Về phương pháp lặp krasnoselskii mann cho ánh xạ không giãn trong không gian hilbert và áp dụng

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

NGUYỄN THỊ NGỌC MAI

VỀ PHƯƠNG PHÁP LẶP
KRASNOSELSKII–MANN CHO ÁNH XẠ KHÔNG GIÃN
TRONG KHÔNG GIAN HILBERT VÀ ÁP DỤNG

LUẬN VĂN THẠC SĨ TOÁN HỌC

THÁI NGUYÊN - 2019


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

NGUYỄN THỊ NGỌC MAI

VỀ PHƯƠNG PHÁP LẶP
KRASNOSELSKII–MANN CHO ÁNH XẠ KHÔNG GIÃN
TRONG KHÔNG GIAN HILBERT VÀ ÁP DỤNG
Chuyên ngành: Toán ứng dụng
Mã số
: 8 46 01 12

LUẬN VĂN THẠC SĨ TOÁN HỌC

NGƯỜI HƯỚNG DẪN KHOA HỌC

TS. Trần Xuân Quý

THÁI NGUYÊN - 2019


ử ử
ỵ







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

✶✳✷

⑩♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✳✶ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✳✷ P❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳
✶✳✶✳✸ ⑩♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✱ →♥❤ ①↕ ✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥
❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳✶ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳✷ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕
❦❤æ♥❣ ❣✐➣♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳
✳
✳

✺
✺
✻

✳ ✾
✳ ✶✵
✳ ✶✵
✳ ✶✶

✷ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ ❝❤♦ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥
tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
✶✹
✷✳✶

✷✳✷

✷✳✸

P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ ❝❤♦ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥
✷✳✶✳✶ ❇➔✐ t♦→♥ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✶✳✷ ❙ü ❤ë✐ tö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❦✐➸✉ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ s✉② rë♥❣ ✳ ✳ ✳ ✳ ✳
✷✳✷✳✶ ❍ë✐ tö ②➳✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷✳✷ ❍ë✐ tö ♠↕♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
Ù♥❣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✸✳✶ Ù♥❣ ❞ö♥❣ ❝❤♦ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ ❉♦✉❣❧❛s✕❘❛❝❤❢♦r❞ ✳
✷✳✸✳✷ Ù♥❣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❧✉➙♥ ♣❤✐➯♥ ❏♦❤♥ ✈♦♥
◆❡✉♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✐

✳
✳
✳
✳
✳
✳
✳
✳

✶✹
✶✺
✶✺
✶✾
✷✵
✷✺
✸✵
✸✵

✳ ✸✷


✐✐

❑➳t ❧✉➟♥

✸✺

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✸✻


ỵ
H
R
R+
N
x
A1
I
C[a, b]
d(x, C)
lim supn xn
lim inf n xn
xn → x0
xn
x0
❋✐①(T )

❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝
t➟♣ ❝→❝ sè t❤ü❝
t➟♣ ❝→❝ sè t❤ü❝ ❦❤ỉ♥❣ ➙♠
t➟♣ ❝→❝ sè tü ♥❤✐➯♥
✈ỵ✐ ♠å✐ x
t♦→♥ tỷ ữủ ừ t tỷ A

t tỷ ỗ t
t ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a, b]
❦❤♦↔♥❣ ❝→❝❤ tø ♣❤➛♥ tû x ✤➳♥ t➟♣ ❤đ♣ C
❣✐ỵ✐ ❤↕♥ tr➯♥ ❝õ❛ ❞➣② số {xn }
ợ ữợ ừ số {xn }
{xn } ❤ë✐ tö ♠↕♥❤ ✈➲ x0
❞➣② {xn } ❤ë✐ tö ②➳✉ ✈➲ x0
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 ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✧ ✈ỵ✐ I ❧➔ t➟♣ ❝❤➾ sè✳ ❇➔✐ t♦→♥ ♥➔②
❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤❛✉ ♥❤÷✿ ①û ❧➼ ↔♥❤✱ ổ ử
t t ỵ ồ
Ci = ❋✐①(Ti )✱ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ Ti ✈ỵ✐
i = 1, 2, . . . , N ✱ ✤➣ ❝â ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ✤÷đ❝ ✤➲ ①✉➜t t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣
❝❤✉♥❣ ❝õ❛ ❤å →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ {Ti }N
i=1 ❞ü❛ tr➯♥ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣
❝ê ✤✐➸♥ ♥ê✐ t✐➳♥❣ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥✱ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❍❛❧♣❡r♥✱
♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛✱ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛♥♦s❡❧s❦✐✐✳ ✳ ✳ ❱✐➺❝ ❝↔✐ t✐➳♥ ✈➔
♠ð rë♥❣ ❝→❝ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ❝❤♦ ❝→❝ ❧ỵ♣ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤❛♥❣ ❧➔ ✤➲
t➔✐ t❤✉ ❤ót ✤÷đ❝ sü q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ t ồ tr
ữợ
ữợ sỹ ữợ ừ r ỵ tổ ồ t ♣❤÷ì♥❣

♣❤→♣ ❧➦♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ ❝❤♦ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥
❍✐❧❜❡rt ✈➔ →♣ ❞ö♥❣✧ ❝❤♦ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ❝õ❛ ♠➻♥❤✳ ▼ö❝ t✐➯✉ ❝õ❛ ❧✉➟♥
✈➠♥ ❧➔ tr➻♥❤ ❜➔② ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕
❦❤ỉ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H tr➯♥ ❝ì sð ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣
❑r❛s♥♦s❡❧s❦✐✐ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥✳ ◆ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔②
tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✳ ❈ư t❤➸ ♥❤÷ s❛✉✿

❈❤÷ì♥❣ ✶✳ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr♦♥❣
❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
t❤ü❝ H ✱ tr➻♥❤ ❜➔② ✈➲ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✱ →♥❤ ①↕ ✤ì♥ ✤✐➺✉✱ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝
✷


✸

tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❝ị♥❣ ♠ët sè t➼♥❤ ❝❤➜t✱ ❣✐ỵ✐ t❤✐➺✉ ✈➲ ❜➔✐ t♦→♥ ✤✐➸♠
❜➜t ✤ë♥❣ ✈➔ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❝ê ✤✐➸♥ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕
❦❤ỉ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H ✳

❈❤÷ì♥❣ ✷✳ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ ❝❤♦ →♥❤ ①↕ ❦❤ỉ♥❣
❣✐➣♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ ①➜♣ ①➾ ✤✐➸♠ ❜➜t
✤ë♥❣ ❝❤♦ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ rt r ự
ỵ sỹ ở ②➳✉✱ ❤ë✐ tư ♠↕♥❤ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❝ị♥❣ ♠ët sè ✈➼ ♠✐♥❤
❤å❛ ❝❤♦ ✤✐➲✉ ❦✐➺♥ ✤➦t r❛ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ởt ự ử ừ ữỡ
rsss ố ợ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ ❉♦✉❣❧❛s✕❘❛❝❤❢♦r❞
✈➔ ♣❤➨♣ ❝❤✐➳✉ ❧✉➟♥ ♣❤✐➯♥ ❏♦❤♥ ✈♦♥ ◆❡✉♠❛♥♥ ❝ơ♥❣ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣
❝❤÷ì♥❣ ♥➔②✳
❚r♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐

❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❡♠ ❧✉ỉ♥ ♥❤➟♥ ✤÷đ❝ sü q✉❛♥ t➙♠ ❣✐ó♣ ✤ï ✈➔ ✤ë♥❣ ✈✐➯♥
❝õ❛ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ ❇❛♥ ●✐→♠ ❤✐➺✉✱ ♣❤á♥❣ ✣➔♦ t↕♦✱ ❑❤♦❛ ❚♦→♥ ✕ ❚✐♥✳
❱ỵ✐ ❜↔♥ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❡♠ ♠♦♥❣ ♠✉è♥ ✤÷đ❝ ❣â♣ ♠ët ♣❤➛♥ ♥❤ä ❝æ♥❣ sù❝ ❝õ❛
♠➻♥❤ ✈➔♦ ✈✐➺❝ ❣➻♥ ❣✐ú ✈➔ ♣❤→t sỹ ỳ ỵ
t ❤å❝ ✈è♥ ❞➽ ✤➣ r➜t ✤➭♣✳ ✣➙② ❝ô♥❣ ❧➔ ♠ët ❝ì ❤ë✐ ❝❤♦ ❡♠ ❣û✐ ❧í✐ tr✐ ➙♥
tỵ✐ t➟♣ t❤➸ ❝→❝ t❤➛② ❝ỉ ❣✐↔♥❣ ✈✐➯♥ ❝õ❛ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✕ ✣↕✐ ❤å❝
❚❤→✐ ◆❣✉②➯♥ ♥â✐ ❝❤✉♥❣ ✈➔ ❑❤♦❛ ❚♦→♥ ✕ ❚✐♥ ♥â✐ r✐➯♥❣✱ ✤➣ tr✉②➲♥ t❤ö ❝❤♦
❡♠ ♥❤✐➲✉ ❦✐➳♥ tự ồ qỵ tr tớ ữủ ❧➔ ❤å❝ ✈✐➯♥
❝õ❛ tr÷í♥❣✳
❚→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉ tr÷í♥❣ ❚❍❙❈ ◗✉❛♥❣
❚r✉♥❣✱ ❚P ❨➯♥ ❇→✐ ❝ị♥❣ t t ỗ t ✤✐➲✉
❦✐➺♥ tèt ♥❤➜t ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ t❤í✐ ❣✐❛♥ ✤✐ ❤å❝ ❈❛♦ ❤å❝❀ ❝↔♠ ì♥ ❝→❝ ❛♥❤
❝❤à ❡♠ ❤å❝ ✈✐➯♥ ợ ồ ỗ ✤➣ tr❛♦ ✤ê✐✱
✤ë♥❣ ✈✐➯♥ ✈➔ ❦❤➼❝❤ ❧➺ t→❝ ❣✐↔ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❧➔♠ ❧✉➟♥ ✈➠♥ t↕✐
tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳
✣➦❝ ❜✐➺t ❡♠ ①✐♥ ữủ tọ ỏ t ỡ s s tợ t r
ỵ ổ q t ❝❤➾ ❜↔♦✱ ✤ë♥❣ ✈✐➯♥ ❦❤➼❝❤ ❧➺✱ ❣✐ó♣ ✤ï t➟♥
t➻♥❤ ✈➔ õ ỵ s s tr sốt q tr ❤å❝ t➟♣ ❝ơ♥❣ ♥❤÷ t❤ü❝
❤✐➺♥ ✤➲ t➔✐✳ ❈❤➦♥❣ ✤÷í♥❣ ✈ø❛ q✉❛ s➩ ❧➔ ♥❤ú♥❣ ❦➾ ♥✐➺♠ ✤→♥❣ ♥❤ỵ ✈➔ ✤➛② þ
♥❣❤➽❛ ✤è✐ ✈ỵ✐ ❝→❝ ❛♥❤ ❝❤à ❡♠ ❤å❝ ✈✐➯♥ ❧ỵ♣ ❑✶✶ ♥â✐ ❝❤✉♥❣ ✈➔ ✈ỵ✐ ❜↔♥ t❤➙♥ ❡♠


✹

♥â✐ r✐➯♥❣✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ t➜t ❝↔ ♥❤ú♥❣ ữớ t ú ù
ỗ ũ tr ❝❤➦♥❣ ✤÷í♥❣ ✈ø❛ q✉❛✳ ▼ët ❧➛♥ ♥ú❛✱ ❡♠ ①✐♥ tr➙♥
trå♥❣ ❝↔♠ ì♥✦

❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✷✷ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✾
❍å❝ ✈✐➯♥

◆❣✉②➵♥ ❚❤à ◆❣å❝ ▼❛✐


❈❤÷ì♥❣ ✶

❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕
❦❤ỉ♥❣ ❣✐➣♥ tr ổ rt
ữỡ ợ t ởt số t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ →♥❤ ①↕
❦❤æ♥❣ ❣✐➣♥✱ →♥❤ ①↕ ✤ì♥ ✤✐➺✉✱ ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tr♦♥❣ ❦❤ỉ♥❣
❣✐❛♥ ❍✐❧❜❡rt ❝ị♥❣ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕
❦❤æ♥❣ ❣✐➣♥✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ ✈✐➳t tr➯♥ ❝ì sð tê♥❣ ❤đ♣ ❦✐➳♥ t❤ù❝
tø ❝→❝ t➔✐ ❧✐➺✉ ❬✷❪✱ ❬✸❪✱ ❬✺❪✱ ❬✽❪ ✈➔ ♠ët sè t➔✐ ❧✐➺✉ ✤÷đ❝ tr➼❝❤ ❞➝♥ tr♦♥❣ ✤â✳

✶✳✶

⑩♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt

❈❤♦ H ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt tỹ ợ t ổ ữợ ., .
. t÷ì♥❣ ù♥❣✳ ❈❤♦ {xn } ❧➔ ♠ët ❞➣② tr♦♥❣ ❦❤ỉ♥❣ H ỵ xn
x
{xn } ❤ë✐ tö ②➳✉ ✤➳♥ x ✈➔ xn → x ♥❣❤➽❛ ❧➔ ❞➣② {xn } ❤ë✐ tö ♠↕♥❤
✤➳♥ x✳

✶✳✶✳✶

▼ët sè t t ừ ổ rt

rữợ t t ✤à♥❤ ♥❣❤➽❛ ✈➲ sü ❤ë✐ tư ②➳✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
t❤ü❝ H ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❉➣② {xn } tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✤÷đ❝ ❣å✐ ❧➔ ❤ë✐
tư ②➳✉ ✈➲ ♣❤➛♥ tû x ∈ H ✱ ♥➳✉

lim xn , y = x, y ,

n→∞

∀y ∈ H.

◆❤➟♥ ①➨t ✶✳✶✳✷✳ ❚ø t➼♥❤ ❧✐➯♥ tử ừ t ổ ữợ s r xn x✱
t❤➻ xn

x✳ ❚✉② ♥❤✐➯♥✱ ✤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❦❤ỉ♥❣ ✤ó♥❣✳
✺


✻

❈❤➥♥❣ ❤↕♥ ①➨t ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
∞
2

|xn |2 < ∞

l := {xn } ⊂ R :
n=1

✈➔ ❣✐↔ sû ❞➣② {en } ⊂ l2 ✤÷đ❝ ❝❤♦ ❜ð✐ en = (0, . . . , 0,
✈ỵ✐ ♠å✐ n 1✳ ❑❤✐ ✤â✱ en

✤➥♥❣ t❤ù❝ ❇❡ss❡❧✱ t❛ ❝â

1

, 0, . . . , 0, . . . ),

✈à tr➼ t❤ù ♥

0✱ ❦❤✐ n → ∞✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠é✐ y ∈ H ✱ tø ❜➜t
∞

| en , y |2 < y

2

< ∞.

n=1

❙✉② r❛ limn→∞ en , y = 0✱ tù❝ ❧➔ en
✈➲ 0✱ ✈➻ en = 1 ✈ỵ✐ ♠å✐ n 1✳

0✳ ❚✉② ♥❤✐➯♥✱ {en } ❦❤ỉ♥❣ ❤ë✐ tư ♠↕♥❤

▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ rt tỹ H ữủ tr tr ờ
ữợ

ờ ✤➲ ✶✳✶✳✸✳ ✭①❡♠ ❬✷❪✮

❈❤♦ H ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✳ ❑❤✐ ✤â✿

(i) x + y

2

x

2

+ 2 x + y, y

(ii) x + y

2

= x

2

+ y

2

+ 2 x, y

(iii) tx + (1 − t)y 2 = t x
✈➔ ♠å✐ x, y ∈ H ✳

2

+ (1 − t) y

∀x, y ∈ H.

✈ỵ✐ ♠å✐ x, y ∈ H ❀
2

− t(1 − t) x − y

2

✈ỵ✐ ♠å✐ t ∈ [0, 1]

▼å✐ ❞➣② ❜à ❝❤➦♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❣✐❛♥ ❍✐❧❜❡rt ✤➲✉
❝❤ù❛ ♠ët ❞➣② ❝♦♥ ❤ë✐ tư ②➳✉✳

❇ê ✤➲ ✶✳✶✳✹✳ ✭①❡♠ ❬✷❪✮
✶✳✶✳✷

P❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ rt

C ởt t ỗ õ rộ tr♦♥❣
❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H ✳ ❑❤✐ ✤â ✈ỵ✐ ♠é✐ x H tỗ t t tỷ
Pc x ∈ C s❛♦ ❝❤♦
x − PC x ≤ x − y
✈ỵ✐ ♠å✐ y ∈ C.
✭✶✳✶✮
❈❤ù♥❣ ♠✐♥❤✳ ❚❤➟t ✈➟②✱ ✤➦t d = uC
inf x u õ tỗ t↕✐ ❞➣② {un } ⊂ C
▼➺♥❤ ✤➲ ✶✳✶✳✺✳ ✭①❡♠ ❬✷❪✮

s❛♦ ❝❤♦ x − un → d ❦❤✐ n → ∞✳ ❚ø ✤â✱

un − um

2

= (x − un ) − (x − um )

2


✼

= 2 x − un

2

≤ 2( x − un

+ 2 x − um
2

+ x − um

un + um
2
2
2
) − 4d → 0,

2

2

−4 x−

❦❤✐ n, m → ∞. ❉♦ ✤â ❞➣② {un } ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
t❤ü❝ H r tỗ t u = lim un C ✳ ❉♦ ❝❤✉➞♥ ❧➔ ❤➔♠ sè ❧✐➯♥ tö❝ ♥➯♥
n→∞

x u = d sỷ tỗ t v C s❛♦ ❝❤♦ x − v = d✳ ❚❛ ❝â
u−v

2

= (x − u) − (x − v)
= 2( x − u

2

2

+ x − v 2) − 4 x −

u+v
2

2

≤ 0.

❙✉② r u = v tỗ t t ♠ët ♣❤➛♥ tû PC x ∈ C s❛♦ ❝❤♦

x − PC x = inf x − u .
u∈C

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✻✳ ✭①❡♠ ❬✷❪✮ P❤➨♣ ❝❤♦ t÷ì♥❣ ù♥❣ ♠é✐ ♣❤➛♥ tû x ∈ H ♠ët
♣❤➛♥ tû PC x ∈ C ①→❝ ✤à♥❤ ♥❤÷ ✭✶✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ ❝❤✐➳✉ H
❧➯♥ C ✳
❙❛✉ ✤➙② ❧➔ ♠ët ✈➼ ❞ö ✈➲ t♦→♥ tû ❝❤✐➳✉✳

❱➼ ❞ö ✶✳✶✳✼✳ ❈❤♦ C = {x ∈ H :

x, u = y} ✈ỵ✐ u = 0✳ ❑❤✐ ✤â ♣❤➨♣ ❝❤✐➳✉

♠➯tr✐❝ ❧➯♥ C ❝❤♦ ❜ð✐

PC (x) = x +

y − x, u
u.
u 2

ữợ t ởt ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ →♥❤ ①↕ PC : H → C
❧➔ ♠ët ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝✳

❈❤♦ C ❧➔ ♠ët t➟♣ ỗ õ rộ ừ
ổ rt tỹ H ✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ →♥❤ ①↕ PC : H → C ❧➔
♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ ❝❤✐➳✉ H ❧➯♥ C ❧➔
x − PC x, PC x − y
0 ✈ỵ✐ ♠å✐ x ∈ H ✈➔ y ∈ C.

✭✶✳✷✮
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû PC ❧➔ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝✳ ❑❤✐ ✤â ✈ỵ✐ ♠å✐ x ∈ H, y ∈ C
▼➺♥❤ ✤➲ ✶✳✶✳✽✳ ✭①❡♠ ❬✸❪✮

✈➔ ♠å✐ t ∈ (0, 1)✱ t❛ ❝â

ty + (1 − t)PC x ∈ C.


✽

❉♦ ✤â✱ tø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝✱ s✉② r❛

x − PC x

2

2

≤ x − ty − (1 − t)PC x

∀t ∈ (0, 1).

❇➜t ✤➥♥❣ t❤ù❝ tr➯♥ t÷ì♥❣ ✤÷ì♥❣ ✈ỵ✐

x − PC x

2

≤ x − PC x

2

− 2t x − PC x, y − PC x + t2 y − PC x 2 ,

✈ỵ✐ ♠å✐ t ∈ (0, 1)✳ ❚ø ✤â✱

x − PC x, PC x − y

−

t
y − PC x
2

2

∀t ∈ (0, 1).

❈❤♦ t → 0+ ✱ t❛ ♥❤➟♥ ✤÷đ❝

x − PC x, PC x − y

0.

◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû

0 ✈ỵ✐ ♠å✐ x ∈ H ✈➔ y ∈ C.

x − PC x, PC x − y

❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ x ∈ H ✈➔ y ∈ C ✱ t❛ ❝â

x − PC x

2

= x − PC x, x − y + y − PC x
= x − PC x, y − PC x + x − PC x, x − y
≤ x−y

2

+ y − PC x, x − PC x + PC x − y

= x−y

2

+ y − PC x, x − PC x − y − PC x

2

≤ x − y 2.
❙✉② r❛ PC ❧➔ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tø H ❧➯♥ C ✳

❈❤♦ C ❧➔ ♠ët t➟♣ ❝♦♥ ỗ õ ừ ổ
rt H PC ❝❤✐➳✉ ♠➯tr✐❝ tø H ❧➯♥ C ✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ x, y ∈ H ✱
t❛ ❝â

❍➺ q✉↔ ✶✳✶✳✾✳ ✭①❡♠ ❬✸❪✮

PC x − PC y

2

≤ x − y, PC x − PC y .

❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠å✐ x, y ∈ H ✱ tø ▼➺♥❤ ✤➲ ✶✳✶✳✽✱ t❛ ❝â
x − PC x, PC y − PC x ≤ 0,
y − PC y, PC x − PC y ≤ 0.
❈ë♥❣ ❤❛✐ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ♥❤➟♥ ✤÷đ❝ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳


✾

✶✳✶✳✸

⑩♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✱ →♥❤ ①↕ ✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥
❍✐❧❜❡rt

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✵✳ ✭①❡♠ ❬✸❪✮ ❈❤♦ C ❧➔ ♠ët t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ ❦❤æ♥❣
❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H ✳

(i) ⑩♥❤ ①↕ T : C → H ✤÷đ❝ ❣å✐ ❧➔ →♥❤ L tử st tr C
tỗ t số L > 0 s❛♦ ❝❤♦
T (x) − T (y) ≤ L x − y

∀x, y ∈ C.

✭✶✳✸✮

(ii) ❚r♦♥❣ ✭✶✳✸✮✱ ♥➳✉ L ∈ [0, 1) t❤➻ T ✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ ❝♦❀ ♥➳✉ L = 1 t❤➻ T
✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳
❙❛✉ ✤➙② ❧➔ ✤à♥❤ ♥❣❤➽❛ ✈➲ t♦→♥ tû ✤ì♥ ✤✐➺✉✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✶✳ ✭①❡♠ ❬✸❪✮ ❈❤♦ C ❧➔ ởt t ỗ õ rộ
tr ổ rt t❤ü❝ H ✳ ❚♦→♥ tû A : C → H ✤÷đ❝ ❣å✐ ❧➔

(i) ✤ì♥ ✤✐➺✉ tr➯♥ C ♥➳✉ A(x) − A(y), x − y
0 ∀x, y ∈ C ❀
✤ì♥ ✤✐➺✉ ❝❤➦t tr➯♥ C ♥➳✉ ❞➜✉ ✧❂✧ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❝❤➾ ①↔② r❛
❦❤✐ x = y ❀
(ii) ✤ì♥ ✤✐➺✉ tr C tỗ t ởt ổ δ(t)✱ ❦❤ỉ♥❣ ❣✐↔♠
✈ỵ✐ t 0✱ δ(0) = 0 ✈➔ t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t
A(x) − A(y), x − y

δ x−y

∀x, y ∈ C;

♥➳✉ δ(t) = βt2 ✱ β ❧➔ ❤➡♥❣ sè ❞÷ì♥❣✱ t❤➻ A ✤÷đ❝ ❣å✐ ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉
♠↕♥❤ tr➯♥ C ✭❤❛② β ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥ C ✮❀

(iii) ỡ ữủ tr C ợ số > 0 ✭❤❛② η ✲✤ì♥ ✤✐➺✉ ♠↕♥❤
♥❣÷đ❝ tr➯♥ C) ♥➳✉
A(x) − A(y), x − y

η A(x) − A(y)

2

∀x, y ∈ C.

❑❤→✐ ♥✐➺♠ t♦→♥ tû ✤ì♥ ✤✐➺✉ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣
ỏ ữủ ổ t ỹ tr ỗ t ♥❤÷ s❛✉✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✷✳ ✭①❡♠ ❬✸❪✮ ❚♦→♥ tû ✤❛ trà A : H → 2H ✤÷đ❝ ❣å✐ ❧➔ ✤ì♥
✤✐➺✉ ♥➳✉

u − v, x − y

0 ∀x, y ∈ H, u ∈ A(x), v ∈ A(y).


✶✵

❚♦→♥ tû A : H → 2H ✤÷đ❝ ❣å✐ ❧➔ ỡ ỹ ỗ t
r(A) := {(x, u) ∈ H × H : u ∈ Ax}
❝õ❛ A ❦❤ỉ♥❣ ự tỹ sỹ tr ỗ t ừ t ý ởt t tỷ ỡ
tr H

ú ỵ ✶✳✶✳✶✸✳ ❚♦→♥ tû A ❧➔ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ♥➳✉ ợ (x, u)
H ì H u − v, x − y ≥ 0 ✈ỵ✐ (y, v) ∈ ●r(A) s✉② r❛ u ∈ A(x)✳
❈❤♦ A ❧➔ t♦→♥ tû λ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ L✲❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③ tø C ✈➔♦ H ✈➔
NC x ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝ tø C t↕✐ x ∈ C ✱ ♥❣❤➽❛ ❧➔

NC x =

y ∈ H : y, x − u ≥ 0, ∀u C

x C;
ữủ

ỵ

Bx =

Ax + NC x,

♥➳✉

x∈C

∅,

♥➳✉

x∈
/ C.

❑❤✐ ✤â B ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳

●✐↔ sû A ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ õ (t1
n A) ở
tử ỗ t NA (0) ❦❤✐ tn → 0 ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ A−1(0) = ∅.
(ii) ◆➳✉ {Bn } ❧➔ ❞➣② ❝→❝ t♦→♥ tû ✤ì♥ ỹ ở tử ỗ t B A ❧➔
t♦→♥ tû ▲✐♣s❝❤✐t③ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✱ t❤➻ (A + Bn) ở tử ỗ t A + B
A + B ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳

❇ê ✤➲ ✶✳✶✳✶✹✳ (i)

−1

✶✳✷
✶✳✷✳✶

❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥
❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣

❚r♦♥❣ ♠ư❝ ♥➔② t❛ ①➨t ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
t❤ü❝ H ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ❈❤♦ C ❧➔ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ H ✈➔ →♥❤ ①↕ T : C → C ✳
✣✐➸♠ x ∈ C ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ừ T T x = x.
ỵ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ❧➔ ❋✐①(T )✱ ♥❣❤➽❛ ❧➔
❋✐①(T ) := x ∈ C :

Tx = x .




C ởt t ỗ õ ré♥❣ ❝õ❛ ❦❤æ♥❣
❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H ✈➔ T : C → H ❧➔ ♠ët →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ❑❤✐ ✤â✱
Fix(T ) ởt t ỗ õ tr H
ự sỷ (T ) = rữợ ❤➳t✱ t❛ ❝❤➾ r❛ Fix(T ) ❧➔ t➟♣ ✤â♥❣✳
▼➺♥❤ ✤➲ ✶✳✷✳✷✳

❚❤➟t ✈➟②✱ ✈➻ T ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ♥➯♥ T ❧✐➯♥ tö❝ tr➯♥ C ✳ ●✐↔ sû {xn } ❧➔ ♠ët

❞➣② ❜➜t ❦ý tr♦♥❣ Fix(T ) t❤ä❛ ♠➣♥ xn → x✱ ❦❤✐ n → ∞✳ ❱➻ {xn } ⊂ Fix(T )✱
♥➯♥
T xn − xn = 0 ∀n ≥ 1.
❚ø t➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ ❝❤✉➞♥✱ ❝❤♦ n → ∞✱ t❛ ♥❤➟♥ ✤÷đ❝ T x − x = 0✱ tù❝ ❧➔
x ∈ Fix(T )✳ ❉♦ ✤â✱ Fix(T ) ❧➔ t➟♣ ✤â♥❣✳
✭❜✮ t t r t ỗ ừ Fix(T ) ●✐↔ sû Fix(T ) = ∅ ✈➔ ❣✐↔ sû
x, y ∈ Fix(T )✳ ❱ỵ✐ λ ∈ [0, 1]✱ ✤➦t z = λx + (1 − λ)y ✳ ❑❤✐ ✤â✱

Tz − z

2

= λ(T z − x) + (1 − λ)(T z − y)
= λ Tz − x

2

= λ Tz − Tx
≤λ z−x

2

2

+ (1 − λ)(T z − y)
2

2

− λ(1 − λ) x − y

+ (1 − λ) (T z − T y)

+ (1 − λ) (z − y)

= λ(z − x) + (1 − λ)(z − y)

2

2

2

2

− λ(1 − λ) x − y

− λ(1 − λ) x − y

2

2

= 0.

❙✉② r❛ T z = z ✈➔ ❞♦ ✤â z ∈ Fix(T ) Fix(T ) ởt t ỗ

t ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ✤÷đ❝ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉✿
❈❤♦ T : C → C ❧➔ →♥❤ ①↕ ổ tứ t ỗ õ rộ C ❝õ❛
❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H ✈➔♦ ❝❤➼♥❤ ♥â ✈ỵ✐ ❋✐①(T ) = ∅✳

❚➻♠ ♣❤➛♥ tû

✶✳✷✳✷

x∗ ∈ ❋✐①(T ).

✭✶✳✹✮

▼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 →♥❤ ổ tứ t C ỗ õ
ré♥❣ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✈➔♦ ❝❤➼♥❤ ♥â✳ ❈❤ó þ r➡♥❣✱ tr♦♥❣ tr÷í♥❣
❤đ♣ H ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈ỉ ❤↕♥ ❝❤✐➲✉ t❤➻ ❞➣② ❧➦♣ ✭✶✳✺✮ ❝❤➾ ❤ë✐ tư
②➳✉ ♠➔ ❦❤ỉ♥❣ ❤ë✐ tư ♠↕♥❤✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ αn = (0, 1) ợ ồ n t
ữỡ ▼❛♥♥ ✭✶✳✺✮ trð t❤➔♥❤ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛♥♦s❡❧s❦✐✐✳

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 ✈➔ 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➢❝ αn =
❧↕✐ ❜à ❧♦↕✐ trø✳

1
n+1


✶✸

P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛

P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ ✤÷đ❝ ✤➲ ①✉➜t s
ợ ữỡ t❤➻ ❞➣② ❧➦♣ {xn } ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐



x0 ∈ C,
✭✶✳✼✮
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 } ❧➔ ❝→❝ số tỹ tr [0, 1]

ú ỵ r trữớ ủ n = 1 ợ ồ n t ữỡ ♣❤→♣ ❧➦♣
■s❤✐❦❛✇❛ ✭✶✳✼✮ trð t❤➔♥❤ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥ ✭✶✳✺✮✳


❈❤÷ì♥❣ ✷

P❤÷ì♥❣ ♣❤→♣ ❧➦♣
❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ ❝❤♦ →♥❤ ①↕

❦❤ỉ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ ①➜♣ ①➾ ✤✐➸♠
❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ▼ư❝ ✷✳✶ tr➻♥❤
❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕
❦❤ỉ♥❣ ❣✐➣♥✳ ▼ư❝ ✷✳✷ tr➻♥❤ ❜➔② sü ❤ë✐ tö ②➳✉ ✈➔ ❤ë✐ tö ♠↕♥❤ ❝õ❛ ♣❤÷ì♥❣
♣❤→♣ ❧➦♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ s✉② rë♥❣✳ ▼ư❝ ✷✳✸ tr➻♥❤ ❜➔② ù♥❣ ❞ư♥❣ ❝õ❛
♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ ✈✐➳t tr➯♥
❝ì sð tê♥❣ ❤đ♣ ❦✐➳♥ t❤ù❝ tr♦♥❣ t➔✐ ❧✐➺✉ ❬✹❪ ✈➔ ❬✻❪✳

✷✳✶

P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ ❝❤♦ →♥❤ ①↕ ❦❤ỉ♥❣
❣✐➣♥

▼ët tr♦♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ♥ê✐ t✐➳♥❣ ❧➔ ♣❤÷ì♥❣ ♣❤→♣
❧➦♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥✳ P❤÷ì♥❣ ♣❤→♣ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✱ ①✉➜t ♣❤→t tø x1 ∈
H t❛ ①➨t ❞➣② ❧➦♣ ♥❤÷ s❛✉

xn+1 = (1 − λn )xn + λn T xn

∀n = 1, 2, . . .

✭✷✳✶✮

✈ỵ✐ λn ∈ [0, 1]✳ ❑➳t q✉↔ ✈➲ sü ❤ë✐ tö tê♥❣ q✉→t ♥❤➜t ✤÷đ❝ ✤÷❛ r❛ ❜ð✐ ❘❡✐❝❤
✭✶✾✼✾✮ ✈➔ ❣✐↔ t❤✐➳t ❋✐①(T ) ❦❤→❝ ré♥❣ ✈➔ λn ✤÷đ❝ ❝❤å♥ s❛♦ ❝❤♦
∞

λn (1 − λn ) = ∞,
n=1

✶✹

✭✷✳✷✮


✶✺

❦❤✐ ✤â ❞➣② {xn } ❤ë✐ tö ②➳✉ ✤➳♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ✱ ð ✤➙② T : C → C
❧➔ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ✈ỵ✐ C H t ỗ õ rộ
ỹ ở tử ♠↕♥❤ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ ❦❤ỉ♥❣ ✤ó♥❣
tr♦♥❣ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t✳

✷✳✶✳✶

❇➔✐ t♦→♥ ✈➔ ♣❤÷ì♥❣ ♣❤→♣

❚r♦♥❣ ♠ư❝ ♥➔② t❛ tr➻♥❤ ❜➔② ♠ët ❦➳t q✉↔ ❝õ❛ ❆✳ ▼♦✉❞❛❢✐ tr♦♥❣ ❬✻❪ ✈➲
♣❤÷ì♥❣ ♣❤→♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣
T tữỡ ự ợ ổ P ❇➔✐ t♦→♥ ✤➦t r❛ ♥❤÷ s❛✉✿
❚➻♠

x¯ ∈ ❋✐①(T ) s❛♦ ❝❤♦

0 ∀x ∈ ❋✐①(T ),

x¯ − P (¯
x), x¯ − x

✭✷✳✸✮

♥❣❤➽❛ ❧➔✱ 0 ∈ (I − P )¯
x + N❋✐①(T ) x¯✱ tr♦♥❣ ✤â ❋✐①(T ) = {¯
x ∈ D; x¯ = T (¯
x)} ❧➔
t➟♣ ❝→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ T : D → D ✈➔ D t
ỗ õ ừ ổ rt H.
♠ð rë♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥✱ t❛ ①➨t ❞➣② ❧➦♣

xn+1 = (1 − αn )xn + αn (σn P xn + (1 − σn )T xn ),

✈ỵ✐ n ≥ 0,

✭✷✳✹✮

ð ✤➙② x0 ∈ D✱ ❝→❝ ❞➣② {σn } ✈➔ {αn } ⊂ (0, 1).

✷✳✶✳✷

❙ü ❤ë✐ tö

◆❤➟♥ ①➨t ✷✳✶✳✶✳ ✭❛✮ ◆➳✉ T : D → D ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr➯♥ D t❤➻
A = I − T ❧➔ ♠ët t♦→♥ tỷ ỡ ỹ tr D ỗ tớ t♦→♥ tû
1/2✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ♥❣÷đ❝✱ ð ✤➙② I ❧➔ →♥❤ ỗ t ừ ổ
rt tỹ H
ỡ ♥ú❛ T ❧➔ ♥û❛ ✤â♥❣ tr➯♥ D t❤❡♦ ♥❣❤➽❛✱ ♥➳✉ ❞➣② {xn } ❤ë✐ tö ②➳✉ ✤➳♥
x tr♦♥❣ D ✈➔ ❞➣② {xn T xn } ❤ë✐ tö ♠↕♥❤ ✤➳♥ 0 t❤➻ x ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣
❝õ❛ →♥❤ ①↕ T ✳

❇ê ✤➲ ✷✳✶✳✷✳ ✭①❡♠ ❬✻❪ ✈➔ t➔✐ ❧✐➺✉ ✤÷đ❝ tr➼❝❤ ❞➝♥ tr♦♥❣ ✤â✮ ❈❤♦ {an } ❧➔ ❞➣②

❝→❝ sè t❤ù❝ ❦❤æ♥❣ ➙♠ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥✿
an+1

tr♦♥❣ ✤â

(1 − αn )an + αn σn + γn ,

n ≥ 1.


✶✻
∞

(a) {αn } ⊂ [0, 1],

αn = ∞;
n=1

(b) lim sup σn

0;

n→∞
∞

(c) γn ≥ 0 (n ≥ 1),

γn < ∞.
n=1

❑❤✐ ✤â αn → 0 ❦❤✐

n → ∞.

❇ê ✤➲ ✷✳✶✳✸✳ ✭①❡♠ ❬✻❪ ✈➔ t➔✐ ❧✐➺✉ ✤÷đ❝ tr➼❝❤ ❞➝♥ tr♦♥❣ ✤â✮
{βn }

❧➔ ❞➣② ❝→❝ sè t❤ü❝ ❦❤æ♥❣ ➙♠ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥
∞

αn < ∞, βn+1

αn + βn

●✐↔ sû {αn} ✈➔

✈ỵ✐ ♠å✐ n = 0, 1, . . . .

n=0

❑❤✐ ✤â ❞➣② {βn} ❤ë✐ tư✳
❙ü ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✭✷✳✹✮ ✤÷đ❝ tr➻♥❤ tr ỵ s

{xn} ❝ỉ♥❣ t❤ù❝ ✭✷✳✹✮ ❤ë✐ tư tỵ✐
✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ T : D → D ✈ỵ✐ ❝→❝ số {n}
{n } tọ
ỵ ✷✳✶✳✹✳ ✭①❡♠ ❬✻❪✮

+∞

(i)

σn < +∞

✈➔

n=0
+∞

(ii)

αn (1 − αn ) = +∞✳

n=0

◆❣♦➔✐ r❛✱ ❞➣② {xn} ❧➔ t✐➺♠ ❝➟♥ ❝❤➼♥❤ q✉② ✱ tù❝ ❧➔
lim ||xn+1 − xn || = 0.

n→+∞

||xn+1 − xn ||
= 0✱ t❤➻ ❞➣② {xn } ❤ë✐ tư
❍ì♥ ♥ú❛ ♥➳✉ t❤➯♠ ✤✐➲✉ ❦✐➺♥ n→+∞
lim
αn σ n
②➳✉ tỵ✐ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✸✮✳
❈❤ù♥❣ ♠✐♥❤✳ ▲➜② x¯ ∈ ❋✐①T ✈➔ ✤➦t Tσ = σnP + (1 − σn)T ✳ ❚ø ❝æ♥❣ t❤ù❝
n

✭✷✳✹✮✱ t❛ ❝â

||xn+1 − x¯||

(1 − αn )||xn − x¯|| + αn ||Tσn xn − T x¯||
||xn − x¯|| + αn ||Tσn (¯
x) − T x¯||


✶✼

= ||xn − x¯|| + αn σn ||P (¯
x) − T x¯||.
❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✸ ✈➔

∞

αn σn < +∞,
n=0

t❛ ❝â ợ


l(
x) = lim ||xn x||
n+

tỗ t ỳ ❤↕♥✳ ❱➟② ❞➣② {xn } ❜à ❝❤➦♥✳
✣➦t x
¯n+1 = (1 − αn )xn + αn T xn ✈➔ G = I − T ✱ t❛ t❤✉ ✤÷đ❝

||xn+1 − x¯n+1 || = αn ||Tσn xn − T xn || = αn σn ||T xn − P xn ||.
▼➦t ❦❤→❝✱ ✈➻ G ❧➔ 1/2✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ♥❣÷đ❝✱ ♥➯♥ t❛ ❝â

||¯
xn+1 − x¯||2 = ||xn − x¯ − αn Gxn ||2
= ||xn − x¯||2 − 2 xn − x¯, Gxn − G¯
x + αn2 ||Gxn ||2
= ||xn − x¯||2 − αn (1 − αn )||Gxn ||2 .
❱➻ ❞➣② {xn } ❜à ❝❤➦♥ ♥➯♥ tỗ t số M > 0 s

||T xn P xn ||

∀n ∈ N

M

✈➔ t❛ ✤÷đ❝

αn (1 − αn )||Gn (xn )||2

||xn − x¯||2 − ||¯
xn+1 − x¯||2
= ||xn − x¯||2 − ||¯
xn+1 − xn+1 + xn+1 − x¯||2
||xn − x¯||2 − ||xn+1 − x¯||2
− 2 x¯n+1 − xn+1 , xn+1 − x¯
||xn − x¯||2 − ||xn+1 − x¯||2 + 2M αn σn .

❚ø ✤➙② s✉② r❛

∞

∞
2

αn (1 − αn )||Gxn ||

2

||x0 − x¯|| + 2M

n=0

αn σn < +∞.
n=0

∞

❱➻

αn (1 − αn ) = +∞, t❛ s✉② r❛
n=0

lim inf ||Gxn || = lim inf ||xn − Tn || = 0.
n→+∞

n→+∞


✶✽

▼➦t ❦❤→❝ ✈ỵ✐ ♠å✐ n t❛ ❝â

T xn+1 − xn+1 = T xn+1 − Tσn xn + (1 − αn )(Tσn xn − xn )
♥➯♥

||xn+1 − Tn+1 || = ||T xn+1 − T xn + T xn − Tσn xn + (1 − αn )(Tσn xn − xn )||
||xn+1 − xn || + ||T xn − Tσn xn || + (1 − αn )||Tσn xn − xn ||
||Tσn xn − xn || + ||T xn − Tσn xn ||
||xn − T xn || + 2M σn .
❉♦

σn < +∞✱ t❤❡♦ ❇ê ✤➲ ✷✳✶✳✸ t❛ ❝â ❞➣② {xn − T xn } ❤ë✐ tö ✈➔ ✈➻ ✈➟②
n

lim ||xn − T xn || = 0.

n→+∞

❱➻ ❞➣② {xn } ❜à ❝❤➦♥✱ s✉② r❛ tỗ t tử x
õ ❝♦♥ {xnk }
❝õ❛ ❞➣② {xn } ❤ë✐ tö ②➳✉ ✈➲ x
¯. ❱➻ →♥❤ ①↕ T ❧➔ ♥û❛ ✤â♥❣ ♥➯♥ t❛ ❝â x¯ ∈ ❋✐①T ✳
❍ì♥ ♥ú❛✱ ✈➻

||xn+1 − xn || = αn ||Tσn xn − xn ||

αn σn ||P xn − T xn || + αn ||xn − T xn ||,

♥➯♥ ❞➣② (xn ) t✐➺♠ ❝➟♥ ❝❤➼♥❤ q✉②✱ ♥❣❤➽❛ ❧➔ t❛ ♥❤➟♥ ✤÷đ❝

lim ||xn+1 − xn || = 0.

n→+∞

P❤➛♥ ❝á♥ ❧↕✐ t❛ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ x
¯ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✸✮✳ ❚❤➟t ✈➟②
tø ✭✷✳✹✮ t❛ ❝â

xn+1 − xn = αn (σn (P xn − xn ) + (1 − σn )(T xn − xn )),
♥❣❤➽❛ ❧➔

1
(xn − xn+1 ) =
α n σn

(I − P ) +

1 − σn
(I − T ) xn .
σn

✭✷✳✻✮

1 − σn
(I − T ) ❤ë✐ tư tỵ✐ N❋✐①(T ) ❝ơ♥❣
σn
1 − σn
t❤❡♦ ❇ê ✤➲ ✶✳✶✳✶✹ t❤➻ (I − P ) +
(I − T ) ❤ë✐ tư tỵ✐ (I − P ) + N❋✐①(T ) .

σn
❚✐➳♣ t❤❡♦✱ t❤❛② n ❜ð✐ nk q✉❛ ❣✐ỵ✐ ❤↕♥ tr♦♥❣ ✤➥♥❣ t❤ù❝ ✭✷✳✻✮ ❦❤✐ k → ∞
1
||xn+1 − xn || → 0 ✈➔ (I − P ) + NF ix(T ) ✤â♥❣ ②➳✉✱ t❛ s✉② r❛
✈➔
αn σn
❚❤❡♦ ❇ê ✤➲ ✶✳✶✳✶✹ t❤➻ ❞➣② t♦→♥ tû

0 ∈ (I − P )¯
x + N❋✐①(T ) x¯.


✶✾

❙✉② r❛ x
¯ ❧➔ ♥❣❤✐➺♠ ❜➔✐ t♦→♥ ✭✷✳✸✮✳ ❇➙② ❣✐í ự tỗ t ổ q
ởt tử tứ ❣✐ỵ✐ ❤↕♥ ✭✷✳✺✮✳ ●✐↔ sû x∗ ❧➔ ✤✐➸♠ tư ②➳✉ ❦❤→❝ ❝õ❛ ❞➣②
{xn }✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ x∗ = x¯. ❚ø

||xn − x∗ ||2 = ||xn − x¯||2 + ||¯
x − x∗ ||2 + 2 xn − x¯, x¯ − x∗ ,

✭✷✳✼✮

t❛ t❤➜② r➡♥❣ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❞➣② { xn x
, x x } tỗ t ✈➔ ❜➡♥❣ 0 ❜ð✐
✈➻ x
¯ ❧➔ ✤✐➸♠ tö ②➳✉ ❝õ❛ ❞➣② {xn }✳ ❱➻ ✈➟②✱ ❣✐ỵ✐ ❤↕♥

l(x∗ ) = l(¯

x) + ||¯
x − x∗ ||2 .
❚❤❛② ✈❛✐ trá ❝õ❛ x∗ ✈➔ x
¯ t❛ ❝ô♥❣ ❝â

l(¯
x) = l(x∗ ) + ||¯
x − x∗ ||2 .
✣✐➲✉ ♥➔② s✉② r❛ x∗ = x
¯✳ ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳
❑➳t q✉↔ t✐➳♣ t❤❡♦ ✤÷đ❝ ❝❤➾ r❛ ❜ð✐ ❨❛♦ ✈➔ ▲✐♦✉ ✭✷✵✵✽✮✱ ❝❤➾ r❛ t➼♥❤ ❤ë✐ tö
②➳✉✱ t➼♥❤ t✐➺♠ ❝➟♥ ❝❤➼♥❤ q✉② ❝õ❛ ❞➣② ❧➦♣ rsss

D t ỗ õ rộ
ừ ổ ❣✐❛♥ ❍✐❧❜❡rt H ✳ ●✐↔ sû P, T : D → D ❧➔ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥
t❤ä❛ ♠➣♥ F ix(T ) = ∅✳ ❳➨t ❞➣② {xn} ①→❝ ✤à♥❤ ♥❤÷ tr♦♥❣ ✭✷✳✹✮✳ ❳➨t ❝→❝ ❞➣②
{αn } ✈➔ {σn } ❝→❝ sè t❤ü❝ ♥➠♠ tr♦♥❣ ❦❤♦↔♥❣ (0, 1) t❤ä❛ ♠➣♥ ❝→❝
ỵ



+

n < +



n=0

n→∞

lim

xn+1 − xn
= 0✳
αn σn

❑❤✐ ✤â ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉
✭❛✮ {xn} ❤ë✐ tư ②➳✉ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✱
✭❜✮ {xn} t✐➺♠ ❝➟♥ ❝❤➼♥❤ q✉②✱ tù❝ ❧➔ n→∞
lim xn+1 − xn
✷✳✷

= 0✳

P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❦✐➸✉ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ s✉② rë♥❣

❚r♦♥❣ ♠ư❝ ♥➔②✱ t❛ ①➨t ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✐♥❡①❛❝t ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ s✉②
rë♥❣ ❝õ❛ ❈✳ ❑❛♥③♦✇ ✈➔ ■✳ ❙❤❡❤✉ tr♦♥❣ ❬✹❪✳




ữ ử trữợ sỹ ở tử ♠↕♥❤ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣
❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ ❦❤ỉ♥❣ ✤ó♥❣ tr♦♥❣ tr÷í♥❣ ❤đ♣ tờ qt
tỗ t ởt số t ❜↔♦ sü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣
❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥✳ ▼ët tr♦♥❣ ❝→❝ ♣❤✐➯♥ ❜↔♥ ✤â ❧➔ ❞➣② ❧➦♣
✭✷✳✽✮

xn+1 := αn xn + βn T xn + δn u,

tr♦♥❣ ✤â T : H → C ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✱ C H rộ õ ỗ
{n } {βn }✱ {δn } ∈ [0, 1] ✤÷đ❝ ❝❤å♥ s❛♦ ❝❤♦ αn + βn + δn = 1✱ ✈➔ u
❧➔ tỷ trữợ tr C
tts ♥❣❤✐➯♥ ❝ù✉ sü ❤ë✐ tö ❝õ❛ ❞➣② ❧➦♣

xn+1 := (1 − λn )xn + λn (T xn + en ),

✭✷✳✾✮

x1 H,

ồ ữỡ t rsss ợ en ❜✐➵✉ ❞✐➵♥ s❛✐ sè
❝õ❛ T xn ✳ ❈♦♠❜❡tt❡s ✤➣ ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tư ②➳✉ ❝õ❛ ❞➣② {xn } ✈ỵ✐ ❣✐↔ t❤✐➳t
❋✐①(T ) ❦❤→❝ ré♥❣✱ λn ∈ (0, 1) t❤ä❛ ♠➣♥ ✭✷✳✷✮✱ ✈➔ t❤➯♠ ✤✐➲✉ ❦✐➺♥
∞

λn ||en || < ∞.
n=1

✷✳✷✳✶

❍ë✐ tö ②➳✉

❚r♦♥❣ ♠ö❝ ♥➔②✱ t❛ ①➨t ❞➣② ❧➦♣ {xn+1 } ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝ỉ♥❣ t❤ù❝

xn+1 := αn xn + βn T xn + rn ,

x1 ∈ H, ∀n ≥ 1,

ỗ tớ tr sỹ ở tử ừ ữỡ

K ởt t ỗ õ ré♥❣ ❝õ❛
❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H ✱ T : H → K ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✈➔ ❣✐↔ t❤✐➳t
❋✐①(T ) ❦❤→❝ ré♥❣✳ ❳➨t ❞➣② {xn } ∈ H ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✶✵✮ tr♦♥❣ ✤â rn ❧➔ ✈➨❝tì
♣❤➛♥ ❞÷✳ ●✐↔ sû {αn} ✈➔ {βn} ⊂ [0, 1] t❤ä❛ ♠➣♥ αn + βn 1 ✈ỵ✐ ♠å✐ n ≥ 1
✈➔ t❤ä❛ ♠➣♥ s
ỵ



n n = ∞;

(a)
n=1
∞

||rn || < ∞;

(b)
n=1


✷✶
∞

(1 − αn − βn ) < ∞.

(c)

n=1

❑❤✐ ✤â ❞➣② {xn} ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✶✵✮ ❤ë✐ tö ②➳✉ ✈➲ ✤✐➸♠ t ở ừ
T
ự ự ỵ ỗ ữợ
ữợ ự tỗ t ợ limn→∞ ||xn − x∗ || ✈ỵ✐ ❜➜t ❦ý

x∗ ∈ ❋✐①(T )✳ ❚❤➟t ✈➟②✱ ❝❤å♥ x∗ ∈ ❋✐①(T )✳ ❑❤✐ ✤â tø ✭✷✳✶✵✮ ✈➔ t➼♥❤ ❦❤æ♥❣
❣✐➣♥ ❝õ❛ →♥❤ ①↕ T t❛ ❝â
||xn+1 − x∗ || = ||αn (xn − x∗ ) + βn (T xn − x∗ ) + rn − (1 − αn − βn )x∗ ||
αn ||xn − x∗ || + βn ||T xn − x∗ || + ||rn − (1 − αn − βn )x∗ ||
(αn + βn )||xn − x∗ || + ||rn − (1 − αn − βn )x∗ ||
(αn + βn )||xn − x∗ || + (1 − αn − βn )||rn − x∗ ||
+ (αn + βn )||rn ||
||xn − x∗ || + (1 − αn − βn )M + ||rn ||,
✈ỵ✐ M > 0 tỗ t t (b) ử ờ ✤➲ ✷✳✶✳✸ ✈➔ ❝→❝ ✤✐➲✉ ❦✐➺♥
(b) ✈➔ (c) ❝õ❛ ✤à♥❤ ỵ t s r lim ||xn x || tỗ t r {xn }
n



ữợ ự ♠✐♥❤ lim inf ||xn − T xn || = 0. ❙û ❞ư♥❣ ❇ê ✤➲ ✶✳✶✳✸ t❛ ❝â✱
n→∞

✈ỵ✐ ❜➜t ❦ý x ∈ ❋✐①(T ) t❤➻
∗

||xn+1 − x∗ ||2 = ||αn (xn − x∗ ) + βn (T xn − x∗ ) + rn − (1 − αn − βn )x∗ ||2
||αn (xn − x∗ ) + βn (T xn − x∗ )||2
+ 2 rn − (1 − αn − βn )x∗ , xn+1 − x∗

= αn (αn + βn )||xn − x∗ ||2
+ βn (αn + βn )||T xn − x∗ ||2 − αn βn ||xn − T xn ||2
+ 2 rn − (1 − αn − βn )x∗ , xn+1 − x∗
(αn + βn )2 ||xn − x∗ ||2 − αn βn ||xn − T xn ||2
+ 2 rn − (1 − αn − βn )x∗ , xn+1 − x∗
||xn − x∗ ||2 − αn βn ||xn − T xn ||2
+ 2 rn − (1 − αn − βn )x∗ , xn+1 − x∗
= ||xn − x∗ ||2 − αn βn ||xn − T xn ||2