Luận văn Thạc sĩ Toán học: Phương pháp lặp Ishikawa cho một họ vô hạn các ánh xạ không giãn

ĐẠI HỌC THÁI NGUYÊN
TRƯỜNG ĐẠI HỌC KHOA HỌC
-------------------------------

NGUYỄN VĂN NGA

PHƯƠNG PHÁP LẶP ISHIKAWA
CHO MỘT HỌ VÔ HẠN CÁC ÁNH XẠ
KHÔNG GIÃN

LUẬN VĂN THẠC SĨ TOÁN HỌC

THÁI NGUYÊN - 2019


ĐẠI HỌC THÁI NGUYÊN
TRƯỜNG ĐẠI HỌC KHOA HỌC
-------------------------------

NGUYỄN VĂN NGA

PHƯƠNG PHÁP LẶP ISHIKAWA
CHO MỘT HỌ VÔ HẠN CÁC ÁNH XẠ
KHÔNG GIÃN
Chuyên ngành: Toán ứng dụng
Mã số
: 8 46 01 12

LUẬN VĂN THẠC SĨ TOÁN HỌC

NGƯỜI HƯỚNG DẪN KHOA HỌC

GS.TS. Nguyễn Bường

THÁI NGUYÊN - 2019




ử ử
ỵ

ổ ❜➜t ✤ë♥❣
✶✳✶

✶✳✷

❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t

✶
✷
✹

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹

✶✳✶✳✶

❑❤æ♥❣ ❣✐❛♥ ỗ trỡ ✳ ✳ ✳ ✳ ✳ ✳ ✹

✶✳✶✳✷

⑩♥❤ ①↕ ✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✼

✣✐➸♠ ❜➜t ✤ë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✶✳✷✳✶

❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✶✳✷✳✷

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛
→♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

✷ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛
❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥
✶✺
✷✳✶

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

✷✳✶✳✶

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

✷✳✶✳✷

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣
❝õ❛ ❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✷✳✷

❈↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✷✳✷✳✶

▼ỉ t↔ ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✷✳✷✳✷

❙ü ❤ë✐ tö

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✸✵
✸✶




ỵ
H

ổ rt tỹ

E

ổ

E

ổ ố ♥❣➝✉ ❝õ❛ E

R

t➟♣ ❝→❝ sè t❤ü❝

∅

t➟♣ ré♥❣

∀x

✈ỵ✐ ♠å✐ x

I

t♦→♥ tû ỗ t

lp , 1 p <

ổ ❞➣② sè ❦❤↔ tê♥❣ ❜➟❝ p

l∞

❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② sè ❜à ❝❤➦♥

Lp [a, b], 1 ≤ p < ∞

❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ t➼❝❤ ❜➟❝ p
tr➯♥ ✤♦↕♥ [a, b]

lim supn→∞ xn

❣✐ỵ✐ ❤↕♥ tr➯♥ ❝õ❛ ❞➣② sè {xn }

lim inf n→∞ xn

ợ ữợ ừ số {xn }

xn x0

{xn } ❤ë✐ tö ♠↕♥❤ ✈➲ x0

xn

❞➣② {xn } ❤ë✐ tö ②➳✉ ✈➲ x0

x0

J

→♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝

❋✐①(T )

t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T


✷

▼ð ✤➛✉
❇➔✐ t♦→♥ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝ →♥❤ ①↕
❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❤❛② ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❧➔ ♠ët
tr÷í♥❣ ❤đ♣ r✐➯♥❣ ❝õ❛ t ỗ ởt tỷ tở
❦❤→❝ ré♥❣ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ ❤❛② ✈æ ❤↕♥ t ỗ õ

{Ci }iI ừ ổ ❍✐❧❜❡rt H ❤❛② ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✧✳ ❇➔✐ t♦→♥
♥➔② ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤❛✉ ữ ỷ
ổ ử t t ỵ ❤å❝✱✳ ✳ ✳
❑❤✐ C = ❋✐①(T )✱ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ T ✱ t❤➻ ✤➣
❝â ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ✤÷đ❝ ✤➲ ①✉➜t ❞ü❛ tr➯♥ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❝ê
✤✐➸♥ ♥ê✐ t✐➳♥❣✳ ✣â ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥ ❬✷❪✱ ■s❤✐❦❛✇❛ ❬✺❪✱ ❍❛❧♣❡r♥
❬✹❪✱ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ♠➲♠ ❬✻❪✳ ◆❤➻♥ ❝❤✉♥❣✱ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❝❤➾
❝â sü ❤ë✐ tư ②➳✉✳ ❱➼ ❞ư✱ ❙✳ ❘❡✐❝❤ ❝❤➾ r❛ r➡♥❣ ♥➳✉ ❦❤ỉ♥❣

E ỗ õ ❋r➨❝❤❡t ✈➔ ♥➳✉ ❞➣② {αn } t❤ä❛ ♠➣♥
∞
n=0 αn (1

− αn ) = ∞ t❤➻ ❞➣② {xn } ✤÷đ❝ t↕♦ r❛ tø ♣❤÷ì♥❣ ♣❤→♣ ▼❛♥♥

❤ë✐ tư ②➳✉ ✤➳♥ ♠ët ♣❤➛♥ tû ❝õ❛ ❋✐①(T )✳ ❱➻ ✈➟②✱ r➜t ♥❤✐➲✉ t→❝ ❣✐↔ ✤➣ ❝↔✐
t✐➳♥ ♣❤÷ì♥❣ ♣❤→♣ ▼❛♥♥ ✈➔ ■s❤✐❦❛✇❛ ✤➸ ❝â ✤÷đ❝ sü ❤ë✐ tư ♠↕♥❤ ❝❤♦ ❝→❝
→♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳ ❈❤♦ ✤➳♥ ♥❛② ✤➣ ❝â ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ✤÷đ❝ ✤÷❛ r❛
❞ü❛ tr➯♥ sü ❝↔✐ ❜✐➯♥ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ❝❤♦ ❝→❝ ❧ỵ♣ ❜➔✐ t♦→♥ ❧✐➯♥
q✉❛♥✳
▼ư❝ t✐➯✉ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ❧➔ t➻♠ ❤✐➸✉ ✈➔ tr➻♥❤ ❜➔② ❧↕✐ ♠ët ❝↔✐ t✐➳♥
❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ✈ỉ
❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ tr ổ ỗ t
trỡ tr ❜➔✐ ❜→♦ ❬✽❪ ❝æ♥❣ ❜è ♥➠♠ ✷✵✶✷✳

◆ë✐ ❞✉♥❣ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✳


✸

❈❤÷ì♥❣ ✶✳ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ✤✐➸♠ ❜➜t ✤ë♥❣
❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❦❤ỉ♥❣
ỗ t trỡ ởt số t t P tự ừ ữỡ ợ
t t ❜➜t ✤ë♥❣✱ tr➻♥❤ ❜➔② ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ✤✐➸♠
❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥✱ ♣❤÷ì♥❣
♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ ❝ị♥❣ ♠ët sè ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣✳

❈❤÷ì♥❣ ✷✳ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣
❝õ❛ ❤å →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❧✉➟♥ ✈➠♥ t➟♣ tr✉♥❣ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t ❦➳t q✉↔ ❝õ❛
❜➔✐ ❜→♦ ❬✽❪ ✈➲ ♠ët ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ t➻♠ ✤✐➸♠ ❜➜t
✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ✈æ ❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳
▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝
❚❤→✐ ◆❣✉②➯♥✳ ❚r♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❚r÷í♥❣
✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ✤➸ t→❝ ❣✐↔ ❤å❝ t➟♣✱
♥❣❤✐➯♥ ❝ù✉✳ ❚→❝ ❣✐↔ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❝→❝
t❤➛②✱ ❝ỉ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✲ ❚✐♥✱ tr♦♥❣ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐
❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ✣➦❝ ❜✐➺t✱ t→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝
tỵ✐ ữớ ữớ t t ữợ t→❝ ❣✐↔ ❤♦➔♥
t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳
❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ✤÷đ❝ ỷ ớ ỡ tợ trữớ P
❇➢❝ ◆✐♥❤✱ ❇➢❝ ◆✐♥❤ ✈➔ t➟♣ t❤➸ ❝→❝ t❤➛② ❝æ ❣✐→♦ tr♦♥❣ tê ❚♦→♥✲
❚✐♥ ❝õ❛ ❚r÷í♥❣ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ t❤í✐ ❣✐❛♥ t→❝ ❣✐↔
t❤❛♠ ❣✐❛ ❤å❝ ❝❛♦ ❤å❝✳

❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✵✹ ♥➠♠ ✷✵✶✾
❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥

◆❣✉②➵♥ ❱➠♥ ◆❣❛


✹

❈❤÷ì♥❣ ✶

❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ✤✐➸♠ ❜➜t
✤ë♥❣
❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ởt số tự ỡ ổ
ỗ ❝❤➦t✱ trì♥ ✤➲✉ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t✳ P❤➛♥ t❤ù ừ ữỡ ợ
t t t ở tr➻♥❤ ❜➔② ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ✤✐➸♠
❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥✱ ♣❤÷ì♥❣
♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ ❝ị♥❣ ♠ët sè ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣✳ ◆ë✐ ❞✉♥❣ ❝õ❛
❝❤÷ì♥❣ ✤÷đ❝ tê♥❣ ❤đ♣ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✲❬✽❪✳

✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t
✶✳✶✳✶ ❑❤æ♥❣ ỗ trỡ
E ởt ổ ✈➔ E ∗ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝õ❛

E ✳ ✣➸ ❝❤♦ ✤ì♥ ❣✐↔♥ ✈➔ t❤✉➟♥ t✐➺♥✱ t❛ sû ❞ư♥❣ ❦➼ ❤✐➺✉ . ✤➸ ❝❤➾ ❝❤✉➞♥
tr➯♥ E ✈➔ E ∗
r t sỷ ử t t ữợ ✤➙② ❝õ❛ ❦❤æ♥❣ ❣✐❛♥
❇❛♥❛❝❤ ♣❤↔♥ ①↕✳

▼➺♥❤ ✤➲ ✶✳✶✳✶ ✭①❡♠ ❬✷❪✱ tr❛♥❣ ✹✶✮ ❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳
❑❤✐ ✤â✱ ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿

✭❛✮ E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕✳
✭❜✮ ▼å✐ ❞➣② ❜à ❝❤➦♥ tr♦♥❣ E ✤➲✉ ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tö ②➳✉✳
❙❛✉ ✤➙② ❧➔ ❦❤→✐ ♥✐➺♠ ✈➔ ♠ët sè ❝➜✉ tró❝ ❤➻♥❤ ❤å❝ ❝→❝ ❦❤ỉ♥❣
ữ t ỗ t trỡ ổ ỗ ổ trỡ


✺

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✤÷đ❝ ❣å✐ ❧➔ ỗ t ợ
ồ x, y E, x = y ♠➔ x = 1, y = 1 t❛ ❝â

x+y
< 1.
2

ú ỵ ỏ õ t t ữợ

tữỡ ữỡ s ổ E ữủ ồ ỗ t ồ

x, y SE t❤ä❛ ♠➣♥

x+y
2

= 1✱ s✉② r❛ x = y ❤♦➦❝ ✈ỵ✐ ♠å✐ x, y ∈ SE ✈➔

x = y t❛ ❝â tx + (1 − t)y < 1 ✈ỵ✐ ♠å✐ t ∈ (0, 1)✱ tr♦♥❣ ✤â
SE = {x ∈ E : x = 1}.

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✤÷đ❝ ồ ỗ ồ

> 0 tỗ t δ(ε) > 0 s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ x, y ∈ E ♠➔ x = 1, y =
1, x − y ≥ ε t❛ ❧✉æ♥ ❝â
x+y
≤ 1 − δ(ε).
2
❉➵ t❤➜② r➡♥❣ E ởt ổ ỗ t õ ổ

ỗ t ữủ ổ ú
t ỗ ừ ổ E ữớ t ữ
ổ ỗ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✿

δE (ε) = inf 1 −

x+y
: x ≤ 1, y ≤ 1, x − y .
2

t ổ ỗ ừ ổ ❣✐❛♥ ❇❛♥❛❝❤ E ❧➔ ❤➔♠ sè ①→❝
✤à♥❤✱ ❧✐➯♥ tö❝ ✈➔ t tr [0; 2] ổ E ỗ ❝❤➦t
❦❤✐ ✈➔ ❝❤➾ ❦❤✐ δE (2) = 1✳ ◆❣♦➔✐ r❛✱ ổ E ỗ
δE (ε) > 0, ∀ε > 0✳

▼➺♥❤ ✤➲ ✶✳✶✳✻ ✭①❡♠ ồ ổ ỗ t ❦❤æ♥❣
❣✐❛♥ ♣❤↔♥ ①↕✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✼ ❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥✳

❈❤✉➞♥ tr➯♥ E ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ●➙t❡❛✉① t↕✐ ✤✐➸♠ x ∈ SE ♥➳✉ ✈ỵ✐ ộ

y SE tỗ t ợ
x + ty − x
d
( x + ty )t=0 = lim
.
t→0
dt
t

✭✶✳✶✮


✻

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✽ ❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥✳
❑❤✐ ✤â✿

✭❛✮ ❈❤✉➞♥ tr➯♥ E ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ●➙t❡❛✉① ♥➳✉ ♥â ❦❤↔ ✈✐ ●➙t❡❛✉①
t↕✐ ♠♦✐ x ∈ SE ✳
✭❜✮ ❈❤✉➞♥ tr➯♥ E ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉ ♥➳✉ ✈ỵ✐ ♠♦✐ y ∈ SE
❣✐ỵ✐ ❤↕♥ tỗ t ợ ồ x SE
❈❤✉➞♥ tr➯♥ E ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ♥➳✉ ợ x SE ợ
tỗ t ợ ồ y SE
tr E ữủ ❣å✐ ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ✤➲✉ ♥➳✉ ❣✐ỵ✐ ❤↕♥ ✭✶✳✶✮ tỗ
t ợ ồ x, y SE

ỵ ✶✳✶✳✾ ✭①❡♠ ❬✷❪✮ ❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❑❤✐ ✤â✱ t❛
❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉✿

✭❛✮ ◆➳✉ E ∗ ổ ỗ t t E ổ trì♥✳

✭❜✮ ◆➳✉ E ∗ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ trì♥ t❤➻ E ổ ỗ t

ổ trỡ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ❧➔ ❤➔♠ sè
①→❝ ✤à♥❤ ❜ð✐

ρE (τ ) = sup{2−1 ( x + y + x − y ) − 1 : x = 1, y = τ }.

◆❤➟♥ ①➨t ✶✳✶✳✶✶ ▼ỉ ✤✉♥ trì♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ❧➔ ❤➔♠ sè
①→❝ ✤à♥❤✱ ❧✐➯♥ tö❝ ✈➔ t tr [0; +)

ỵ ữợ t ❜✐➳t ✈➲ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ♠æ ✤✉♥ ❝õ❛ ❦❤æ♥❣
❣✐❛♥ E ợ ổ ỗ ừ E ữủ

ỵ E ổt ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❑❤✐ ✤â t❛
❝â

τε
− δX (ε) : ε ∈ [0, 2] , τ > 0✳
2
τε
∗
− δX
(ε) : ε ∈ [0, 2] , τ > 0✳
✭❜✮ ρE (τ ) = sup
2

✭❛✮ ρE ∗ (τ ) = sup

◆❤➟♥ ①➨t ✶✳✶✳✶✸ ứ ỵ s r
0 (E) =

0 (E )
ε0 (E)
✈➔ ρ0 (E ∗ ) =
,
2
2

tr♦♥❣ ✤â ε0 (E) = sup ε : δE (ε) = 0 , ρ0 (E) = limτ →0 ρEτ(τ ) .


✼

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✹ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✤÷đ❝ ❣å✐ ❧➔ trì♥ ✤➲✉ ♥➳✉
ρE (τ )
= 0.
τ →0
τ
lim

❚ø ◆❤➟♥ ①➨t ✶✳✶✳✶✸✱ t õ ỵ ữợ

ỵ ❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❑❤✐ ✤â t❛
❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉✿

✭❛✮ ◆➳✉ E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ trì♥ t E ổ ỗ
E ổ ỗ t E ❦❤ỉ♥❣ ❣✐❛♥ trì♥ ✤➲✉✳

✶✳✶✳✷ ⑩♥❤ ①↕ ✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✻ ⑩♥❤ ①↕ J : E → 2E ✭♥â✐ ❝❤✉♥❣ ❧➔ ✤❛ trà✮ ①→❝ ✤à♥❤

∗

❜ð✐

Jx = {u ∈ E ∗ : x, u = x

u , u = x },

✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✳

❱➼ ❞ư ✶✳✶✳✶✼ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✱ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝
❧➔ →♥❤ ①↕ ✤ì♥ ✈à I ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✽ ⑩♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ J : E → 2E

∗

❝õ❛ ❦❤ỉ♥❣

❣✐❛♥ ❇❛♥❛❝❤ E ✤÷đ❝ ❣å✐ ❧➔

✭✐✮ ❧✐➯♥ tư❝ ②➳✉ t❤❡♦ ❞➣② ♥➳✉ J ✤ì♥ trà ✈➔ ✈ỵ✐ ♠å✐ ❞➣② {xn } ❤ë✐ tö ②➳✉
✤➳♥ x t❤➻ Jxn ❤ë✐ tö ②➳✉ ✤➳♥ Jx t❤❡♦ tæ♣æ ②➳✉∗ tr♦♥❣ E ∗ ✳
✭✐✐✮ ❧✐➯♥ tư❝ ♠↕♥❤✲②➳✉ t❤❡♦ ❞➣② ♥➳✉ J ✤ì♥ trà ✈➔ ✈ỵ✐ ♠å✐ ❞➣② {xn } ❤ë✐
tö ♠↕♥❤ ✤➳♥ x t❤➻ Jxn ❤ë✐ tư ②➳✉ ✤➳♥ Jx t❤❡♦ tỉ♣ỉ ②➳✉∗ tr♦♥❣ E ∗ ✳

◆❤➟♥ ①➨t ✶✳✶✳✶✾ ❑❤æ♥❣ ❣✐❛♥ lp, 1 < p < ∞ ❝â →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥

t➢❝ ❧✐➯♥ tö❝ ②➳✉ t❤❡♦ ❞➣②✳ ⑩♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥

Lp [a, b], 1 < p < ∞ ❦❤æ♥❣ t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t ♥➔②✳
❚➼♥❤ ✤ì♥ trà ❝õ❛ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ❝â ♠è✐ ❧✐➯♥ ❤➺ ✈ỵ✐ t➼♥❤
❦❤↔ ✈✐ ❝õ❛ ❝❤✉➞♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♥❤÷ ❦❤➥♥❣ ✤à♥❤ tr
ỵ s

ỵ E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈ỵ✐ →♥❤ ①↕ ✤è✐
∗

♥❣➝✉ ❝❤✉➞♥ t➢❝ J : E → 2E ✳ ❑❤✐ ✤â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿


✽

✭✐✮ E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ trì♥❀
✭✐✐✮ J ❧➔ ✤ì♥ trà❀
✭✐✐✐✮ ❈❤✉➞♥ ❝õ❛ E ❧➔ ❦❤↔ ✈✐ ●➙t❡❛✉① ∇ x = x

1

Jx

ú ỵ ũ j →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ✤ì♥

trà✳

✶✳✷ ✣✐➸♠ ❜➜t ✤ë♥❣
✶✳✷✳✶ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶ ❈❤♦ C ❧➔ ♠ët t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥

❇❛♥❛❝❤ E ✳

✭✐✮ ⑩♥❤ ①↕ T : C → E ✤÷đ❝ ❣å✐ ❧➔ L tử st tỗ
t số L ≥ 0 s❛♦ ❝❤♦

Tx − Ty ≤ L x − y ,

✭✶✳✷✮

∀x, y ∈ C.

✭✐✐✮ ❚r♦♥❣ ✭✶✳✷✮✱ ♥➳✉ L ∈ [0, 1) t❤➻ T ✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ ❝♦❀ ♥➳✉ L = 1
t❤➻ T ✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ ổ
ỵ (T ) := {x C : T x = x} ❧➔ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤
①↕ T ✳ ❚❛ ❝â ❦➳t q✉↔ s❛✉ ✈➲ t➼♥❤ t ừ t (T )

ỵ C t rộ ỗ õ tr
ổ ỗ t E T : C E ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ❑❤✐
✤â t➟♣ ❋✐①(T ) ❧➔ t ỗ õ

A : C

E ữủ ồ

t ợ ♠é✐ x, y ∈ D(A)✱ t➟♣ ①→❝ ✤à♥❤ ❝õ❛ →♥❤ A tỗ t

j(x y) J(x y) s❛♦ ❝❤♦
Ax − Ay, j(x − y) ≤ x − y

2

− λ x − y − (Ax − Ay)

✭✶✳✸✮

2

✈ỵ✐ ♠é✐ λ ∈ (0, 1)✳ ❚r♦♥❣ ✭✶✳✸✮✱ ♥➳✉ λ = 0 t❤➻ T ✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ ❣✐↔
❝♦✳
❚❛ t❤➜② ✭✶✳✸✮ ❝â t❤➸ ✤÷đ❝ ✈✐➳t ❧↕✐ ♥❤÷ s❛✉

(I − A)x − (I − A)y, j(x − y) ≥ λ (I − A)x − (I − A)y
− λ x − y − (Ax − Ay) 2 .

2

✭✶✳✹✮




ỵ C t ủ tt ①↕ ❝♦ tr➯♥ C ✱ tù❝ ❧➔

πC = {f |f : C → C ❧➔ →♥❤ ①↕ ❝♦}.
▼ët tr♦♥❣ ❝→❝ ❝→❝❤ ♥❣❤✐➯♥ ❝ù✉ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ❧➔ sû ❞ư♥❣ →♥❤ ①↕ ❝♦
✤➸ ①➜♣ ①➾ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳ ❈ư t❤➸✱ t❛ ❧➜② t ∈ (0, 1) ✈➔ ✤à♥❤ ♥❣❤➽❛ →♥❤
①↕ ❝♦ Tt : C → C ❜ð✐

Tt x = tf (x) + (1 − t)T x,

∀x ∈ C

✭✶✳✺✮

tr♦♥❣ ✤â f ∈ πC ✳ ◆❣✉②➯♥ ❧➼ →♥❤ ①↕ ❝♦ ❝õ❛ ❇❛♥❛❝❤ ❜↔♦ ✤↔♠ r➡♥❣ Tt ❝â
❞✉② ♥❤➜t ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ xt tr♦♥❣ C ✳

❇➔✐ t♦→♥✳ ❈❤♦ T : C → C ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tø t➟♣ ❝♦♥ ỗ õ
rộ C ừ ổ E ❝❤➼♥❤ ♥â ✈ỵ✐ ❋✐①(T ) = ∅✳
❚➻♠ ♣❤➛♥ tû x∗ (T )

ú ỵ T

tr➯♥ C t❤➻ ❞➣② ❧➦♣ P✐❝❛r❞ ①→❝ ✤à♥❤

❜ð✐ x0 ∈ C, xn+1 = T (xn ) ❤ë✐ tö ♠↕♥❤ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❝õ❛

T ✳ ❚✉② ♥❤✐➯♥ ✤✐➲✉ ♥➔② ❧➔ ❦❤ỉ♥❣ ❝á♥ ✤ó♥❣ ✤è✐ ✈ỵ✐ ❧ỵ♣ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳

✶✳✷✳✷ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤
①↕ ❦❤ỉ♥❣ ❣✐➣♥
P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥
◆➠♠ ✶✾✺✸✱ ❲✳❘✳ ▼❛♥♥ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ✤➲ ①✉➜t ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣

xn+1 = αn xn + (1 − αn )T (xn ), x1 ∈ C,

n ≥ 1.

✭✶✳✻✮

➷♥❣ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣✱ ♥➳✉ ❞➣② {αn } ✤÷đ❝ ❝❤å♥ t❤ä❛ ♠➣♥

∞

αn (1 − αn ) = ∞

✭▲✶✮

n=1

t❤➻ ❞➣② {xn } ①→❝ ✤à♥❤ ❜ð✐ ✭✶✳✻✮ s➩ ❤ë✐ tö ②➳✉ ✈➲ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛
→♥❤ ①↕ T ✱ ð ✤➙② T : C → C ❧➔ ♠ët →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tø t➟♣ C
ỗ õ rộ ừ ổ rt H õ ú ỵ
r tr trữớ ủ H ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈æ ❤↕♥ ❝❤✐➲✉ t❤➻
❞➣② ❧➦♣ ✭✶✳✻✮ ❝❤➾ ❤ë✐ tư ②➳✉ ♠➔ ❦❤ỉ♥❣ ❤ë✐ tư ♠↕♥❤✳ r trữớ ủ

n = (0, 1) ợ ồ n t❤➻ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥ ✭✶✳✻✮ trð t❤➔♥❤
♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛♥♦s❡❧s❦✐✐✳


✶✵

◆➠♠ ✶✾✼✾✱ ❙✳ ❘❡✐❝❤ ✤➣ ♠ð rë♥❣ ❦➳t q✉↔ ❝õ❛ ▼❛♥♥ ❝❤♦ tr÷í♥❣ ❤đ♣

T : C → C ❧➔ ♠ët tứ ởt t C rộ ỗ õ ừ
ởt ổ ỗ ợ ✈✐ ❋r➨❝❤❡t ✈➔♦ C ✈➔ ỉ♥❣
❝ơ♥❣ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ ♥➳✉ ❞➣② {αn } ✤÷đ❝ ❝❤å♥ t❤ä❛ ♠➣♥ ✤✐➲✉
❦✐➺♥ limn→∞ αn = 0 t❤➻ ❞➣② {xn } s➩ ❤ë✐ tö ②➳✉ ✈➲ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣
❝õ❛ →♥❤ ①↕ T ✳
◆➠♠ ✷✵✵✸✱ ❑✳ ◆❛❦❛❥♦ ✈➔ ❲✳ ❚❛❦❛❤❛s❤✐ ✤➣ ✤➲ ①✉➜t ♠ët ❝↔✐ t✐➳♥ ❝õ❛
♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✭✶✳✻✮ ❝❤♦ tr÷í♥❣ ❤đ♣ T ❧➔ ♠ët →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣
❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❞↕♥❣ s❛✉✿



x0 ∈ C,







y = αn xn + (1 − αn )T (xn ),

 n
Cn = {z ∈ C : yn − z ≤ xn − z },





Qn = {z ∈ C : xn − z, x0 − xn ≥ 0},




x
n+1 = PCn ∩Qn (x0 ),

✭✶✳✼✮

tr♦♥❣ ✤â PK ❧➔ ♣❤➨♣ tr tứ H ởt t ỗ õ K

❝õ❛ H ✳ ❍å ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ ♥➳✉ ❞➣② {αn } t❤ä❛ ♠➣♥ ✤✐➲✉
❦✐➺♥ {αn } ⊆ [0, 1) t❤➻ ❞➣② ❧➦♣ {xn } ①→❝ ✤à♥❤ ❜ð✐ ✭✶✳✼✮ ❤ë✐ tö ♠↕♥❤ ✈➲

P❋(T ) (x0 )✳
◆➠♠ ✷✵✵✺✱ ❑✐♠ ✈➔ ❳✉ ✤➣ ♠ð rë♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥ ✭✶✳✻✮ tr➯♥
❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❞↕♥❣✿



x ∈ C,

 0
yn = αn xn + (1 − αn )T (xn ),



x
= β u + (1 )T (y ).n 0.
n+1

n

n



n

ỵ C ởt t ỗ õ ừ ổ ❣✐❛♥ ❇❛♥❛❝❤
E ✳ ❈❤♦ T : C → C ❧➔ ♠ët →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ✈ỵ✐ ❋✐①(T ) = ∅✳ ❱ỵ✐ u ∈ C

✈➔ ❝→❝ ❞➣② sè {αn }, {βn } ⊂ (0, 1) t❤ä❛ ♠➣♥
✭✐✮ αn → 0, βn → 0❀
✭✐✐✮

∞
n=0 αn

✭✐✐✐✮

∞
n=0 |αn

= ∞,

∞
n=0 βn

− αn+1 | < ∞,

= ∞❀
∞
n=0 |βn

− βn+1 | < ∞✳

❑❤✐ ✤â✱ ❞➣② ❧➦♣ {xn } ①→❝ ✤à♥❤ ❜ð✐ ✭✶✳✽✮ ❤ë✐ tö ♠↕♥❤ ✈➲ ♠ët ✤✐➸♠ ❜➜t
✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ✳

✶✶

P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❍❛❧♣❡r♥
P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❝õ❛ ❇✳ ❍❛❧♣❡r♥ ❬✹❪ ✤÷đ❝ ✤➲ ①✉➜t ♥➠♠ ✶✾✻✼ ❞↕♥❣✿

xn+1 = αn u + (1 − αn )T (xn ),

n ≥ 0,

✭✶✳✾✮

tr♦♥❣ ✤â u, x0 ∈ C, {xn } ⊂ (0, 1) ✈➔ T ởt ổ tứ t
ỗ õ C ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✈➔♦ C ✳ ➷♥❣ ✤➣ ❝❤ù♥❣ ♠✐♥❤
♥➳✉ αn = n−α , α ∈ (0, 1) t❤➻ ❞➣② {xn } ①→❝ ✤à♥❤ ❜ð✐ ✭✶✳✾✮ s➩ ❤ë✐ tö ♠↕♥❤
✈➲ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ✳
◆➠♠ ✶✾✼✼✱ P✳▲✳ ▲✐♦♥s ✤➣ ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tö ♠↕♥❤ ❝õ❛ ❞➣② {xn }
✈➲ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ♥➳✉ ❞➣② sè {αn }
t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿

(C1)

lim αn = 0,

n→∞
∞

(C2)

αn = +∞,
n=1

(C3)

|αn+1 − αn |
= 0.
2
n→∞
αn+1
lim

❚✉② ♥❤✐➯♥✱ ✈ỵ✐ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❍❛❧♣❡r♥ ✈➔ ▲✐♦♥s t❤➻ ❞➣② ❝❤➼♥❤ t➢❝
1
αn =
❧↕✐ ❜à ❧♦↕✐ trø✳ ◆➠♠ ✶✾✾✷✱ ❘✳ ❲✐tt♠❛♥♥ ✤➣ ♠ð rë♥❣ ❦➳t q✉↔
n+1
❝õ❛ ❍❛❧♣❡r♥ ✈➔ ❣✐↔✐ q✉②➳t ✤÷đ❝ ✈➜♥ ✤➲ tr➯♥✳ ➷♥❣ ✤➣ ❝❤➾ r❛ r➡♥❣ ♥➳✉ ❞➣②
sè {αn } t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ (C1), (C2) ✈➔ ✤✐➲✉ ❦✐➺♥
∞

|αn+1 − αn | < ∞,

(C4)
n=1

t❤➻ ❞➣② ❧➦♣ {xn } ①→❝ ✤à♥❤ ❜ð✐ ✭✶✳✾✮ ❤ë✐ tö ♠↕♥❤ ✈➲ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣
❝õ❛ T ✳ ❙ü ❤ë✐ tö ♠↕♥❤ ❝õ❛ ❞➣② ❧➦♣ ✭✶✳✾✮ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕
❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❝ơ♥❣ ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❝❤ù♥❣
♠✐♥❤✳ ❙✳ ❘❡✐❝❤ ✤➣ ❝❤➾ r❛ sü ❤ë✐ tö ♠↕♥❤ ❝õ❛ ❞➣② ❧➦♣ ✭✶✳✾✮ ❦❤✐ ❞➣② sè

{αn } t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ (C1), (C2) ✈➔ ✤✐➲✉ ❦✐➺♥

(C5) {αn } ❧➔ ♠ët ❞➣②
t ữủ ỵ sü ❤ë✐ tö ♠↕♥❤ ❝õ❛ ❞➣② ❧➦♣
✭✶✳✾✮ ♥➳✉ ❞➣② {αn } t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ (C1), (C2) ✈➔ ✤✐➲✉ ❦✐➺♥

(C6)

lim

n→∞

αn − αn+1
= 0.
αn+1


✶✷

❚✉② ♥❤✐➯♥✱ ❧✐➺✉ r➡♥❣ ❞➣② sè {αn } t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ (C1), (C2) ❝â
❧➔ ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ ✤↔♠ ❜↔♦ sü ❤ë✐ tö ♠↕♥❤ ❝õ❛ ❞➣② ❧➦♣ ✭✶✳✾✮ ✈➲ ✤✐➸♠
❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ T ❤❛② ❦❤æ♥❣❄ ✣➙② ✈➝♥ ❝á♥ ❧➔ ♠ët ❝➙✉
❤ä✐ ♠ð✳
▼ët ♣❤➛♥ ❝➙✉ ❤ä✐ ♥➔② ✤➣ ✤÷đ❝ ❣✐↔✐ q✉②➳t ♠ët ❝→❝❤ ✤ë❝ ❧➟♣ ❜ð✐ ❝→❝
t→❝ ❣✐↔ ❈✳❊✳ ❈❤✐❞✉♠❡ ✈➔ ❈✳❖✳ ❈❤✐❞✉♠❡ ✈➔ ❚✳ ❙✉③✉❦✐✳ ❍å ✤➣ ①→❝ ✤à♥❤
❞➣② ❧➦♣ {xn } ❜ð✐

xn+1 = αn u + (1 − αn )((1 − δ)xn + δT (xn )),

✭✶✳✶✵✮

tr♦♥❣ ✤â δ ∈ (0, 1) ✈➔ t❤✉ ✤÷đ❝ sü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ❞➣② ❧➦♣ ✭✶✳✶✵✮ ❦❤✐

❞➣② sè {αn } t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ (C1), (C2)✳
◆➠♠ ✷✵✵✽✱ ▲✳●✳ ❍✉ ✤➣ ♠ð rë♥❣ ❦➳t q✉↔ ❝õ❛ ❍❛❧♣❡r♥ ✈➔ ❝õ❛ ▼❛♥♥
❞↕♥❣

xn+1 = αn u + βn xn + γn T (xn ),

n ≥ 0,

✭✶✳✶✶✮

tr♦♥❣ ✤â {αn }, {βn } ✈➔ {γn } ❧➔ ❝→❝ ❞➣② sè ♥➡♠ tr♦♥❣ ❦❤♦↔♥❣ (0, 1)✳ ❚❛
t❤➜② r➡♥❣ ♥➳✉ βn = 0 ✈ỵ✐ ♠å✐ n t❤➻ ❞➣② ❧➦♣ ✭✶✳✶✶✮ trð t❤➔♥❤ ❞➣② ❧➦♣
✭✶✳✾✮ ✈➔ ♥➳✉ αn = 0 ✈ỵ✐ ♠å✐ n t❤➻ ❞➣② ❧➦♣ ✭✶✳✶✶✮ trð t❤➔♥❤ ❞➣② ❧➦♣ ✭✶✳✻✮✳
❑➳t q✉↔ ❝õ❛ ▲✳●✳ ữủ ỵ s

ỵ C t rộ ỗ õ ừ ❦❤ỉ♥❣ ❣✐❛♥
❇❛♥❛❝❤ E ✈ỵ✐ ❝❤✉➞♥ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉✳ ❈❤♦ T : C → C ❧➔ ♠ët →♥❤

①↕ ❦❤æ♥❣ ❣✐➣♥ ✈ỵ✐ ❋✐①(T ) = ∅✳ ●✐↔ sû {zt } ❤ë✐ tö ♠↕♥❤ ✈➲ ♠ët ✤✐➸♠
❜➜t ✤ë♥❣ z ❝õ❛ T ✱ tr♦♥❣ ✤â {zt } ❧➔ ♣❤➛♥ tû ❞✉② ♥❤➜t ❝õ❛ C t❤ä❛ ♠➣♥

zt = tu + (1 − t)T (zt ), t ∈ (0, 1)✳ ●✐↔ sû {αn }, {βn } ✈➔ {γn } ❧➔ ❝→❝ ❞➣②
sè ♥➡♠ tr♦♥❣ ❦❤♦↔♥❣ (0, 1) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ (C1) ✈➔ (C2)✳ ❑❤✐
✤â✱ ✈ỵ✐ x0 ∈ C ✱ ❞➣② {xn } ①→❝ ✤à♥❤ ❜ð✐ ✭✶✳✶✶✮ ❤ë✐ tö ♠↕♥❤ ✈➲ ♠ët ✤✐➸♠
❜➜t ✤ë♥❣ ❝õ❛ T ✳

P❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ♠➲♠
◆➠♠ ✷✵✵✵✱ ▼♦✉❞❛❢✐ ❬✻❪ ✤➣ ✤➲ ①✉➜t ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ♠➲♠✱ ✤➸ t➻♠
✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ ✤➣
❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❦➳t q✉↔ s❛✉✿

✭✶✮ ❉➣② {xn } ⊂ C ①→❝ ✤à♥❤ ❜ð✐✿

x0 ∈ C, xn =

1
εn
T xn +
f (xn ),
1 + εn
1 + εn

∀n ≥ 0,


✶✸

❤ë✐ tö ♠↕♥❤ ✈➲ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✿

x ∈ ❋✐①(T ) s❛♦ ❝❤♦ (I − f )(x), x − x ≤ 0,

∀x ∈ ❋✐①(T ),

tr♦♥❣ ✤â {εn } ❧➔ ♠ët ❞➣② sè ❞÷ì♥❣ ❤ë✐ tư ✈➲ 0✳
✭✷✮ ❱ỵ✐ ♠é✐ ♣❤➙♥ tû ❜❛♥ ✤➛✉ z0 ∈ C ✱ ①→❝ ✤à♥❤ ❞➣② {zn } ⊂ C ❜ð✐✿
1
εn
zn+1 =
T zn +
f (zn ), ∀n ≥ 0,
1 + εn

1 + εn
∞
n=1 εn

◆➳✉ limn→∞ εn = 0,

= +∞ ✈➔ limn→∞

1
εn+1

−

1
εn

= 0 t❤➻

{zn } ❤ë✐ tö ♠↕♥❤ ✈➲ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✿
x ∈ ❋✐①(T ) s❛♦ ❝❤♦ (I − f )(x), x − x ≤ 0,

∀x ∈ ❋✐①(T ),

ð ✤➙② f : C → C ❧➔ ởt trữợ ợ số c ∈ [0, 1)✳
❚ù❝ ❧➔

f (x) − f (y) ≤ c x y ,

x, y C.

ú ỵ ❑❤✐ f (x) = u ✈ỵ✐ ♠å✐ x ∈ C ✱ t❤➻ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ♠➲♠
❝õ❛ ▼♦✉❞❛❢✐ trð ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❝õ❛ ❍❛❧♣❡r♥✳

P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛
P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ ✤÷đ❝ ✤➲ ①✉➜t ❜ð✐ ❙✳ ■s❤✐❦❛✇❛ ❬✺❪ ✈➔♦ ♥➠♠
✶✾✼✹✳ ợ ữỡ t {xn } ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐



x ∈ C,

 0
✭✶✳✶✷✮
yn = βn xn + (1 − βn )T (xn ),



x
= α u + (1 − α )T (y ), n ≥ 1
n+1

n

n

n

tr♦♥❣ ✤â {αn } ✈➔ {βn } ❧➔ ❝→❝ ❞➣② sè t❤ü❝ tr [0, 1]

ú ỵ r trữớ ủ n = 1 ợ ồ n t ữỡ

s trð t❤➔♥❤ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥ ✭✶✳✻✮✳ ❚✉② ♥❤✐➯♥✱ ❙✳❆✳
▼✉t❛♥❣❛❞✉r❛ ✈➔ ❈✳❊✳ ❈❤✐❞✉♠❡ ✤➣ ①➙② ❞ü♥❣ ♠ët ✈➼ ❞ö ❝❤♦ tr÷í♥❣ ❤đ♣

T ❧➔ ♠ët →♥❤ ①↕ ▲✐♣s❝❤✐t③ ❣✐↔ ❝♦ t❤➻ ❞➣② ❧➦♣ ■s❤✐❦❛✇❛ ❤ë✐ tö ✈➲ ♠ët
✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ♥❤÷♥❣ ❞➣② ❧➦♣ ▼❛♥♥ ❧↕✐ ❦❤ỉ♥❣ ❤ë✐ tư✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✾ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✤÷đ❝ ❣å✐ ❧➔ t❤ä❛ ♠➣♥ ✤✐➲✉

❦✐➺♥ ❝õ❛ ❖♣✐❛❧ ♥➳✉ ✈ỵ✐ ❜➜t ❦➻ ❞➣② {xn } tr♦♥❣ E ❤ë✐ tö ②➳✉ ✈➲ x ∈ E t❤➻

lim inf xn − x < lim inf xn − y ,
n→∞

n→∞

∀y ∈ E, y = x.


✶✹

❙ü ❤ë✐ tö ②➳✉ ❝õ❛ ❞➣② ❧➦♣ ■s❤✐❦❛✇❛ ✈➲ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕
❦❤æ♥❣ ❣✐➣♥ T tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❝❤ù♥❣
♠✐♥❤ ❜ð✐ ❑✳❑✳

ỵ E

ởt ổ ỗ tọ

ừ ❝â ❝❤✉➞♥ ❦❤↔ ✈✐ ❋r➨❝❤❡t✱ C ❧➔ ♠ët t➟♣ ❝♦♥

rộ ỗ õ ừ E T : C → C ❧➔ ♠ët →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✱

{αn } ✈➔ {βn } ❧➔ ❝→❝ ❞➣② sè tr♦♥❣ ✤♦↕♥ [0, 1] s❛♦ ❝❤♦
∞,

∞
n=1 βn (1

∞
n=1 αn (1 − αn )

=

− βn ) < ∞ ✈➔ lim supn→∞ βn < 1✳ ❑❤✐ ✤â✱ ❞➣② {xn }

①→❝ ✤à♥❤ ❜ð✐ ✭✶✳✶✷✮ ❤ë✐ tö ②➳✉ ✈➲ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ✳
◆➠♠ ✷✵✵✺✱ ◆✳ ❙❤❛❤③❛❞ ✤➣ ❝↔✐ t✐➳♥ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ s trữớ
ủ C ởt t ỗ õ ❝♦ rót ❦❤ỉ♥❣ ❣✐➣♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤

E ❞↕♥❣
xn+1 = P ((1 − αn )xn + αn T P ((1 − αn )xn + βn T (xn ))),

n ≥ 1, ✭✶✳✶✸✮

tr♦♥❣ ✤â x1 ∈ C ✈➔ {αn }, {βn } ❧➔ ❝→❝ ❞➣② sè t❤ü❝ tr♦♥❣ ✤♦↕♥ [ε, 1 − ε]✱

ε ∈ (0, 1)✳ ◆✳ ❙❤❛❤③❛❞ ✤➣ ❝❤➾ r❛ r➡♥❣ ♥➳✉ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ E ∗ ❝õ❛
❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ❝â t➼♥❤ ❝❤➜t ❑❛❞❡❝✕❑❧❡❡ t❤➻ ❞➣② {xn } ①→❝ ✤à♥❤
❜ð✐ ✭✶✳✶✸✮ ❤ë✐ tö ②➳✉ ✈➲ ♠ët ♣❤➛♥ tû x∗ ∈ ❋✐①(T )✳ ❍ì♥ ♥ú❛✱ ♥➳✉ →♥❤ ①↕
❦❤ỉ♥❣ ❣✐➣♥ T t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥

x − T x ≥ f (d(x, ❋✐①(T ))),

∀x ∈ C,

✭✶✳✶✹✮

tr♦♥❣ ✤â f : [0, ∞) → [0, ∞) t❤ä❛ ♠➣♥ f (0) = 0 ✈➔ f (r) > 0 ✈ỵ✐ ♠å✐

r > 0 t❤➻ ❞➣② ❧➦♣ ✭✶✳✶✸✮ ❤ë✐ tö ♠↕♥❤ ✈➲ ♠ët ♣❤➛♥ tû x∗ ∈ ❋✐①(T )✳


✶✺

❈❤÷ì♥❣ ✷

P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ t➻♠
✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ồ
ổ
ữỡ ợ t t t ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å
→♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✱ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❜➔✐ t♦→♥
♥➔②✳ ❚r➻♥❤ ❜➔② ♠ët ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ t➻♠ ✤✐➸♠ ❜➜t
✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ✈æ ❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳ ◆ë✐
❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ ✈✐➳t tr➯♥ ❝ì sð ❜➔✐ ❜→♦ ❬✽❪ ✈➔ ♠ët sè t➔✐ ❧✐➺✉
✤÷đ❝ tr➼❝❤ ❞➝♥ tr♦♥❣ ✤â✳

✷✳✶ ✣✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å →♥❤ ①↕ ❦❤æ♥❣
❣✐➣♥
✷✳✶✳✶ ✣✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥
❚❛ ❜✐➳t r➡♥❣ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ T tr ổ

ỗ t E rộ t ởt t ỗ õ
õ t t ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝ →♥❤ ①↕
❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ❧➔ ♠ët trữớ ủ t ừ
t ỗ ờ t✐➳♥❣ s❛✉✿
❳→❝ ✤à♥❤ ♣❤➛♥ tû x∗ ∈ C = ∩N
i=1 Ci = ∅✱
tr♦♥❣ ✤â Ci , i = 1, 2, . . . , N t ỗ õ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤

E✳


✶✻

❚r♦♥❣ tr÷í♥❣ ❤đ♣ E ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H t t
ỗ tr tữỡ ữỡ ợ t♦→♥ t➻♠ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣
❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ Ti , i = 1, 2, . . . , N, ✈ỵ✐ Ti ❧➔
❝→❝ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tø H ❧➯♥ Ci ✳ ❉♦ ✤â✱ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♣❤÷ì♥❣
♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝
→♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✈➔ tê♥❣ q✉→t ❤ì♥ ❧➔ tr➯♥
❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❧➔ ♠ët ♥↔② s✐♥❤ t➜t ②➳✉✳

P❤→t ❜✐➸✉ ❜➔✐ t♦→♥
❚➻♠ ♠ët ♣❤➛♥ tû x∗ ∈ S = ∩N
i=1 ❋✐①(Ti ) = ∅,

✭✷✳✶✮

tr♦♥❣ ✤â Ti : E → E, i = 1, 2, . . . , N ❧➔ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tø ❦❤æ♥❣
❣✐❛♥ ❇❛♥❛❝❤ E õ

ú ỵ t õ ♥❤✐➲✉ ❜✐➳♥ t❤➸ ❦❤→❝ ♥❤❛✉✱ ❝❤➥♥❣ ❤↕♥

♥❤÷✿ Ti : C → C, Ti : Ci → Ci ❤❛② T : Ci → E tr♦♥❣ ✤â C, Ci , i =

1, 2, . . . , N ❧➔ ❝→❝ t➟♣ ❝♦♥ ỗ õ ừ E
ỵ ữợ sỹ tỗ t t ở ừ
ồ ổ

ỵ E

ởt ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ K ❧➔ ♠ët t➟♣ ❝♦♥

❦❤→❝ ré♥❣✱ ❝♦♠♣❛❝t ỗ ừ E F ởt ồ ❣✐❛♦ ❤♦→♥ ❤ú✉ ❤↕♥
❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tø K ✈➔ K ✱ t❤➻ ❤å F ❝â ➼t ♥❤➜t ♠ët t

ở tr K

ỵ E ởt ổ ỗ K ởt t
rộ ỗ õ ừ E ✳ ❈❤♦ {Tλ } ❧➔ ♠ët ❤å ❣✐❛♦
❤♦→♥ ❤ú✉ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tø K ✈➔ K ✱ ❦❤✐ ✤â ❤å {Tλ } ❝â ➼t
♥❤➜t ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ tr♦♥❣ K ✳

✷✳✶✳✷ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛
❤å →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥
P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❍❛❧♣❡r♥ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët
❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥

◆➠♠ ✶✾✾✻✱ ❍✳ ❍✳ ❇❛✉s❝❤❦❡ ❬✸❪ ✤➣ ♠ð rë♥❣ ❦➳t q✉↔ ❝õ❛ ❘✳ ❲✐tt♠❛♥♥

❝❤♦ ❜➔✐ t♦→♥ ①→❝ ✤à♥❤ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥

❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳




ỵ H ởt ổ rt C
t rộ ỗ ✈➔ ✤â♥❣ ❝õ❛ H ✳ ❈❤♦ {T1 , . . . , Tr } ❧➔ ♠ët ❤å ❤ú✉
❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tø C ✈➔♦ C ✈ỵ✐ F = ∩ri=1 ❋✐①(Ti ) = ∅ ✈➔

F = ❋✐①(Tr Tr−1 . . . T1 ) = ❋✐①(T1 Tr . . . T2 ) = · · · = ❋✐①(Tr−1 . . . T1 Tr ).
❈❤♦ {λn } ❧➔ ♠ët ❞➣② sè t❤ü❝ tr♦♥❣ ✤♦↕♥ [0, 1) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥
∞

λn = ∞,

lim λn = 0,

n→∞

∞

n=0

|λn+1 − λn | < ∞.
n=0

◆➳✉ y, x0 ∈ C ✱ ❞➣② {xn } ①→❝ ✤à♥❤ ❜ð✐

xn+1 = λn y + (1 − λn )Tn+1 xn ,
tr♦♥❣ ✤â Tn = Tn(

mod r) ✱

n ≥ 0,

✭✷✳✷✮

t❤➻ {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ PF u✳

◆➠♠ ✷✵✵✸✱ ❏✳ ●✳ ❖✬ ❍❛r❛✱ P✳ P✐❧❧❛② ✈➔ ❍✳ ❑✳ ❳✉ ✤➣ t❤❛② ✤✐➲✉ ❦✐➺♥
λn
∞
|λ
−
λ
|
<
∞
❜ð✐
✤✐➲✉
❦✐➺♥
lim
= 1 ✈➔ ❝ơ♥❣ t❤✉ ✤÷đ❝
n+1
n
n→∞
n=0
λn+r
sü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ❞➣② ❧➦♣ ✈➲ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ T1 , . . . , Tr

ỵ ❈❤♦ H ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ ❝❤♦ C t

rộ ỗ õ ừ H ❈❤♦ {T1 , . . . , Tr } ❧➔ ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝
→♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tø C ✈➔♦ C ✈ỵ✐ F = ∩ri=1 ❋✐①(Ti ) = ∅ ✈➔

F = ❋✐①(Tr Tr−1 . . . T1 ) = ❋✐①(T1 Tr . . . T2 ) = · · · = ❋✐①(Tr−1 . . . T1 Tr ).
❈❤♦ {λn } ❧➔ ♠ët ❞➣② sè t❤ü❝ tr♦♥❣ ✤♦↕♥ [0, 1) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥
∞

λn = ∞,

lim λn = 0,

n→∞

n=0

λn
= 1.
n→∞ λn+r
lim

◆➳✉ y, x0 ∈ C ✱ ❞➣② {xn } ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✷✮ t❤➻ {xn } ❤ë✐ tö ♠↕♥❤ ✈➲

PF u✳
◆➠♠ ✷✵✵✺✱ ❏✳ ❙✳ ❏✉♥❣ ✤➣ ♠ð rë♥❣ ❦➳t q✉↔ ❝õ❛ ❏✳ ●✳ ❖✬ ❍❛r❛ tr♦♥❣
❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ➷♥❣ ự ỵ s

ỵ E ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ trì♥ ✤➲✉ ✈ỵ✐ →♥❤ ①↕
✤è✐ ♥❣➝✉ j : E → E ∗ ❧✐➯♥ tö❝ ②➳✉ t❤❡♦ ❞➣② ✈➔ ❝❤♦ C ❧➔ ♠ët t➟♣ ❝♦♥

❦❤→❝ ré♥❣✱ ỗ õ ừ E T1 , . . . , TN tø C ✈➔♦ ❝❤➼♥❤ ♥â ✈ỵ✐

F = ∩N
i=1 ❋✐①(Ti ) = ∅ ✈➔
F = ❋✐①(Tr Tr−1 . . . T1 ) = ❋✐①(T1 Tr . . . T2 ) = · · · = ❋✐①(Tr−1 . . . T1 Tr ).


✶✽

❈❤♦ {λn } ❧➔ ♠ët ❞➣② sè t❤ü❝ tr♦♥❣ ✤♦↕♥ [0, 1) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥
∞
n→∞

λn
= 1.
n→∞ λn+r

λn = ∞,

lim λn = 0,

lim

n=0

◆➳✉ y, x0 ∈ C ✱ ❞➣② {xn } ①→❝ ✤à♥❤ ❜ð✐

xn+1 = λn y + (1 − λn )Tn+1 xn ,
tr♦♥❣ ✤â Tn = Tn(

mod r) ✱

n ≥ 0,

✭✷✳✸✮

t❤➻ {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ QF u✱ ð ✤➙② QF ❧➔ ♠ët

❝♦ rót ❦❤ỉ♥❣ ❣✐➣♥ t❤❡♦ t✐❛ tø C ❧➯♥ F ✳
◆➠♠ ✷✵✵✼✱ ❝→❝ t→❝ ❣✐↔ ❙✳ ❙✳ ❈❤❛♥❣✱ ❏✳ ❈✳ ❨❛♦✱ ❏✳ ❑✳ ❑✐♠ ✈➔ ▲✳ ❨❛♥❣
✤➣ ♠ð rë♥❣ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❇❛✉s❝❤❦❡ ❬✸❪ ✈➔ ❖✬ ❍❛r❛ ✤➸ ❣✐↔✐ q✉②➳t ❜➔✐
t♦→♥ ✭✷✳✶✮✱ ❤å ✤➣ ✤➲ ①✉➜t ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣

xn+1 = P (αn+1 f (xn ) + (1 − αn+1 )Tn+1 xn ),

n ≥ 0,

tr♦♥❣ ✤â x0 ∈ E, f : C → C ❧➔ ởt trữợ Tn = Tn(


mod N )

✈➔ P ❧➔ ♠ët ❝♦ rót ❦❤ỉ♥❣ ❣✐➣♥ t❤❡♦ t✐❛ tứ E C

ỵ E ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕ ✈ỵ✐ →♥❤ ①↕
✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ j ❧✐➯♥ tö❝ ②➳✉ t❤❡♦ ❞➣② tø E E K ởt

t ỗ ✤â♥❣ ✈➔ ❝♦ rót ❦❤ỉ♥❣ ❣✐➣♥ t❤❡♦ t✐❛ ❝õ❛ E ✈ỵ✐ P ❧➔ →♥❤ ①↕
❝♦ rót ❦❤ỉ♥❣ ❣✐➣♥ t❤❡♦ t✐❛ tø E ❧➯♥ K ✳ ❈❤♦ f : K → K ❧➔ →♥❤ ①↕ ❝♦

✈ỵ✐ ❤➡♥❣ sè ❝♦ ❧➔ β ∈ (0, 1) ✈➔ Ti : E → E, i = 1, 2, . . . , N ❧➔ ❝→❝ →♥❤ ①↕
❦❤æ♥❣ ❣✐➣♥ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿
✭✐✮ ∩N
i=1 (❋✐①(Ti ) ∩ K) = ∅✱
✭✐✐✮ ∩N
i=1 ❋(Ti ) = ❋✐①(TN TN −1 . . . T1 ) = · · · = ❋✐①(TN −1 . . . T1 TN ) =
❋✐①(S)✱ ✈ỵ✐ S = TN TN −1 . . . T1 ✱

✐✐✐✮ →♥❤ ①↕ S : K → E t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✐♥✇❛r❞ ②➳✉✳
❱ỵ✐ x0 ∈ K ❜➜t ❦ý✱ {xn } ❧➔ ❞➣② ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✹✮✳ ◆➳✉ ❝→❝ ✤✐➲✉
❦✐➺♥ s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥
✭❛✮ limn→∞ αn = 0,
✭❜✮

∞
n=0 |αn+1

∞
n=0 αn

= ∞✱

− αn | < ∞ ❤♦➦❝ limn→∞

αn
= 1✱
αn+r

t❤➻ ❞➣② {xn } ❤ë✐ tư ♠↕♥❤ tỵ✐ ♠ët ✤✐➸♠ p ∈ ∩N
i=1 (❋✐①(Ti ) ∩ K)✱ ✤â ❝ô♥❣

❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥

p − f (p), j(p − u) ≤ 0, ∀u ∈ ∩N
i=1 (❋✐①(Ti ) ∩ K).




ú ỵ r trữớ ủ E ởt ổ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ T1, . . . , TN
❧➔ ❝→❝ ổ tứ t ỗ õ C ❝õ❛ E ✈➔♦ ❝❤➼♥❤ ♥â ✈➔

f : C → C t❤ä❛ ♠➣♥ f (x) = u, ∀x ∈ C t❤➻ ❞➣② ❧➦♣ ✭✷✳✹✮ ❝❤➼♥❤ ❧➔ ❦➳t
q✉↔ ❝õ❛ ❍✳ ❍✳ ❇❛✉s❝❤❦❡ r

ú ỵ K ởt t ỗ õ rộ ừ ổ

E ✱ ✈ỵ✐ ♠é✐ x ∈ K t❛ ①→❝ ✤à♥❤ t➟♣

IK (x) = {x + λ(z − x), z ∈ K,

λ ≥ 0}

✈➔ ❣å✐ ❧➔ t➟♣ inward ❝õ❛ x ✤è✐ ✈ỵ✐ K ✳ ▼ët →♥❤ ①↕ S : K → E ✤÷đ❝ ❣å✐
❧➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ inward ②➳✉✱ ♥➳✉ S(x) IK (x) ợ ộ x K

Pữỡ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët
❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥
◆➠♠ ✷✵✵✻✱ ❙✳ P❧✉❜t✐❡♥❣ ✈➔ ❑✳ ❯♥❣❝❤✐ttr❛❦♦♦❧ ❬✼❪ ✤➣ ♠ð rë♥❣ ♣❤÷ì♥❣

♣❤→♣ ❧➦♣ ✭✶✳✶✸✮ ❝❤♦ ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ❈❤♦ K ❧➔

♠ët t ỗ õ rộ rút ổ ừ ổ
ỗ E T1 , T2 , . . . , TN : K → E ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ❳→❝
✤à♥❤ ❞➣② {xn } ❜ð✐ x1 ∈ K ✈➔



x1n = P (αn1 T1 xn + βn1 xn + γn1 u1n ),




x2 = P (α2 T2 x1 + β 2 xn + γ 2 u2 ),
n
n
n
n
n n
✳✳


✳




x
N
N
N
N

N N
n+1 = xn = P (αn TN xn + βn xn + γn un ),

✭✷✳✺✮

✈ỵ✐ n ≥ 1✱ tr♦♥❣ ✤â P ❧➔ ♠ët →♥❤ ①↕ ❝♦ rót ❦❤ỉ♥❣ ❣✐➣♥ tø E ❧➯♥ K ❀

{αn1 }, {αn2 }, . . . , {αnN }, {βn1 }, {βn2 }, . . . , {βnN }, {γn1 }, {γn2 }, . . . , {γnN } ❧➔ ❝→❝
❞➣② sè tr♦♥❣ ✤♦↕♥ [0, 1] t❤ä❛ ♠➣♥ αni +βni +γni = 1 ✈ỵ✐ ♠å✐ i = 1, 2, . . . , N
✈➔ ♠å✐ n ≥ 1 ✈➔ {u1n }, {u2n }, . . . , {uN
n } ❧➔ ❝→❝ ❞➣② ❜à ❝❤➦♥ tr♦♥❣ K ✳
▼ët ❤å ❤ú✉ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ T1 , T2 , . . . , TN : K E ợ

F = N
i=1 (Ti ) ữủ ồ tọ

tỗ t ởt ❤➔♠

❦❤æ♥❣ ❣✐↔♠ f : [0, ∞) → [0, ∞) t❤ä❛ ♠➣♥ f (0) = 0 ✈➔ f (r) > 0 ✈ỵ✐ ♠å✐

r > 0 s❛♦ ❝❤♦
max { x − Ti x } f (d(x, F )),

1iN

x K.

ỵ ✷✳✶✳✶✵ ✭①❡♠ ❬✼❪✮ ❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ỗ K

ởt t ỗ õ rộ ✈➔ ❝♦ rót ❝õ❛ E ✳ ❈❤♦ T1 , T2 , . . . , TN :

✷✵

K → E ❧➔ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥
❞➣② ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✺✮ ✈ỵ✐

∞
i
n=1 γn

✭❇✮✳ ◆➳✉ {xn} ❧➔

< ∞ ✈➔ {αni } ⊂ [ε, 1 − ε] ✈ỵ✐ ♠å✐

i = 1, 2, . . . , N ✈➔ ε ∈ (0, 1) t❤➻ {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣
❝❤✉♥❣ ❝õ❛ T1 , T2 , . . . , TN ✳

✷✳✷ ❈↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛
✷✳✷✳✶ ▼ỉ t↔ ♣❤÷ì♥❣ ♣❤→♣
P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❧➔ ❝→❝❤ ♣❤ê ❜✐➳♥ ✤➸ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤
①↕ ❦❤ỉ♥❣ ❣✐➣♥✳ ◆❤➢❝ ❧↕✐ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ tổ tữớ ữủ ợ
t Pữỡ s ữủ ợ t
s P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ tê♥❣ q✉→t ❤ì♥
♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥✳ ◆❤÷♥❣ ❝→❝ ♥❣❤✐➯♥ ❝ù✉ ❧↕✐ t➟♣ tr✉♥❣ ✈➔♦ ♣❤÷ì♥❣
♣❤→♣ ❧➦♣ ▼❛♥♥ ❝â t❤➸ ❧➔ ❞♦ ❦➳t q✉↔ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ✤ì♥ ❣✐↔♥ ❤ì♥
❦➳t q✉↔ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ ỵ sỹ ở tử ừ ữỡ
õ t ỵ sỹ ở tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣
❧➦♣ ■s❤✐❦❛✇❛✳ ❚✉② ♥❤✐➯♥✱ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ ❝ô♥❣ ❝â t➼♥❤ ❝❤➜t
r✐➯♥❣ ❝õ❛ ♥â✳ ❱➜♥ ✤➲ t❤ü❝ t➳ ❧➔ ❝â ♥❤✐➲✉ tr÷í♥❣ ❤đ♣ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣

▼❛♥♥ ❝â t❤➸ ❦❤ỉ♥❣ ❤ë✐ tư ♥❤÷♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ ❧↕✐ ❤ë✐ tư✱
❝❤➥♥❣ ❤↕♥ tr÷í♥❣ ❤đ♣ →♥❤ ①↕ ❣✐↔ ❝♦ ▲✐♣s❝❤✐t③ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳
◆❤➻♥ ❝❤✉♥❣✱ ❝↔ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛
❝❤➾ ❝❤♦ sü ❤ë✐ tư ②➳✉✳ ❱➻ ✈➟②✱ r➜t ♥❤✐➲✉ t→❝ ❣✐↔ ✤➣ ❝↔✐ t✐➳♥ ♣❤÷ì♥❣ ♣❤→♣
❧➦♣ ▼❛♥♥ ✈➔ ■s❤✐❦❛✇❛ ✤➸ ❝â ✤÷đ❝ sü ❤ë✐ tư ♠↕♥❤ ❝❤♦ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣
❣✐➣♥✳
❱➜♥ ✤➲ ①➜♣ ①➾ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ♠ët ❤å ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥
✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐ r➜t ♥❤✐➲✉ ❝→❝ t→❝ ❣✐↔✳ ◆➠♠ ✷✵✵✶✱ ❙❤✐♠♦❥✐ ✈➔


✷✶

❚❛❦❛❤❛s❤✐ ✤➲ ①✉➜t →♥❤ ①↕ Wn ♥❤÷ s❛✉


Un,n+1 = I






Un,n = rn Tn Un,n+1 + (1 − rn )I,






Un,n−1 = rn−1 Tn−1 Un,n + (1 − rn−1 )I,








···


Un,k = rk Tk Un,k+1 + (1 − rk )I,





Un,k−1 = rk−1 Tk−1 Un,k + (1 − rk−1 )I,






···







Un,2 = r2 T2 Un,3 + (1 − r2 )I,





Wn = Un,1 = r1 T1 Un,2 + (1 − r1 )I,

✭✷✳✻✮

tr♦♥❣ ✤â r1 , r2 , · · · ❧➔ ❝→❝ sè t❤ü❝ s❛♦ ❝❤♦ 0 ≤ rn ≤ 1, T1 , T2 , . . . , Tn ❧➔ ❤å
✈æ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tø C ✈➔♦ ❝❤➼♥❤ ♥â✳

❇ê ✤➲ ✷✳✷✳✶ ❈❤♦ C t ỗ õ rộ ừ ổ ỗ
T1 , T2 , . . . ❧➔ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ❝õ❛ C ✈➔♦ ❝❤➼♥❤ ♥â
t❤ä❛ ♠➣♥

∞
n=1 ❋✐①(T )

❦❤æ♥❣ ré♥❣ ✈➔ r1 , r2 , . . . ❧➔ ❝→❝ sè t❤ü❝ s❛♦ ❝❤♦

0 < rn ≤ γ < 1 ✈ỵ✐ n ≥ 0 ❜➜t ❦➻✳ ❑❤✐ ✤â✱ ✈ỵ✐ x ∈ C ✈➔ k ∈ N t t
ợ limn Un,k x tỗ t
ỷ ❞ö♥❣ ❇ê ✤➲ ✷✳✷✳✶✱ t❛ ❝â t❤➸ ✤à♥❤ ♥❣❤➽❛ →♥❤ ①↕ W tø C ✈➔♦ ❝❤➼♥❤
♥â ♥❤÷ s❛✉✿

W x = lim Wn x = lim Un,1 x,
n→∞

n→∞

∀x ∈ C

✭✷✳✼✮

◆❤÷ ✈➟② →♥❤ ①↕ W ✤÷đ❝ ❣å✐ ❧➔ W ✲→♥❤ ①↕ ❝❤♦ ❜ð✐ T1 , T2 , . . . ✈➔ r1 , r2 , . . . ✳

❇ê ✤➲ ✷✳✷✳✷ ❈❤♦ {xn} ✈➔ {yn} ❧➔ ❝→❝ ❞➣② ❜à ❝❤➦♥ tr♦♥❣ ❦❤æ♥❣

ỗ t T1 , T2 , . . . ❧➔ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tø C ✈➔♦ ❝❤➼♥❤
♥â s❛♦ ❝❤♦

∞
n=1 ❋✐①(Tn )

❦❤æ♥❣ ré♥❣ ✈➔ r1 , r2 , . . . ❧➔ ❝→❝ sè t❤ü❝ s❛♦ ❝❤♦

0 < rn ≤ γ < 1 ✈ỵ✐ n ≥ 1 ❜➜t ❦➻✳ ❑❤✐ ✤â ❋✐①(W ) =

∞
n=1 ❋✐①(Tn )✳

❉ü❛ tr➯♥ →♥❤ ①↕ W ✱ t❛ ①➙② ❞ü♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ s❛✉ ✤➙② t➻♠ ✤✐➸♠
❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ✈æ ❤↕♥ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥
❇❛♥❛❝❤✿ ❈❤♦ C t ỗ õ rộ ừ ổ ❇❛♥❛❝❤

E ✱ Ti : C → C ❧➔ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ✈ỵ✐ i ∈ N+ ✈➔ f : C → C ❧➔ →♥❤

✷✷

①↕ α✲❝♦✱ {αn }✱ {βn } ✈➔ {γn } ❧➔ ❝→❝ ❞➣② sè t❤ü❝ tr♦♥❣ (0, 1)✱ ✈ỵ✐ x0 ∈ C
tũ ỵ {xn } ữủ ỹ ữ s



z = γn Wn xn + (1 − γn )xn ,

 n
yn = βn Wn zn + (1 − βn )xn ,



x
n+1 = αn f (xn ) + (1 − αn )yn ,

✭✷✳✽✮

∀n ≥ 0,

tr♦♥❣ ✤â Wn ✤÷đ❝ ❝❤♦ ❜ð✐ ❝ỉ♥❣ t❤ù❝ ✭✷✳✻✮✳

❇ê ✤➲ ✷✳✷✳✸ ❈❤♦ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ trỡ C t ỗ
õ ừ E T : C → C ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✈ỵ✐ ❋✐①(T ) = ∅ ✈➔ ❝❤♦

f ∈ ΠC ✳ ❑❤✐ ✤â ❞➣② {xt } ❝❤♦ ❜ð✐
xt = tf (xt ) + (1 − t)T xt
❤ë✐ tö ♠↕♥❤ ✤➳♥ ♠ët ✤✐➸♠ tr♦♥❣ ❋✐①(T )✳ ◆➳✉ t❛ ✤à♥❤ ♥❣❤➽❛ →♥❤ ①↕

Q : Πc → ❋✐①(T ) ❜ð✐
Q(f ) := lim xt ,
t→0

∀f ∈ ΠC

t❤➻ Q(f ) t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉✿

(I − f )Q(f ), J(Q(f ) − p) ≤ 0,

f ∈ ΠC ,

p ∈ ❋✐①(T ).

❇ê ✤➲ ✷✳✷✳✹ ❈❤♦ {xn} ✈➔ {yn} ❧➔ ❝→❝ ❞➣② ❜à ❝❤➦♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥

❇❛♥❛❝❤ E ✈➔ {βn } ❧➔ ♠ët ❞➣② tr♦♥❣ [0, 1] ✈ỵ✐ 0 < lim inf n→∞ βn ≤

lim supn→∞ βn < 1✳ ●✐↔ sû xn+1 = (1 − βn )yn + βn xn ✈ỵ✐ ♠å✐ n ≥ 0 ✈➔
lim sup( yn+1 − yn − xn+1 − xn ) ≤ 0.
n→∞

❑❤✐ ✤â limn→∞ yn − xn = 0✳

❇ê ✤➲ ✷✳✷✳✺ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✱ t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ s❛✉✿
x+y

2

≤ x

2

+ 2 y, j(x + y) ,

∀x, y ∈ E,

tr♦♥❣ ✤â j(x + y) ∈ J(x + y)✳

❇ê ✤➲ ✷✳✷✳✻ ●✐↔ sû r➡♥❣ {αn} ❧➔ ❞➣② ❝→❝ sè t❤ü❝ ❦❤æ♥❣ ➙♠ s❛♦ ❝❤♦
αn+1 ≤ (1 − γn )αn + δn ,

∀n ≥ 0,

tr♦♥❣ ✤â {γn } ❧➔ ♠ët ❞➣② tr♦♥❣ (0, 1) ✈➔ {δn } ❧➔ ❝→❝ ❞➣② t❤ä❛ ♠➣♥✿