♠ët ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ T ✳
❚❛ ❜✐➳t r➡♥❣✱ ♥➳✉ t♦→♥ tû T ❧➔ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ t❤➻ Jk ❧➔ →♥❤ ①↕ ✤ì♥
trà✱ ❦❤æ♥❣ ❣✐➣♥✱ D(Jk ) = H ✈➔ F ix(Jk ) = ZerT ✳ ❱➻ ✈➟②✱ ♠ët tr♦♥❣ ♥❤ú♥❣
÷✉ ✤✐➸♠ ♥ê✐ ❜➟t ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❧➔ ✤➣ ✤÷❛ ❜➔✐ t♦→♥ ✤❛ trà
✈➲ ❜➔✐ t♦→♥ ✤ì♥ trà ✤➸ ❣✐↔✐✳ ❚✉② ♥❤✐➯♥✱ ♥➠♠ ✶✾✾✶✱ ●✐✐❧❡r ❬✸❪ ✤➣ ❝❤➾ r❛ r➡♥❣
PP▼ ❝❤➾ ✤↕t ✤÷đ❝ sü ❤ë✐ tư ②➳✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈ỉ ❤↕♥ ❝❤✐➲✉✳
◆➠♠ ✶✾✾✷✱ ❊❝❦st❡✐♥ ✈➔ ❇❡rts❡❦❛s ❬✹❪ ✤÷❛ r❛ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲
tê♥❣ q✉→t ❝❤♦ ❜➔✐ t♦→♥ ✭✶✳✶✮ ✈ỵ✐ ❞➣② ❧➦♣ xk ❝â ❞↕♥❣✿

xk+1 = (1 − tk )xk + tk wk , k ≥ 1,

✭✶✳✺✮

ð ✤â ∥wk − Jk xk ∥ ≤ εk , ∀k ≥ 1, εk ✱ {tk }✱ {rk } ⊂ (0, ∞) t❤ä❛ ♠➣♥

εk < ∞, inf tk > 0, sup tk < 2, inf rk > 0.
k≥1

k≥1

k≥1

k≥1

❑❤✐ ✤â✱ ♥➳✉ ZerT ̸= Ø t❤➻ ❞➣② xk ❤ë✐ tư ②➳✉ tỵ✐ ♠ët ✤✐➸♠ t❤✉ë❝ ZerT ✳

◆➳✉ ZerT = Ø t❤➻ ❞➣② xk ❦❤æ♥❣ ❜à ❝❤➦♥✳
P❤÷ì♥❣ ♣❤→♣ ✭✶✳✺✮ ❝â t❤➸ ✈✐➳t ❧↕✐ ❧➔✿

xk+1 = (1 − tk )xk + tk Jk xk + ek , k ≥ 1,

✭✶✳✻✮

✈ỵ✐ ek ❧➔ ❞➣② s❛✐ sè ✈➔ ∥ek ∥ ≤ εk .
◆➠♠ ✷✵✵✹✱ ▼❛r✐♥♦ ✈➔ ❳✉ ❬✺✺❪ ự sỹ ở tử ừ ữỡ
ợ ✤✐➲✉ ❦✐➺♥ ✤➦t ❧➯♥ ❞➣② t❤❛♠ sè {tk } ♥❤➭ ❤ì♥ ✈➔ t❤➯♠ ✤✐➲✉ ❦✐➺♥
✤è✐ ✈ỵ✐ t❤❛♠ sè rk ✳ õ ữủ ồ ỵ s

ỵ ✶✳✶ ✭❬✺✺❪✮ ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➲✉ t❤ä❛ ♠➣♥✿
∥ek ∥ < ∞✱

(A1)
k=1
∞

(A2)

tk (2 − tk ) = ∞✱

k=1

(A3) tỗ t số 0 < c1 c2 s ❝❤♦ c1 ≤ rk ≤ c2 ✱ ∀k ≥ 1 ✈➔
∞

|rk+1 − rk | < ∞✳

k=1

❑❤✐ ✤â✱ ❞➣② xk
✧

①→❝ ✤à♥❤ ❜ð✐ ✭✶✳✻✮ ❤ë✐ tư ②➳✉ tỵ✐ ♠ët ✤✐➸♠ t❤✉ë❝ ZerT.


✶✻

✣➸ ♥❤➟♥ ✤÷đ❝ sü ❤ë✐ tư ♠↕♥❤✱ ♠ët sè ❝↔✐ ừ ữỡ
ữủ ợ t t ủ ữỡ
ợ ữỡ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❚✐❦❤♦♥♦✈✱ ▲❡❤❞✐❧✐ ✈➔ ▼♦✉❞❛❢✐ ❬✺❪ ✤➣ ✤➲ ①✉➜t
♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❤✐➺✉ ❝❤➾♥❤ ❚✐❦❤♦♥♦✈ ✈➔♦ ♥➠♠ ✶✾✾✻✱ ✈ỵ✐ ❞➣②

xk ❝❤♦ ❜ð✐✿
xk+1 = JkTk xk ,

✭✶✳✼✮

ð ✤â JkTk = (I + rk Tk )−1 , Tk = T + µk I, µk > 0✳
❙ü ❤ë✐ tư ♠↕♥❤ ừ ữỡ ữủ ỵ s

ỵ t tỷ T : H 2H ❧➔ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳ ●✐↔ sû

❞➣② sè {rk } ợ ở {àk } > 0 tọ ♠➣♥
+∞

µk = ∞, ✈➔ lim

lim µk = 0,

k→+∞

❑❤✐ ✤â ❞➣②

xk

k→+∞

k=1

1
µk+1

−

1
= 0.
àk



ở tử tợ ởt tû t❤✉ë❝

ZerT ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t✳
▼÷í✐ ♥➠♠ s❛✉✱ ❳✉ ❬✸✸❪ ✤➣ ♠ð rë♥❣ ❦➳t q✉↔ ❝õ❛ ▲❡❤❞✐❤✐ ✈➔ ▼♦✉❞❛❢✐ ❬✺❪
❜➡♥❣ ❝→❝❤ ✤÷❛ r❛ ❞➣② xk ❝â ❞↕♥❣✿

xk+1 = Jk [tk u + (1 − tk )xk + ek ], k ≥ 1,

✭✶✳✾✮

✈ỵ✐ u ∈ H ✳ ❑❤✐ ✤â✱ ❞➣② xk ❤ë✐ tư ♠↕♥❤ tỵ✐ PZerT u ♥➳✉ ❝→❝ ❞➣② t❤❛♠ sè

{rk } , {tk } ✈➔ ek t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (A1) , (A3)✱
∞

(A4)

|tk+1 − tk | < ∞✱

k=1

∞

(A5) tk ∈ (0, 1), ∀k ≥ 1, lim tk = 0,
k→∞

tk = ∞✳

k=1

❍ì♥ ♥ú❛✱ ❳✉ ❝❤➾ r❛ r➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✶✳✼✮ ❧➔ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛
tk
♣❤÷ì♥❣ ♣❤→♣ ✭✶✳✾✮ ♥➳✉ ❧➜② u = 0, ek = 0 ✈➔ µk = , ∀k ≥ 1.
rk
◆➠♠ ✷✵✶✷✱ ❇♦✐❦❛♥②♦ ✈➔ ▼♦r♦s❛♥✉ ❬✼❪ ✤➣ tê♥❣ q✉→t ❤â❛ ♣❤÷ì♥❣ ♣❤→♣
✭✶✳✾✮ ❜➡♥❣ ❝→❝❤ ❞ị♥❣ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ S : H → H ✱ ð ✤â ❞➣② ❧➦♣ xk
✤÷đ❝ ❝❤♦ ❜ð✐✿

xk+1 = Jk [tk u + λk xk + δk Sxk + ek ], k ≥ 1,

✭✶✳✶✵✮

✈ỵ✐ u ∈ H ✱ rk ∈ (0, +∞)✱ tk , λk , δk ∈ [0, 1] ✈➔ tk + λk + δk = 1✳
❘ã r➔♥❣✱ tr♦♥❣ ✭✶✳✶✵✮ ♥➳✉ ❧➜② S ≡ I ❧➔ →♥❤ ①↕ ✤ì♥ ✈à t❤➻ ✭✶✳✶✵✮ trð t❤➔♥❤


✶✼

♣❤÷ì♥❣ ♣❤→♣ ✭✶✳✾✮✳ ❈→❝ t→❝ ❣✐↔ ✤➣ ❝❤➾ r❛ sü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ❞➣② xk tỵ✐

PZerT u ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ (A5)✱ (A1) ❤♦➦❝
∥ek ∥
′
(A1 ) lim
= 0 ✈➔
k→∞ tk
′
(A3 ) ∃ε > 0 t❤♦↔ ♠➣♥ rk , k 1
ỵ r tr t q ❝õ❛ ❇♦✐❦❛♥②♦ ✈➔ ▼♦r♦s❛♥✉ ❬✼❪✱ ❝→❝ t→❝ ❣✐↔ ✤➣
❧♦↕✐ ❜ä ✤÷đ❝ ❝→❝ ✤✐➲✉ ❦✐➺♥ (A3)✱ (A4) ✈➔ sû ❞ư♥❣ ✤✐➲✉ ❦✐➺♥ tê♥❣ q✉→t ❤ì♥
′

✤è✐ ✈ỵ✐ ✈ỵ✐ ❞➣② ek ✱ ✤â ❧➔ ❞➣② ek t❤ä❛ ♠➣♥ (A1) ❤♦➦❝ (A1 )✳❚ø ✤â✱ ❝❤♦
t❤➜② ♠å✐ ❞➣② ek

♠➔ ∥ek ∥ → 0 ✤➲✉ t❤ä❛ ♠➣♥ ❝❤♦ sü ❤ë✐ tư ♠↕♥❤ ❝õ❛

♣❤÷ì♥❣ ♣❤→♣ ✭✶✳✶✵✮ ♥➳✉ t❛ ❝❤å♥ ❞➣② {tk } t❤➼❝❤ ❤ñ♣✳
◆➠♠ ✷✵✶✸✱ ❚✐❛♥ ✈➔ ❙♦♥❣ ❬✺✻❪ ❝ơ♥❣ ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ✭✶✳✾✮
tỵ✐ PZerT u ✈ỵ✐ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ ♥❤➭ ❤ì♥ ❝õ❛ ❳✉ ❬✸✸❪✱ ✤â ❧➔ ❝→❝ ✤✐➲✉ ❦✐➺♥

(A1)✱ (A5) ✈➔
′′
(A3 ) lim inf rk > 0.
k→∞

▼ët ❝↔✐ ❜✐➯♥ ❦❤→❝ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠
❣➛♥ ❦➲ ❝♦ ✤÷đ❝ ✤➲ ①✉➜t ❜ð✐ ❑❛♠✐♠✉r❛ ✈➔ ❲✳❚❛❦❛❤❛s❤✐ ❬✽❪ ✈➔♦ ♥➠♠
ợ xk ữủ

y k Jk xk , ∥y k − Jk xk ∥ ≤ δk , xk+1 = tk u + (1 − tk )y k ,
u ❧➔ ♠ët ♣❤➛♥ tû ❝è ✤à♥❤ tr♦♥❣ H {rk } (0, )
ỵ s t q ở tử ừ ữỡ





k < .
k=1

ỵ ✶✳✸ ✭❬✽❪✮ ❈❤♦ t♦→♥ tû T : H → 2H ❧➔ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✱ u ∈ H ✱
xk ❧➔ ❞➣② t↕♦ ❜ð✐ ✭✶✳✶✶✮✱ ❝→❝ ❞➣② t❤❛♠ sè {tk }✱ {rk } t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉

❦✐➺♥ (A5) ✈➔
′′′

(A3 ) lim rk = ∞✳
k→∞

❑❤✐ ✤â✱ ❞➣② xk

❤ë✐ tư ♠↕♥❤ tỵ✐ PZerT u ❦❤✐ k → ∞ ♥➳✉ ZerT ̸= Ø✳

◆➠♠ ✷✵✵✹✱ ▼❛r✐♥♦ ✈➔ ❳✉ ❬✺✺❪ ❝ơ♥❣ ✤÷❛ r❛ ❞➣② ❧➦♣ ①→❝ ✤à♥❤ ❜ð✐ ♣❤÷ì♥❣
♣❤→♣ ✤✐➸♠ ❣➛♥ ❝♦ ❝â ❞↕♥❣

xk+1 = tk u + (1 − tk )Jk xk + ek , k ≥ 1.

✭✶✳✶✷✮

❈→❝ t→❝ ❣✐↔ ❝❤➾ r❛ t❤✉➟t t♦→♥ ✭✶✳✶✷✮ ❤ë✐ tư ♠↕♥❤ tỵ✐ PZerT u ✈ỵ✐ {tk } ⊂

(0, 1)✱ {rk } ⊂ (0, +∞)✱ ek ⊂ H ✈➔ t❤ä❛ ♠➣♥ (A1)✱ (A3)✱ (A5) ✈➔ (A4)
❤♦➦❝
tk
′
= 1✳
(A4 ) lim
k→∞ tk+1


✶✽

❚r♦♥❣ ❬✾❪✱ ❨❛♦ ✈➔ ◆♦♦r ✤➣ ♠ð rë♥❣ ❦➳t ❦➳t q✉↔ ❝õ❛ ▼❛r✐♥♦ ✈➔ ❳✉ ❬✺✺❪
✈ỵ✐ t❤✉➟t t♦→♥✿
✭✶✳✶✸✮

xk+1 = tk u + βk xk + γk Jk xk + ek , k ≥ 1,

ð ✤â tk , βk , γk ∈ (0, 1) ✈➔ tk + βk + γk = 1. ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❞➣② xk ❤ë✐ tư
′

♠↕♥❤ tỵ✐ PZerT u ♥➳✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t❤ä❛ ♠➣♥✿ (A1), (A3 ), (A5)✱

(A6) βk ∈ [a, b] ⊂ (0, 1)✱
(A7) rk+1 − rk → 0✳
❙♦ ✈ỵ✐ ❦➳t q✉↔ ❝õ❛ ▼❛r✐♥♦ ✈➔ ❳✉ ❬✺✺❪✱ ❨✳❨❛♦ ✈➔ ◆♦♦r ❬✾❪ ✤➣ t❤❛② t❤➳ ✤✐➲✉
❦✐➺♥ (A3) ❜ð✐ ✤✐➲✉ ❦✐➺♥ ♥❤➭ ❤ì♥ (A7)✱ ✈➻ rã r➔♥❣

∞

|rk+1 − rk | < ∞ ❦➨♦

k=1

t❤❡♦ rk+1 − rk → 0 ỗ tớ t ụ ọ ữủ ❝→❝ ✤✐➲✉ ❦✐➺♥
′

(A4) ✈➔ (A4 ) ♥❤÷♥❣ t❤➯♠ ✈➔♦ ✤✐➲✉ ❦✐➺♥ (A6)✳
◆➠♠ ✷✵✶✷✱ tr♦♥❣ ❬✺✼❪✱ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✱ ✤â ❧➔
t➻♠ p∗ ∈ H s❛♦ ❝❤♦

p∗ ∈ C : ⟨F p∗ , p∗ − p⟩ ≤ 0, ∀p ∈ C,

✭✶✳✶✹✮

✈ỵ✐ C = ZerT ❧➔ t➟♣ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ T ✱ F ❧➔ →♥❤ ①↕ L✲ ❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③
✈➔ η ✲ ✤ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥ H ✱ ❙✳ ❲❛♥❣ ✤➣ ✤➲ ①✉➜t ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣
✭✶✳✶✺✮

xk+1 = Jk [(I − tk F )xk + ek ], k ≥ 1.

❚→❝ ❣✐↔ ✤➣ ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tö ♠↕♥❤ ừ ữỡ tợ

p C ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✶✳✶✹✮
✈ỵ✐ ✤✐➲✉ ❦✐➺♥ (A3′′ )✳
❚❛ t❤➜②✱ sü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ❝→❝ ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥
′

′′

′′′

❦➲ ✤↕t ✤÷đ❝ ✤➲✉ ❝➛♥ ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ (A3), (A3 ), (A3 ) ✈➔ (A3 )✳
❈→❝ ✤✐➲✉ ❦✐➺♥ ♥➔② ✤➲✉ ❞➝♥ ✤➳♥ t❤❛♠ sè rk ❝õ❛ t♦→♥ tû ❣✐↔✐ ♣❤↔✐
ữợ ởt số ợ ỡ ✷✵✶✼✱ tr♦♥❣ ❬✶✶❪✱ ◆✳ ❇÷í♥❣✱ P✳❚✳❚✳
❍♦➔✐✱ ◆✳❉✳ ◆❣✉②➵♥ ✤➣ ✤➲ t t ợ ừ ữỡ
õ ❞↕♥❣ ❣✐è♥❣ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❤✐➺✉ ❝❤➾♥❤ ❚✐❦❤♦♥♦✈ ✭✶✳✾✮
✈➔ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❝♦ ✭✶✳✶✷✮ ♥❤÷♥❣ tr♦♥❣ ❦➳t q✉↔ ♥➔② t♦→♥ tû
❣✐↔✐ Jk ✤÷đ❝ t❤❛② ❜ð✐ ❤đ♣ ❝õ❛ k t♦→♥ tû ❣✐↔✐✳ ❈❤ó♥❣ tỉ✐ t❤✉ ✤÷đ❝ sỹ ở
tử ừ ữỡ ợ t sè rk ❝õ❛ t♦→♥ tû ❣✐↔✐ ❞➛♥
tỵ✐ ✵✱ ❝ư t❤➸ ❧➔ rk t❤♦↔ ♠➣♥

∞

k=1

rk < +∞✱ ✤➙② ❧➔ ✤✐➲✉ ❦✐➺♥ ♠ỵ✐ s♦ ✈ỵ✐ ❝→❝