Về phương pháp lặp tìm điểm bất động của ánh xạ không giãn trong không gian banach

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TRƯỜNG ĐẠI HỌC KHOA HỌC
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PHẠM QUANG DŨNG

VỀ PHƯƠNG PHÁP LẶP
TÌM ĐIỂM BẤT ĐỘNG CỦA ÁNH XẠ KHƠNG GIÃN
TRONG KHƠNG GIAN BANACH

LUẬN VĂN THẠC SĨ TỐN HỌC

THÁI NGUN - 2019


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

PHẠM QUANG DŨNG

VỀ PHƯƠNG PHÁP LẶP
TÌM ĐIỂM BẤT ĐỘNG CỦA ÁNH XẠ KHƠNG GIÃN
TRONG KHƠNG GIAN BANACH
Chun ngành: Tốn 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


▼ư❝ ❧ư❝
▼ð ✤➛✉
❈❤÷ì♥❣ ✶✳ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣
✶✳✶

✶✳✷

❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷
✹
✹

✶✳✶✳✶

❑❤æ♥❣ ❣✐❛♥ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✶✳✷

❑❤æ♥❣ ❣✐❛♥ ỗ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣

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

✼

✶✳✷✳✶

⑩♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✶✳✷✳✷

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

✾

❈❤÷ì♥❣ ✷✳ ❱➲ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕
❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
✶✶
✷✳✶

✷✳✷

✷✳✸

❈❤➼♥❤ q✉② t✐➺♠ ❝➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

✷✳✶✳✶

❉→♥❣ ✤✐➺✉ ❝❤➼♥❤ q✉② t✐➺♠ ❝➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

✷✳✶✳✷

❉→♥❣ ✤✐➺✉ ❝❤➼♥❤ q✉② t✐➺♠ ❝➟♥ ✤➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹

❙ü ❤ë✐ tö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳





ỵ ở tö ♠↕♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳





ỵ ở tử ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✸

P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❦✐➸✉ ❍❛❧♣❡r♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✹

✷✳✸✳✶

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

✷✹

✷✳✸✳✷

❙ü ❤ë✐ tö

✷✹

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

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

✸✸
✸✹


▼ð ✤➛✉
❇➔✐ t♦→♥ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ✤➣ ✈➔ ✤❛♥❣ ❧➔ ♠ët ❝❤õ ✤➲ t❤✉ ❤ót
sü q✉❛♥ t ừ t ồ tr ữợ
ởt tr ỳ ữợ ự t t ✤ë♥❣ ❧➔ ①➙② ❞ü♥❣
♣❤÷ì♥❣ ♣❤→♣ t➻♠ ✭①➜♣ ①➾✮ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
❤♦➦❝ ❦❤æ♥❣ ❣✐❛♥ t q tợ ữỡ ①➾

♥➔② ✤➣ ✤÷đ❝ ✤➦t r❛ ✈➔ ❣✐↔✐ q✉②➳t ❝❤♦ tø♥❣ ❧ỵ♣ →♥❤ ①↕ ❦❤→❝ ♥❤❛✉✱ ❝❤➥♥❣ ❤↕♥
❧ỵ♣ →♥❤ ①↕ ❝♦✱ ❧ỵ♣ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✱✳ ✳ ✳ ❱ỵ✐ ❧✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣ t❤↕❝ s➽✱ tæ✐
❧ü❛ ❝❤å♥ ♠ët ♣❤➛♥ tr♦♥❣ ❜➔✐ t♦→♥ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤♦ ❧ỵ♣ →♥❤ ①↕ ổ
tr ổ ữợ sỹ ữợ ừ r ỵ
tổ ồ t ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤
①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✧✳ ◆ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔②
tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✱ ❝ư t❤➸ ♥❤÷ s❛✉✿
❈❤÷ì♥❣ ✶✿ ❚r➻♥❤ ổ ỗ ỗ t ❜➔✐ t♦→♥
t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳
❈❤÷ì♥❣ ✷✿ ❚r➻♥❤ ❜➔② ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤
①↕ ❦❤æ♥❣ ❣✐➣♥ tr ổ ũ ỵ ở tử ②➳✉✱ ❤ë✐ tư
♠↕♥❤ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣✳
❚r♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ t↕✐ ❚r÷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 ❤✉② sỹ ỳ ỵ t ❤å❝ ✈è♥
❞➽ ✤➣ r➜t ✤➭♣✳ ✣➙② ❝ơ♥❣ ❧➔ ♠ët ❝ì ❤ë✐ ❝❤♦ ❡♠ ❣û✐ ❧í✐ tr✐ ➙♥ tỵ✐ t➟♣ t❤➸ ❝→❝
t❤➛② ❝ỉ ❣✐↔♥❣ ✈✐➯♥ ❝õ❛ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✕ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ♥â✐
✷


✸

❝❤✉♥❣ ✈➔ ❑❤♦❛ ❚♦→♥ ✕ ❚✐♥ ♥â✐ r✐➯♥❣✱ ✤➣ tr✉②➲♥ tử tự
ồ qỵ tr t❤í✐ ❣✐❛♥ ❡♠ ✤÷đ❝ ❧➔ ❤å❝ ✈✐➯♥ ❝õ❛ tr÷í♥❣✳
❚→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉ tr÷í♥❣ ❚❍P❚ ❚❤❛♥❤ ❚❤õ②✱
P❤ó ❚❤å ❝ị♥❣ t♦➔♥ t❤➸ ❝→❝ ❛♥❤ ❝❤à ❡♠ ỗ t tốt t
t tr♦♥❣ t❤í✐ ❣✐❛♥ ✤✐ ❤å❝ ❈❛♦ ❤å❝❀ ❝↔♠ ì♥ ❝→❝ ❛♥❤ ❝❤à ❡♠ ❤å❝ ✈✐➯♥
❧ỵ♣ ❈❛♦ ❤å❝ ❚♦→♥ ❑✶✶ ✈➔ ỗ tr ờ ở ❦❤➼❝❤

❧➺ t→❝ ❣✐↔ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❧➔♠ ❧✉➟♥ ✈➠♥ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛
❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳
✣➦❝ ❜✐➺t ❡♠ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s s tợ ữợ
r ỵ ✤➣ ❧✉æ♥ q✉❛♥ t➙♠ ➙♥ ❝➛♥ ❝❤➾ ❜↔♦✱ ✤ë♥❣ ✈✐➯♥
ú ù t t õ ỵ s s ❝❤♦ ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ❝ơ♥❣
♥❤÷ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✳ ❈❤➦♥❣ ✤÷í♥❣ ✈ø❛ q✉❛ s➩ ❧➔ ♥❤ú♥❣ ợ
ỵ ố ợ ❛♥❤ ❝❤à ❡♠ ❤å❝ ✈✐➯♥ ❧ỵ♣ ❑✶✶ ♥â✐ ❝❤✉♥❣ ✈➔ ✈ỵ✐ ❜↔♥
t❤➙♥ ❡♠ ♥â✐ r✐➯♥❣✳ ❉➜✉ ➜♥ ➜② ❤✐➸♥ ♥❤✐➯♥ ❦❤ỉ♥❣ t❤➸ t❤✐➳✉ sü ❤é trđ✱ s➫ ❝❤✐❛
✤➛② ②➯✉ t❤÷ì♥❣ ❝õ❛ ❝❤❛ ♠➭ ❤❛✐ ❜➯♥ ✈➔ ❝→❝ ❛♥❤ ❝❤à ❡♠ ❝♦♥ ❝❤→✉ tr♦♥❣ ❣✐❛
✤➻♥❤✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ t➜t ỳ ữớ t ú ù ỗ
ũ ❡♠ tr➯♥ ❝❤➦♥❣ ✤÷í♥❣ ✈ø❛ q✉❛✳ ▼ët ❧➛♥ ♥ú❛✱ ❡♠ ①✐♥ tr➙♥ trå♥❣
❝↔♠ ì♥✦

❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✷✷ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✾

❍å❝ ✈✐➯♥

P❤↕♠ ◗✉❛♥❣ ❉ô♥❣


❈❤÷ì♥❣ ✶

❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠
❜➜t ✤ë♥❣
❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❤➻♥❤ ❤å❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ❜➔✐
t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❑✐➳♥ t❤ù❝ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝
tê♥❣ ❤đ♣ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪ ✈➔ ❬✹❪✳

✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤

✶✳✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ỗ
X ổ x0 X trữợ ỵ Sr (x0 ) t
t x0 ❜→♥ ❦➼♥❤ r > 0✱

Sr (x0 ) := {x ∈ X : ||x − x0 || = r}.

✣à♥❤ ♥❣❤➽❛ ổ X ữủ ồ ỗ

(0, 2]
t ý tỗ t = ( ) > 0 s❛♦ ❝❤♦ ♥➳✉ x, y ∈ X ✈ỵ✐ ||x|| = 1, ||y|| = 1 ✈➔
1
||x − y|| ≥ t
(x + y) 1 .
2
t q ữợ ởt ử ổ ỗ

ỵ ổ Lp[a, b] ợ 1 < p < ổ ỗ
ỵ sỷ X ổ ỗ õ ợ t ý
d > 0,

>0

tỡ tũ ỵ x, y ∈ X ✈ỵ✐ ||x|| ≤ d, ||y|| ≤ d, ||x − y|| ≥ ✱
✹




tỗ t > 0 s
1

(x + y) 1 − δ
d.
2
d
❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ❜➜t ❦ý x, y ∈ X ✱ ①➨t z1 = xd , z2 = yd ✱ ✈➔ t➟♣ ¯ = d ✳ ❍✐➸♥
1
♥❤✐➯♥ ¯ > 0. ❍ì♥ ♥ú❛✱ ||z1 || ≤ 1, ||z2 || ≤ 1 ✈➔ ||z1 − z2 || = ||x − y|| = .
d
d
ứ t ỗ t õ = δ

d

> 0,

1
(z1 + z2 ) 1 − δ(¯),
2
♥❣❤➽❛ ❧➔

1
(x + y) ≤ 1 − δ
,
2d
d
s✉② r❛

1
(x + y) ≤ 1 − δ
2

d

d.

❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤

▼➺♥❤ ✤➲ ✶✳✶✳✹✳ ❈❤♦ X ổ ỗ sỷ α ∈ (0, 1)✱
> 0✳

❑❤✐ ✤â ✈ỵ✐ ❜➜t ❦ý d > 0✱ ♥➳✉ x, y ∈ X t❤ä❛ ♠➣♥ ||x|| ≤ d✱ ||y|| ≤ d✱
||x − y|| ≥ ✱ t❤➻ tỗ t =
> 0 s
d
||x + (1 − α)|| ≤ 1 − 2δ

d

min{α, 1 − α} d.

✶✳✶✳✷ ổ ỗ t
ổ X ữủ ồ ỗ t ợ ồ
x, y ∈ X ✱ x = y, ||x|| = ||y|| = 1✱ t❛ ❝â ||λx + (1 − λ)y|| < 1 (0, 1).

ỵ ồ ổ ỗ ổ ỗ t
ỵ r ởt ợ ổ ỗ t ổ
ồ ổ ỗ t ữợ ởt ử ổ
ỗ t ữ ỗ

ử trữợ à > 0 t C[0, 1] ợ ||.||à ữ
s

1

x2 (t)dt

||x||à := ||x||0 + à

1
2

0

ợ ||.||0 s õ

||x||0 ||x||µ (1 + µ)||x||0 ,

x ∈ C[0, 1],

✈➔ ❤❛✐ ❝❤✉➞♥ tữỡ ữỡ ||.||à ||.||0 ợ à (C[0, 1], ||.||0 )
ổ ỗ tr ợ t ý à > 0, (C[0, 1], ||.||à ) ỗ t
ợ t ý
x+y

2

(0, 2] tỗ t x, y C[0, 1] ợ ||x||à = ||y||à = 1, ||x y|| =
tũ ỵ 1 (C[0, 1], ||.||à ) ổ ỗ

ử t µ0 ✈➔ c0 = c0(N) ✈ỵ✐ ❝❤✉➞♥ ||.||µ ①→❝ ✤à♥❤ ợ x = {xn} c0

ữ s



||x||à := ||x||c0 + µ
i=1

xi
i

2

1
2

tr♦♥❣ ✤â ||.||c0 ❧➔ ❝❤✉➞♥ t❤ỉ♥❣ t❤÷í♥❣✳ ◆❤÷ tr♦♥❣ ✈➼ ử tr (c0 , ||.||à ) ợ

à > 0 ỗ t ữ ổ ỗ tr c0 ợ tổ tữớ
ổ ỗ t
t tr ữủ ồ ừ t ỗ ừ ổ
X ỵ X : (0, 2] [0, 1]) ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠
♥➔②

▼➺♥❤ ✤➲ ✶✳✶✳✾✳ (a) ❱ỵ✐ ♠å✐ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ❤➔♠ δX ( ) ổ

tr (0, 2].
(b) s ừ t ỗ ừ ổ tử

ỗ
(c) r ổ ỗ X s ừ t ỗ ừ ổ
X t➠♥❣ t❤ü❝ sü✳
❚r♦♥❣ ♠ư❝ ♥➔② tr➻♥❤ ❜➔② ❝→❝ ✤➦❝ tr÷♥❣ ừ ổ ỗ

ỵ ổ X ỗ X ( ) > 0 ✈ỵ✐

♠å✐

∈ (0, 2].




q r ổ ỗ X ừ t

ỗ t t
ỵ X ổ ỗ t X ❧➔ ❦❤æ♥❣ ❣✐❛♥
♣❤↔♥ ①↕✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✸✳

✶✳ ❈❤✉➞♥ ❝õ❛ ❦❤æ♥❣ X ữủ ồ

t ợ ♠é✐ y ∈ SX t❤➻ ❣✐ỵ✐ ❤↕♥

lim
t→0

x + ty − x

t

tỗ t ợ x SX ỵ y,

x ✳ ❑❤✐ ✤â

✭✶✳✶✮

x ✤÷đ❝ ❣å✐ ❧➔ ✤↕♦

❤➔♠ ●➙t❡❛✉① ❝õ❛ ❝❤✉➞♥✳
✷✳ ❈❤✉➞♥ ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉ ♥➳✉ ✈ỵ✐ ♠é✐ y ∈ SX ✱ ❣✐ỵ✐
❤↕♥ ✭✶✳✶✮ t ữủ ợ ồ x SX
ừ X ữủ ồ rt ợ ộ x SX ợ
tỗ t ợ ồ y SX
ừ X ữủ ❣å✐ ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ✤➲✉ ♥➳✉ ❣✐ỵ✐ ❤↕♥ ✭✶✳✶✮ tỗ t
ợ ồ x, y SX

t t ở

ổ
ỵ 2X ❧➔ t➟♣ t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ X ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ⑩♥❤ ①↕ J s : X → 2X ✱ s > 1 ✭♥â✐ ❝❤✉♥❣ ❧➔ ✤❛ trà✮ ①→❝
∗

✤à♥❤ ❜ð✐

J s (x) = x∗ ∈ X ∗ : x∗ , x = x∗

x , x∗ = x

s−1

x ∈ X,

✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ ✤è✐ ♥❣➝✉ tê♥❣ q✉→t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ X ✳ ❑❤✐ s = 2✱
→♥❤ ①↕ J 2 ữủ ỵ J ữủ ồ ố t ừ

X
ỵ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ✤ì♥ trà ❧➔ j ✳


✽

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

❚➼♥❤ ✤ì♥ trà ❝õ❛ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ❝â ♠è✐ ❧✐➯♥ ❤➺ ✈ỵ✐ t➼♥❤ ❦❤↔ ✈✐
❝õ❛ ❝❤✉➞♥ ❝õ❛ ổ ữ tr ỵ s

ỵ X ổ ✈ỵ✐ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝

❑❤✐ ✤â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
(i) X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ trì♥❀
(ii) J ❧➔ ✤ì♥ trà❀
(iii) ❈❤✉➞♥ ❝õ❛ X ❧➔ ❦❤↔ ✈✐ t ợ
x = x 1 Jx
ỵ sỷ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ t❤ü❝✱ ✈➔ →♥❤ ①↕ Jp :
X −→ 2X , 1 < p < ∞✱ ❧➔ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝✳ ❑❤✐ ✤â ✈ỵ✐ ❜➜t ❦ý

x, y ∈ X, t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ s❛✉
∗

J : X → 2X .

∗

||x + y||p ≤ ||x||p + p y, jp (x + y)

✭✶✳✷✮

✈ỵ✐ ♠å✐ jp(x + y) ∈ Jp(x + y). ✣➦❝ ❜✐➺t ♥➳✉ p = 2 t❤➻
||x + y||2 ≤ ||x||2 + 2 y, j( x + y)

✭✶✳✸✮

✈ỵ✐ ♠å✐ j(x + y) ∈ J(x + y)

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✺✳ ❈❤♦ C ❧➔ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✳

(i) ⑩♥❤ ①↕ T : C → E ✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ L✲❧✐➯♥ tư❝ st tỗ t
số L 0 s
Tx − Ty ≤ L x − y

∀x, y ∈ C.

✭✶✳✹✮

(ii) ❚r♦♥❣ ✭✶✳✹✮✱ ♥➳✉ L ∈ [0, 1) t❤➻ T ✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ ❝♦❀ ♥➳✉ L = 1 t❤➻ T
✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳

▼å✐ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✱ ✈ỵ✐ t➟♣ ❝→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ ❦❤→❝ ré♥❣ ❧➔ tü❛ ❦❤æ♥❣
❣✐➣♥✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✻✳ ●✐↔ sû K ❧➔ t➟♣ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤
❝❤✉➞♥ X ✳ ❳➨t T : K → E ❧➔ →♥❤ ①↕ t❤ä❛ ♠➣♥ ❋✐①(T ) = ∅✳ ⑩♥❤ ①↕ T ✤÷đ❝

❣å✐ ❧➔ tü❛ ❦❤ỉ♥❣ ❣✐➣♥ ♥➳✉ T x − T x∗ ≤ x − x∗ t❤ä❛ ♠➣♥ ✈ỵ✐ ♠å✐ x ∈ K
✈➔ x∗ ∈ F (T )✳




t t ở
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➟♣ (T )
t ỗ õ
t t ở ữủ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉✳

❇➔✐ t♦→♥✳ ❈❤♦ T : C → C ổ tứ t ỗ ✤â♥❣✱ ❦❤→❝
ré♥❣ C ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✈➔♦ ❝❤➼♥❤ ♥â ✈ỵ✐ ❋✐①(T ) = ∅✳ ❚➻♠ ♣❤➛♥ tû

x∗ ∈ (T )

ỵ ỵ ●✐↔ sû (M, p) ❧➔ ❦❤æ♥❣ ❣✐❛♥

♠❡tr✐❝ ✤➛② ✤õ ✈➔ T : M → M ❧➔ →♥❤ ①↕ ❝♦✳ ❑❤✐ õ T õ t t
ở tỗ t↕✐ ❞✉② ♥❤➜t x∗ ∈ M t❤ä❛ ♠➣♥ T x∗ = x∗✳ ◆❣♦➔✐ r❛✱ ✈ỵ✐ ❜➜t

❦ý x0 ∈ M ✱ ❞➣② {xn} ①→❝ ✤à♥❤ ♥❤÷ s❛✉ xn+1 = T xn, n ≥ 0✱ ❤ë✐ tư tỵ✐ ✤✐➸♠
❜➜t ✤ë♥❣ ❝õ❛ →♥❤ T
ỵ t X ổ K t ỗ
õ ❝❤➦♥ ❝õ❛ X ✈ỵ✐ ❝➜✉ tró❝ ❝❤✉➞♥✳ ●✐↔ sû T : K → K ❧➔ →♥❤ ①↕ ❦❤æ♥❣
❣✐➣♥✳ ❑❤✐ ✤â T ❝â ✤✐➸♠ ❜➜t ✤ë♥❣✳
❑❤ỉ♥❣ ❣✐è♥❣ ♥❤÷ tr÷í♥❣ ❤đ♣ tr♦♥❣ ỵ ởt
ử t tữớ ❝❤➾ r❛ ♠ët ❞➣② ①→❝ ✤à♥❤ ♥❤÷ s❛✉ xn+1 = T xn , x0 ∈ K, n ≥ 0,
✭tr♦♥❣ ✤â K t rộ ỗ õ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
t❤ü❝ X ✱ ✈ỵ✐ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐❛♥ ①↕ T : K → K ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣✱
❝â t❤➸ ❦❤ỉ♥❣ ❤ë✐ tư ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣✳
◆➠♠ ✶✾✺✺✱ ❑r❛s♥♦s❡❧s❦✐✐✱ ✤➣ ❝❤➾ tr❛ tr♦♥❣ ✈➼ ❞ö✱ t❛ ❝â t❤➸ t❤✉ ✤÷đ❝ ♠ët
1
❞➣② ❤ë✐ tư ♥➳✉ t❤❛② t❤➸ T ❜ð✐ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ (I + T )✱ tr õ I
2
ỗ t ❞➣② ①➜♣ ①➾ ❧✐➯♥ t✐➳♣ ①→❝ ✤à♥❤ ❜ð✐ x0 ∈ K ✱ ✈➔
1
✭✶✳✺✮
xn+1 = (xn + T xn ), n = 0, 1, ...
2
❞➣② ❧➦♣ ♥➔② t❤÷í♥❣ ✤÷đ❝ ❣å✐ ❧➔ ❞➣② ❧➦♣ P✐❝❛r❞✱ xn+1 = T xn , x0 ∈ K, n ≥ 0.
1
❈→❝ →♥❤ ①↕ T ✈➔ (I + T ) ❝â ❝ò♥❣ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣✱ ✈➻ ✈➟② ❣✐ỵ✐ ❤↕♥ ❝õ❛
2
❞➣② ①→❝ ✤à♥❤ ❜ð✐ ❞➣② ❧➦♣ ✭✶✳✺✮ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ✳


✶✵

❚ê♥❣ q✉→t ❤ì♥✱ ♥➳✉ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ K t
ỗ ừ X ♠ët ♠ð rë♥❣ ❝õ❛ ❞➣② ❧➦♣ ✭✶✳✺✮ tr♦♥❣ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣

❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ T : K K ú tỗ t ữ s

x0 ∈ K,
xn+1 = (1 − λ)xn + λT xn , n = 0, 1, ...; λ ∈ (0, 1)

✭✶✳✻✮

✈ỵ✐ λ ❧➔ ❤➡♥❣ sè✳
❚✉② ♥❤✐➯♥✱ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t ♥❤➜t ❧➔ ❞➣② ❧➦♣ ❧♦↕✐ ▼❛♥♥ ①→❝ ✤à♥❤ ♥❤÷
s❛✉✿ x0 ∈ K ✱

xn+1 = (1 − Cn )xn + Cn T xn , n = 0, 1, 2, ...

✭✶✳✼✮

tr♦♥❣ ✤â {Cn }∞
n=1 ⊂ (0, 1) ❧➔ ❞➣② sè t❤ü❝ t❤ä❛ ♠➣♥ ♠ët sè ✤✐➲✉ ❦✐➺♥ t❤➼❝❤
❤đ♣✳ ❱ỵ✐ ❝→❝ ❣✐↔ t❤✐➳t ❜ê s✉♥❣ ♥❤÷ s❛✉
✭✐✮ lim Cn=0 ❀
✭✐✐✮

n

C n = ∞✱

i=1

❞➣② {xn } ①→❝ ✤à♥❤ tr♦♥❣ ✭✶✳✼✮ ❧➔ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t ❝õ❛ ❞➣② ❧➦♣ ▼❛♥♥
✭✶✾✺✸✮✳ ❉➣② ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝ tr✉② ỗ ữủ ồ ổ tự
rsss t t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳ P❤➛♥ t✐➳♣

t❤❡♦✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ q✉❛♥ trå♥❣ ✤➸ ①➜♣ ①➾ ✤✐➸♠
❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳


❈❤÷ì♥❣ ✷

❱➲ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ t➻♠ ✤✐➸♠ ❜➜t
✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr♦♥❣
❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✈➲ t➼♥❤ ❝❤➼♥❤ q✉② t✐➺♠ ❝➟♥ ❝õ❛ →♥❤
①↕ ❦❤ỉ♥❣ ❣✐➣♥ ❝ị♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❍❛❧♣❡r♥ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤
①↕ ❦❤æ♥❣ ❣✐➣♥✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ tê♥❣ ❤đ♣ tø ❝→❝ ❜➔✐ ❜→♦ ❬✸❪✕❬✶✷❪✳

✷✳✶ ❈❤➼♥❤ q✉② t✐➺♠ ❝➟♥

✷✳✶✳✶ ❉→♥❣ ✤✐➺✉ ❝❤➼♥❤ q✉② t✐➺♠ ❝➟♥
●✐↔ sû T : K → K ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr t K ỗ ừ ổ
X ✣➦t

Sλ := λI + (1 − λ)T,

λ ∈ (0, 1),

tr♦♥❣ õ I ỗ t tr K ợ x0 K trữợ t
{Sλn (x0 )} ❜ð✐

Sλn (x0 ) = λxn + (1 − λ)T xn ,

xn = Sλn−1 (x0 ).

❚r♦♥❣ ❬✽❪✱ ❑r❛s♥♦s❡❧s❦✐✐ ✤➣ ❝❤➾ r❛ r➡♥❣✱ ♥➳✉ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ỗ
K t t t ợ t ý x0 ∈ K ✱ ❞➣② ❧➦♣

S n1 (x0 )
2

∞

n=1

, ✈ỵ✐

S 12 = 12 (I + T ) ❤ë✐ tö ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ✳ ❙❝❤❛❡❢❡r ❝❤➾ r❛ ✤✐➲✉ ♥➔②
✶✶


✶✷

✈➝♥ ✤ó♥❣ ✈ỵ✐ ❜➜t ❦ý Sλ = λI + (1 − λ)T tr♦♥❣ ✤â 0 < λ < 1✱ ✈➔ st
r ữủ ỗ t ừ ổ ❣✐❛♥ ❇❛♥❛❝❤ X ❧➔ ✤✐➲✉ ❦✐➺♥ ✤õ✳ ▼ët
❝➙✉ ❤ä✐ ✤➦t r❛✱ ❧➔ ❧✐➺✉ ❝â t❤➸ ❧÷đ❝ ❜ä ✤✐➲✉ ❦✐➺♥ ❦❤ỉ♥❣ X ỗ t
ổ ọ ♥➔② ✤➣ ✤÷đ❝ ❝❤➾ r❛ ❜ð✐ ■s❤✐❦❛✇❛ ✭①❡♠ ❬✼❪✮ tr♦♥❣
✤à♥❤ ỵ ữợ

ỵ sỷ K ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤

✈➔ T : K → X ❧➔ ♠ët →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳ ❱ỵ✐ x0 ∈ K ✱ ❞➣② ❧➦♣ {xn}∞n=1
①→❝ ✤à♥❤ ❜ð✐
X

xn+1 = (1 − Cn )xn + Cn T xn ,

n = 0, 1, 2, . . .

✭✷✳✶✮

tr♦♥❣ ✤â ❞➣② sè t❤ü❝ {Cn}∞n=0 t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿
∞

(a)

Cn
n=0

♣❤➙♥ ❦ý✱

✈ỵ✐ ♠å✐ sè ữỡ n
xn K ợ ồ số ❞÷ì♥❣ n✳ ❑❤✐ ✤â✱ ♥➳✉ ❞➣② {xn}∞n=1 ❜à ❝❤➦♥✱ t❤➻
xn − T xn → 0 ❦❤✐ n → ∞✳
(b) 0 ≤ Cn ≤ b ≤ 1

▼ët ❤➺ q✉↔ ❝õ❛ ✣à♥❤ ỵ t K ỗ t t ❞➣② {xn }
①→❝ ✤à♥❤ ♥❤÷♥❣ tr♦♥❣ ✭✷✳✶✮ ❤ë✐ tư ♠↕♥❤ t ở ừ T
ỵ ữợ ởt q ừ ỵ K
t ỗ T tø K ✈➔♦ t➟♣ ❝♦♥ ✤â♥❣ ❜à ❝❤➦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤

X t❤➻ →♥❤ ①↕
Sλ = (1 − λ)I + λT,

λ ∈ (0, 1)

❧➔ ❝❤➼♥❤ q✉② t✐➺♠ ❝➟♥ x✱ ♥❣❤➽❛ ❧➔✱

Sλn+1 x − Sλn x → 0 ❦❤✐ n → ∞.
❑❤→✐ ♥✐➺♠ ❝❤➼♥❤ q✉② t✐➺♠ ❝➟♥ ✤÷đ❝ ✤÷❛ r❛ ❜ð✐ ❇r♦✇❞❡r ✈➔ P❡tr②s❤②♥ tr♦♥❣
❬✸❪✳ ❚➼♥❤ ❝❤➼♥❤ q✉② t✐➺♠ ❝➟♥ ❧✐➯♥ q tợ t tỗ t t ở ừ
T ữủ t t q ữợ

ỵ ✷✳✶✳✷✳ ●✐↔ sû M ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈➔ T : M → M ❧➔ →♥❤ ①↕

❧✐➯♥ tö❝ ✈➔ ❝❤➼♥❤ q✉② t✐➺♠ ❝➟♥ t↕✐ x0 ∈ M ✳ ❑❤✐ ✤â ❜➜t ❦ý ✤✐➸♠ tö ②➳✉ ❝õ❛
❞➣② {T n(x0)}∞n=1 ✤➲✉ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ✳


✶✸

◆❤÷ ✈➟②✱ tø t➼♥❤ ❧✐➯♥ tư❝ ❝õ❛ →♥❤ ①↕ T ✱ t➼♥❤ ❝❤➼♥❤ q✉② t✐➺♠ ❝➟♥ ❝õ❛ Sλ t↕✐

x0 ∈ K ❜➜t ❦ý s✉② r❛ Sλ (p) = p ✈ỵ✐ ♠å✐ ✤✐➸♠ tö p ❝õ❛ ❞➣② {Sλn (x0)}∞
n=1 ✳ ❚➼♥❤
❝❤➼♥❤ q✉② t✐➺♠ ❝➟♥ ❦❤æ♥❣ ❝❤➾ ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣ tr ự t
tỗ t t ở ừ ①↕ ♠➔ ❝á♥ ❝❤➾ r❛ ❝→❝ tr÷í♥❣ ❤đ♣ ❝ư t❤➸✱ ❞➣②
❧➦♣ ❤ë✐ tö ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ♠ët →♥❤ ử t ữủ tr tr
ữợ

✤➲ ✷✳✶✳✸✳ ❈❤♦ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ G : E → E ❧➔ ♠ët →♥❤ ①↕
t✉②➳♥ t➼♥❤ tø X ✈➔♦ X ✳ ●✐↔ sû G ❜à ❝❤➦♥ ♠↕♥❤✱ ♥❣❤➽❛ ❧➔✱ ✈ỵ✐ k ≥ 0, Gn ≤
k ✱ n = 1, 2, . . .✱ ✈➔ ❝❤➼♥❤ q✉② t✐➺♠ ❝➟♥✳ ◆➳✉ ✈ỵ✐ ♠é✐ x0 ∈ E ✱ co {Gn (x0 )} ❝❤ù❛
✤✐➸♠ ❜➜t ✤ë♥❣ x∗ ❝õ❛ G✱ t❤➻ ❞➣② {Gn(x0)} ❤ë✐ tö ♠↕♥❤ ✈➲ x∗✳
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ε > 0 trữợ y co {Gn(x0)} ợ x y < 2(k ε+ 1) ✳

✣➦t y =

m

λj Gj (x0 ) tø t➼♥❤ t✉②➳♥ t➼♥❤ ❝õ❛ →♥❤ ①↕ G t❛ ❝â

j=1

Gn (x0 − x∗ ) = Gn (x0 − y) + Gn (y − x∗ )
m
n

λj Gj (x0 )

x0 −

=G

+ +Gn (y − x∗ )

j=1
m

λj Gn (x0 ) − Gn+j (x0 ) + Gn (y − x∗ ),

=
j=1

✈➻

m

λj = 1✳ ❱➻ ✈➟②✱

j=1
m
∗

n

λj Gn (x0 ) − Gn+j (x0 )

G (x0 − x ) ≤
j=1

+

kε
2(k + 1)

kε
✳
2(k + 1)
❚ø t➼♥❤ ❝❤➼♥❤ q✉② t ừ G tỗ t số N0 > 0 s❛♦ ❝❤♦ ✈ỵ✐
♠å✐ n ≥ N0 , t❛ ❝â
ε
Gn (x0 ) − Gn+j (x0 ) ≤ , (j = 1, 2, . . . , m).
2
❞♦ Gn (y − x∗ ) ≤ Gn . (y − x∗ ) ≤

❱➻ ✈➟②✱

m
n

∗

G (x0 − x ) <

λj
j=1

ε
ε
+ =ε
2
2

∀n ≥ N0 .


✶✹

❚❛ s✉② r❛ Gn (x0 − x∗ ) = Gn (x0 ) − x∗ → 0 ❦❤✐ n → ∞✳

◆❤➟♥ t ứ ỵ t t r X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
✈➔ t➟♣ ❝♦♥ K ❝õ❛ X ❧➔ t➟♣ ❝♦♠♣❛❝t ②➳✉ t❤➻✱ tr♦♥❣ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t✱ ❞➣②

{Sλn (x0 )} s➩ ❦❤ỉ♥❣ ❝â ✤✐➸♠ tư ♠↕♥❤✱ ♥❤÷ tr ử ữợ

ử ỗ t t ỗ õ K tr ổ rt
l2 ♠ët →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ T : K → K ✈➔ ✤✐➸♠ x0 ∈ K t❤ä❛ ♠➣♥
❦❤ỉ♥❣ ❤ë✐ tư t❤❡♦ ❝❤✉➞♥✳

S n1 (x0 )
2

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✻✳ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ X ữủ ồ ổ

ợ ồ {xn }∞
n=0 tr♦♥❣ X s❛♦ ❝❤♦ {xn }n=0 ❤ë✐ tö ②➳✉ ✈➲ x ∈ X t❤➻ ❜➜t

✤➥♥❣ t❤ù❝

lim inf xn − y > lim inf xn − x
n→∞

n→∞

t❤ä❛ ♠➣♥ ✈ỵ✐ ♠å✐ y = x✳

◆❤➟♥ ①➨t ✷✳✶✳✼✳ ❱ỵ✐ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ t ý X sỹ tỗ t

ừ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❧✐➯♥ tö❝ ②➳✉ s✉② r❛ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❖♣✐❛❧✱ ♥❤÷♥❣

✤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❦❤ỉ♥❣ ✤ó♥❣✳ ✣➦❝ ❜✐➳t✱ lp (1 < p < ∞) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❖♣✐❛❧✱
♥❤÷♥❣ Lp (1 < p < ∞✱ p = 2) ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❖♣✐❛❧✳
●✐↔ sû K ❧➔ t➟♣ ỗ t ừ ổ tỹ X ✈➔

T : K → K ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ❚r♦♥❣ ❱➼ ❞ư ✷✳✶✳✺ ❝❤➾ r❛ r➡♥❣✱ ♥â✐ ❝❤✉♥❣✱

t❛ ❦❤ỉ♥❣ ♥❤➟♥ ✤÷đ❝ sü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ❞➣② ❧➦♣ ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✶✮ tỵ✐ ✤✐➸♠
❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ỵ ữợ r ♥➔② ❤ë✐ tư ②➳✉
tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ♥➳✉ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❖♣✐❛❧✳

✷✳✶✳✷ ❉→♥❣ ✤✐➺✉ ❝❤➼♥❤ q✉② t✐➺♠ ❝➟♥ ✤➲✉
✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✽✳ ❈❤♦ K ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ X ✳
⑩♥❤ ①↕ U : K → X ✤÷đ❝ ❣å✐ ❧➔

❝❤➼♥❤ q✉② t✐➺♠ ❝➟♥ ✤➲✉ ợ t ý > 0

tỗ t số N > 0 s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ x0 ∈ K ✈➔ ✈ỵ✐ ♠å✐ n ≥ N, t❛ ❝â

U n+1 x0 − U n x0 < ✳

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✾✳ ❈❤♦ t➟♣ ❤đ♣ A ✈➔ x0 ∈ A✳ ❉➣② {xn}∞n=0 ✤÷đ❝ ❣å✐
ữủ tỗ t ởt ổ t {Cn }∞
n=0 tr♦♥❣ ❦❤♦↔♥❣ (0, 1) s❛♦
❝❤♦ ✭✷✳✶✮ t❤ä❛ ♠➣♥✳


✶✺

❚✐➳♣ t❤❡♦✱ t❛ tr➻♥❤ ❜➔② ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè t q s

ỵ K ởt t ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ X ✈➔

❧➔ ♠ët →♥❤ ổ sỷ ợ x0 K tỗ t↕✐ ♠ët ❞➣②
❝❤➜♣ ♥❤➟♥ ✤÷đ❝ {xn}∞n=0 ⊆ K ❜à ❝❤➦♥✳ ❑❤✐ ✤â n→∞
lim xn+1 − xn = 0✳ ◆❣♦➔✐
r❛✱ ♥➳✉ K ❧➔ t➟♣ ❝♦♥ ❜à ❝❤➦♥ ❝õ❛ X ✱ t❤➻ ợ tr

ỵ ợ K X f ữủ ữ tr ỵ ợ
x0 K sỷ tỗ t ♥❤➟♥ ✤÷đ❝ {xn }∞
n=0 ⊆ K ❜à ❝❤➦♥✱ s❛♦ ❝❤♦
❞➣② ❦❤ỉ♥❣ t➠♥❣ {Cn}∞n=0 ❝ơ♥❣ t❤ä❛ ♠➣♥ 0 < a ≤ Cn < 1 ✈ỵ✐ ♠å✐ n ≥ 1✳ ❑❤✐
✤â n→∞
lim xn f (xn ) = 0
ỵ K ❧➔ t➟♣ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ X ✈➔ →♥❤
①↕ ❦❤æ♥❣ ❣✐➣♥ f : K → X ✳ sỷ tỗ t t A K s ợ ộ x0 A
tỗ t ữủ {xn}n=0 A sỷ tỗ t > 0 s ợ
ộ số ữỡ N, ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ {xn}∞n=0 ⊆ A✱ ❝â
f :K →X

✭✻✳✹✮

sup xk+1 − xk > δ.
k≥N

❑❤✐ ✤â✱ A ❦❤æ♥❣ ❜à ❝❤➦♥✳
❚❛ ❝❤ù♥❣ ỵ ỵ tứ ỵ

ự ự ỵ s trỹ t

tứ ỵ tự ♥❤➜t ❜➡♥❣ ❝→❝❤ ✤➦t {xn }∞
n=0 = A tr♦♥❣ ✤à♥❤ ỵ
tự t K = A.

ự ỵ f ổ t❛ ❝â✱
xn+1 − xn = (1 − Cn )(xn − f (xn )) + f (xn ) − f (xn+1 )
≤ (1 − Cn ) (xn − f (xn ))
+ xn − ((1 − Cn )xn + Cn f (xn ))

= xn − f (xn ) .
❱➻ ❞➣② { xn − f (xn ) }∞
n=0 ❦❤æ♥❣ t➠♥❣ ✈➔ ❜à ❝❤➦♥ ữợ lim xn f (xn )
tỗ t ứ xn+1 = (1 − Cn )xn + Cn f (xn )✱

1
1
xn+1 − xn ≤ lim xn+1 − xn = 0,
n→∞ Cn
a n→∞

lim xn − f (xn ) = lim

n→∞

n→∞




t ỵ ữủ {xn }
n=0 t õ
ỵ t t ự ỵ

ự ỵ ✷✳✶✳✶✷ ●✐↔ sû ♥❣÷đ❝ ❧↕✐ A ❜à ❝❤➦♥ ✈➔ ①➨t

xn p

ợ ộ n t M số ữỡ trữợ tọ (M 1) > 2p + 1✳
❈❤å♥ N ✱ ✈ỵ✐ N > max M, [(2p − δ)M/(1 − c1 )M C1 ]

✭tr♦♥❣ ✤â [x] ♣❤➛♥

♥❣✉②➯♥ ❝õ❛ x✮ s❛♦ ❝❤♦ ✈ỵ✐ δ > 0 ✈➔ x0 ∈ A✱ ❞➣② ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ {xn }∞
n=0 ⊂ A
t❤ä❛ ♠➣♥ xN +1 − xN

> δ ✳ ❙û ❞ư♥❣ t➼♥❤ ❦❤ỉ♥❣ ❣✐➣♥ ❝õ❛ →♥❤ ①↕ f ✱ t❛ ❞➵

❞➔♥❣ t❤✉ ✤÷đ❝

xn+1 − xn = Cn (1 − Cn−1 )xn−1 − f (xn−1 ) + f (xn−1 ) − f (xn )
Cn (1 − Cn−1 ) xn−1 − f (xn−1 )+
+ xn−1 − [(1 − Cn−1 )xn−1 + Cn−1 f (xn−1 )]
Cn
xn−1 − xn
xn−1 − xn
=
C−1
❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ❝ò♥❣ s✉② r❛ tø ✭✷✳✶✮ ✈ỵ✐ n t❤❛② ❜ð✐ n − 1 ✈➔ T t❤❛② ❜ð✐ f ✱
✈➔ ✈➻ ❞➣② {Cn }∞
n=1 ❦❤æ♥❣ t➠♥❣✳ ❱➻ xi+1 − xi > δ ✈ỵ✐ ♠å✐ i

N ✱ ✈➔

δ < xN +1 − xN ≤ xN − xN −1 ≤ · · · ≤ x2 − x1 ≤ 2p,
f (xi + 1) − f (xi ) ≤ xi+1 − xi , ∀i = 0, 1, . . . , N,

✭✷✳✷✮
✭✷✳✸✮

✈➔ xi+1 = (1 − Ci )xi + Ci f (xi ) s❛♦ ❝❤♦

f (xi ) =

xi+1
−
Ci

1 − Ci
Ci

xi , i = 1, 2, . . . , N ;

✭✷✳✹✮

s✉② r❛

1
1
{xi+1 − (1 − Ci xi )} −
{xi − (1 − Ci−1 )xi−1 }
Ci
Ci−1
= f (xi ) − f (xi−1 ) ≤ xi − xi−1 ,
✤✐➲✉ ♥➔② ❞➝♥ tỵ✐

1
[xi+1 − xi ] −
Ci

1 − Ci−1
Ci−1

[xi − xi−1 ] ≤ xi − xi−1

✭✷✳✺✮


✶✼

✈ỵ✐ ♠å✐ i = 1, 2, . . . , N ✳ ❚✐➳♣ t❤❡♦ t❛ ✤➦t I = [(2p − δ)/(1 − C1 )M C1 ] ✈➔ ①➨t
❤å I ❝→❝ ❦❤♦↔♥❣ [sk , sk+1 ] tr♦♥❣ ✤â

δ + k(1 − C )M C , k = 0, 1, . . . , I − 1,
1
1
Sk =
2P,
k = I.
❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ♠ët tr♦♥❣ ❝→❝ ❦❤♦↔♥❣ ♣❤↔✐ ❝❤ù❛ ➼t ♥❤➜t M sè tr♦♥❣ ❝→❝
N −1
sè { xi − xi+1 }i=0
⊆ [δ, 2p]✳ ◆➳✉ ✤✐➲✉ ♥➔② ❦❤æ♥❣ ①↔② r❛✱ t❤➻

N < MI = M

2p − δ
(1 − C1 )M C1

✤✐➲✉ ♥❛② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❝→❝❤ ❝❤å♥ N ✳ ❱➻ ✈➟② ✈ỵ✐ r✱ ✈➔ s = sk ∈ [δ, 2p]✱ t❛ ❝â

xr+i+1 − xr+i ∈ [s, s + (1 − C1 )M C1 ]

✭✷✳✻✮

✈ỵ✐ i = 0, 1, . . . , (M − 1)✳ ✣➦t ∆xi = xi − xi−1 , i = 1, 2, . . . , N ✳ ❚❤❛② t❤➳ i
tr♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✺✮ ❜➡♥❣ r + M − j − 1, (j = 0, 1, . . . , M − 1) t❛ ❝â ❜➜t
✤➥♥❣ t❤ù❝ ✭✷✳✺✮ ✈➔ ✭✷✳✻✮ s✉② r❛

1
Cr+M −j−1

∆xr+M −j −

1 − Cr+M −j−2
Cr+M −j−2

∆xr+M −j−1 ≤ s + (1 − C1 )M C1 .
✭✷✳✼✮

❈❤å♥ f ∗ ∈ X ∗ ✱ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝õ❛ X ✱ ✈ỵ✐

f ∗ = 1 ✈➔ f ∗ (∆xr+M ) = ∆xr+M .
❚ø ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✼✮ t❛ t❤✉ ✤÷đ❝

1 − Cr+M −j−2
f ∗ (∆xr+M −j−1 )
Cr+M −j−1
Cr+M −j−2

1 − Cr+M −j−2
1
≤ f∗ .
∆xr+M −j −
∆xr+M −j−1
Cr+M −j−1
Cr+M −j−2
1

f ∗ (∆xr+M −j ) −

≤ s + (1 − C1 )M C1 .
❚ø ❜➜t ✤➥♥❣ t❤ù❝ ♥➔② t❛ ❝â

f ∗ (∆xr+M −j−1 ) ≥
−

Cr+M −j−2
Cr+M −j−1

Cr+M −j−2
1 − Cr+M −j−2

1
1 − Cr+M −j−2

f ∗ (∆xr+M −j )

f ∗ (∆xr+M −j ) (s + (1 − C1 )M C1 ).

✭✷✳✽✮


✶✽
−1
❱➻ ❞➣② {Ci }∞
≤ (1 − C1 )−1 ✈➔
i=0 ❦❤æ♥❣ t➠♥❣✱ ♥➯♥ ✈ỵ✐ ♠å✐ i ≥ 1, (1 − Ci )

Ci (1 − Ci )−1 ≤ C1 (1 − C1 )−1 ✳ ❱ỵ✐ j = 0✱ tø
f ∗ (∆xr+M ) = ∆xr+M ∈ [s, s + (1 − C1 )M C1 ]
✈➔ ✭✷✳✽✮ t❛ t❤✉ ✤÷đ❝✱

f ∗ (∆xr+M −1 ) ≥

1
1 − Cr+M −2

Cr+M −2
1 − Cr+M −2

s−

s + (1 − C1 )M C1 )

≥ s − C12 (1 − C1 )M −1 .

✭✷✳✾✮

❚❛ s➩ ❝❤➾ r❛ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✾✮ s✉② r❛

j
∗

M −1

f (∆xr+M −j−1 ) ≥ s − (1 − C1 )

1
1 − C1

C12
t=0

t

✭✷✳✶✵✮

,

✈ỵ✐ j = 1, 2, . . . , M − 1✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ q✉② ♥↕♣✳ ❱ỵ✐ j = 0✱ t❤➻ ❜➜t ✤➥♥❣
t❤ù❝ ✭✷✳✶✵✮ ❝❤➼♥❤ ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✾✮✳ ●✐↔ sû ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✵✮ ✤ó♥❣
✈ỵ✐ ♠å✐ j ≤ k ✱ ✈ỵ✐ k ∈ 1, 2, 3, . . . , M − 2✳ ❑❤✐ ✤â tø ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✽✮ ✈➔
❜➡♥❣ q✉② ♥↕♣ t❛ t❤✉ ✤÷đ❝

f ∗ (∆xr+M −(k+1)−1 ) = f ∗ (∆xr+M −k−2 )
≥

≥

1

Cr+m−k−3
f ∗ (∆xr+m−k−2 )
Cr+m−k−2
1 − Cr+m−k−3
Cr+m−k−3
−
s + (1 − C1 )M C1 )
1 − Cr+m−k−3
k

1

M −1

s − (1 − C1 )

1 − Cr+m−k−3
−

≥s−
−

t=0

Cr+m−k−3
1 − Cr+m−k−3

1
1 − Cr+m−k−3
k

1
1 − C1

(1 − C1 )

C1
1 − C1

(1 − C1 )M C1

M −1

k+1

= s − (1 − C1 )

C12
t=0

1
1 − C1

t

.

1
1 − C1

t

s + (1 − C1 )M C1 )

1
1 − C1

C12
t=0

M −1

C12

t


✶✾

❉♦ f ∗ ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ✈➔ ❧➜② tê♥❣ ❤❛✐ ✈➳ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✵✮ t❤❡♦

j = 0, . . . , (M − 2) t❛ t❤✉ ✤÷đ❝✿
f ∗ (xr+M −1 − xr ) = f ∗ (xr+M −1 ) − f ∗ (xr )
≥ (M − 1)s − (1 − C1 )M −1 C12 1 + 1 +
+ ··· +

1
1+
+ ··· +
1 − C1

1
1 − C1

1
1 − C1

M −2

.

✣➦t λ = 1 − C1 ✱

f ∗ (xr+M −1 − xr ) = (M − 1)s − λM −1 (1 − λ)2 [1 +

λ+1
λ

λM −2 + ... + λ + 1
+ ··· +
]
λM −2
1 − λ 1 − λ2
+
= (M − 1)s − λ(1 − λ)[λM −1 {
λ
λ2
1 − λM −1
}]
+ ··· +

λM −1
≥ (M − 1)s − 1,
✈➻
M −1

λ(1 − λ) λ

1−λ
λ

+

1 − λ2
λ2

+ ··· +

1 − λM −1
λM −1

< λ(1 − λ)(λM −2 + · · · + λ + 1) ≤ 1.
▼➔ s ≥ δ s✉② r❛ (M − 1)s ≥ (M − 1)δ > 2p + 1✱ ✈➻ ✈➟②

f ∗ (xr+M −1−x−r ) > p.
❚❛ ❝ô♥❣ ❝â

f ∗ (xr+M −1−xr ) ≤ |f ∗ (xr+M −1−xr )| ≤ f ∗ . xr+M −1 − xr
= xr+M −1 − xr .
❱➻ ✈➟② xr+M −1 − xr > 2p✱ ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t xn ≤ p ✈ỵ✐
♠é✐ n✱ ✈➟② t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳

✷✵

✷✳✷ ❙ü ❤ë✐ tö
❚r♦♥❣ ♠ö❝ ♥➔② t❛ s➩ tr➻♥❤ ❜➔② sü ❤ë✐ ♠↕♥❤ ✈➔ ❤ë✐ tö ②➳✉ ❝õ❛ ❞➣② ❧➦♣ ✭✷✳✶✮
tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳

✷✳✷✳✶ ✣à♥❤ ỵ ở tử
ỷ ử tt ự ỵ t õ t ự ữủ
t q s

ỵ K t ừ ổ ❇❛♥❛❝❤ t❤ü❝ X ✈➔ →♥❤ ①↕
❦❤æ♥❣ ❣✐➣♥ f

: K → X

sỷ x0 K tữỡ ự ợ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝
{xn }∞
n=0 ⊆ K ❝â ✤✐➸♠ tư q ∈ K ✳ ❑❤✐ ✤â f (q) = q ✈➔ xn → q ✳ ✣➦❝ ❜✐➺t✱ ♥➳✉
♠✐➲♥ ❣✐→ trà ❝õ❛ f ❝❤ù❛ tr♦♥❣ t➟♣ ❝♦♥ ❝♦♠♣❛❝t ❝õ❛ K t❤➻ ❞➣② {xn}∞n=0 ❤ë✐ tư
♠↕♥❤ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ✳
❈❤ù♥❣ ♠✐♥❤✳ ◆➠♠ ✶✾✻✹ ❊❞❡❧st❡✐♥✱ ✤➣ ❝❤➾ r❛ ✤÷đ❝ r➡♥❣ q ❝ị♥❣ ❧➔ ✤✐➸♠ tư ❝õ❛
{f n (q)} ✈➔ f n+1 (q) − f n (q) = f (q) − q ✈ỵ✐ ♠å✐ n✳ ❱➻ ✈➟② ♥➳✉ f i (q) := xi

✈➔ ∆xi = xi − xi−1 , i = 1, 2, . . . t❤➻ ∆xi+1 = ∆xi ✈ỵ✐ ♠å✐ i ữ tr
ự ừ ỵ t õ

Ci
xi − xi−1

Ci − 1
= ∆xi ,

∆xi + 1 = xi+1 − xi ≤
≤ xi − xi−1

✈➔ s✉② r❛ Ci = Ci−1 ✈ỵ✐ ♠å✐ i ✈➻ ∆xi−1 = ∆xi ✳ ❱➻ ✈➟② tø

xi+1 − xi = (1 − Ci )xi + Ci f (xi ) − (1 − Ci − 1)xi−1 − Ci−1 f (xi−1 )
t❛ ❝â

∆xi+1 ≤ (1 − Ci ) ∆xi + Ci ∆f (xi )
≤ (1 − Ci ) ∆xi + Ci ∆xi
= ∆xi ,
tø ❦➳t q✉↔ ♥➔② t❛ ❝â ∆xi = ∆f (xi ) ✳ ●✐↔ sû ♥❣÷đ❝ ❧↕✐ ❧➔✱

∆xi = ∆f (xi ) = β > 0, i = 1, 2, . . .

✭✷✳✶✶✮


✷✶

❈❤å♥ N, K ∈ N + ✤õ ❧ỵ♥✳ ❚ø ✤➥♥❣ t❤ù❝ ✭✷✳✶✶✮ t❛ ❝â✱ ✈ỵ✐ i = N + K ✱

∆xN +K = ∆f (xN +K ) = β > 0
❳➨t f ∗ ∈ X ∗ s❛♦ ❝❤♦

= 1 ✈➔ f ∗ (∆xN +K ) =

f

✭✷✳✶✷✮

∆xN +K ✳ ❑❤✐ ✤â ✈ỵ✐

j = 0, 1, 2, . . . ✱
f ∗ (∆f (xN +K−j )) ≤ f ∗ . ∆f (xN +K−j ) = ∆f (xN +K−j ) = s.

✭✷✳✶✸✮

❚ø xN +K−j+1 = (1 − CN +K−j )xN +K−j + CN +K−j f (xN +K−j ) ✈➔ Ci = Ci−1 ✈ỵ✐
♠å✐ i✱ t❛ ❝â

∆xN +K−j+1 = (1 − CN +K−j )∆xN +K−j + CN +K−j ∆f (xN +K−j ).

✭✷✳✶✹✮

❚✐➳♣ t❤❡♦ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤✿

f ∗ (∆xN +K−j ) ≥ β

✈ỵ✐ j = 0, 1, . . . .

✭✷✳✶✺✮

❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ t tự q t trữợ ❤➳t t❛
❝â f ∗ (∆xN +K ) = ∆xN +K = β t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✺✮ ✈ỵ✐ j = 0✳
❈á♥ ♥➳✉ j = 1✱ tø ✤➥♥❣ t❤ù❝ ✭✷✳✶✹✮ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✸✮ t❛ ❝â

f ∗ (∆xN +K−1 ) =

≥

1
f ∗ (∆xN +K )
1 − CN +K−1
CN +K−1
−
f ∗ (∆f (xN +K−1 ))
1 − CN +K−1
CN +K−1
1
β−
β = β.
1 − CN +K−1
1 − CN +K−1

●✐↔ sû ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✺✮ ✤ó♥❣ ✈ỵ✐ j = 0, 1, . . . , t✳ ❑❤✐ ✤â →♣ ❞ö♥❣ ❜➜t ✤➥♥❣
t❤ù❝ ✭✷✳✶✸✮ ✈➔ ✤➥♥❣ t❤ù❝ ✭✷✳✶✹✮ ✈➔ ❣✐↔ t❤✐➳t q✉② ♥↕♣ t❛ ❝â

f ∗ (∆xN +K−t−1 ) =

≥

1

f ∗ (∆xN +K−t )

1 − CN +K−t−1

CN +K−t−1
−
f ∗ (∆f (xN +K−t−1 ))
1 − CN +K−t−1
1
CN +K−t−1
β−
β = β.
1 − CN +K−t−1
1 − CN +Kt1

ữỡ tỹ ự ỵ tê♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✺✮
t❤❡♦ j = 0, 1, . . . , K − 1 t❛ t❤✉ ✤÷đ❝

xN +K − xN ≥ f ∗ (xN +K − xN ) ≥ Kβ,

✭✷✳✶✻✮


✷✷
∞
❤❛② ❞➣② {xi }∞
i=0 ❦❤ỉ♥❣ ❝â ❞➣② ❝♦♥ ❤ë✐ tư✱ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t {xn }n=0 ❝â

✤✐➸♠ tư✳ ❉♦ ✤â β = 0 ✈➔ f (q) = q ✳ ◆❣❤➽❛ ❧➔ xn → q ✈➻ f ❧➔ →♥❤ ①↕ ❦❤æ♥❣
❣✐➣♥✳
✣➲ tr➻♥❤ ❜➔② ❝→❝ ❦➳t q✉↔ t✐➳♣ t❤❡♦✱ t❛ ❝➛♥ ❦❤→✐ ♥✐➺♠ s❛✉✳

✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✷✳ ❈❤♦ C ❧➔ t➟♣ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥

t❤ü❝ X ✳ ⑩♥❤ ①↕ f : C → X ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ❝♦♠♣❛❝t ✭❞❡♠✐❝♦♠♣❛❝t✮ t↕✐

h ∈ X ♥➳✉✱ ✈ỵ✐ ❜➜t ❦ý ❞➣② ❜à ❝❤➦♥ {xn }∞
n=0 tr♦♥❣ C s❛♦ ❝❤♦ xn − f (xn ) h
n tỗ t ❞➣② ❝♦♥ {xnj }∞
j=0 ✈➔ x ∈ C s❛♦ ❝❤♦ xnj → x ❦❤✐ j → ∞
✈➔ x − f (x) = h✳

❍➺ q✉↔ ✷✳✷✳✸✳ ❈❤♦ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝✱ C ❧➔ t➟♣ ❝♦♥ ✤â♥❣ ❜à ❝❤➦♥

❝õ❛ X ✈➔ f : C → C ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ●✐↔ sû✱ ❤♦➦❝ (i) f ❧➔ ♥û❛ ❝♦♠♣❛❝t
t↕✐ 0✱ ❤♦➦❝ (ii) (I − f ) →♥❤ ①↕ t➟♣ ❝♦♥ ✤â♥❣ ❜à ❝❤➦♥ ❝õ❛ X ✈➔♦ t➟♣ ❝♦♥ ✤â♥❣
❝õ❛ X ✳ ❱ỵ✐ x0 ∈ C ①➨t {xn}∞n=0 ⊆ C ❧➔ ữủ ự ợ số
tỹ {Cn}n=0 ỗ t❤í✐ t❤ä❛ ♠➣♥ 0 < a ≤ Cn ≤ b < 1 ✈ỵ✐ ♠å✐ n ≥ 1✳ ❑❤✐ ✤â
{xn }∞
n=0 ❧➔ ❞➣② ❤ë✐ tư ♠↕♥❤ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f tr♦♥❣ C ✳
❈❤ù♥❣ ♠✐♥❤✳ (i) ❚ø xn+1 = (1 − Cn)xn + Cnf (xn) t❛ ❝â
xn − f (xn ) =

1
{xn − xn−1 }.
Cn

∞
❱➻ t➟♣ C ❜à ❝❤➦♥✱ ♥➯♥ ❞➣② {xn }∞
n=0 ❜à ❝❤➦♥ ✈➔ ❞➣② {Cn }n=1 ❝ô♥❣

ữợ 0 ỵ s✉② r❛ {xn − f (xn )} ❤ë✐ tư tỵ✐

0 ✈➻ ✈➟② tø t➼♥❤ ♥û❛ ❝♦♠♣❛❝t ❝õ❛ →♥❤ ①↕ f t↕✐ 0✱ ❞➣② {xn }∞

n=0 ❝â ✤✐➸♠ tö
tr♦♥❣ C ✳ ỵ t õ ự

(ii) ◆➳✉ q ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ f ✱ t❤➻ ❞➣② { xn − q }∞
n=0 ❦❤æ♥❣ t➠♥❣
t❤❡♦ n õ t r tỗ t ❝õ❛ ❞➣② {xn }∞
n=0 ❤ë✐ tư ♠↕♥❤
tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ f ✳ ❱ỵ✐ x0 ∈ C ✱ ①➨t K ❧➔ ❜❛♦ ✤â♥❣ ♠↕♥❤
❝õ❛ t➟♣ {xn }∞
n=0 ✳ ỵ {(I f )(xn )} ❤ë✐ tư ♠↕♥❤ tỵ✐

0 ❦❤✐ n → ∞✳ ❱➻ ✈➟②✱ 0 t❤✉ë❝ ✈➔♦ ❜❛♦ ✤â♥❣ ♠↕♥❤ ❝õ❛ t➟♣ (I − f )(K)✱
(I − f )(K) ❧➔ t➟♣ ✤â♥❣ ✭✈➻ t➟♣ K ✤â♥❣ ✈➔ ❜à ❝❤➦♥✮✱ ✈➻ ✈➟② 0 ∈ (I f )(K)
tỗ t {xnj }
j=0 s xnj à C ợ à t❤ä❛ ♠➣♥

(I − f )µ = 0✳ ❉♦ ✤â xn → µ✳




ỵ ở tử
→♥❤ ①↕ ♥û❛ ✤â♥❣✳

✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✹✳ ⑩♥❤ ①↕ T : K → X ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ✤â♥❣ t↕✐ y ♥➳✉ ✈ỵ✐
∞
❜➜t ❦ý ❞➣② {xn }∞
n=0 ⊆ K ❤ë✐ tư ②➳✉ tỵ✐ x ∈ K ✱ ❞➣② {T (xn )}n=0 ❤ë✐ tư ♠↕♥❤

tỵ✐ y ∈ K s✉② r❛ T x = y

ỵ X ổ ❖♣✐❛❧ ✈➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ f : K → K
ợ K t ỗ t tr X ✳ ❇➜t ❦ý x0 ∈ K ✱ ①➨t {xn}∞n=0 ⊆ K
ữủ ự ợ ổ t {Cn}∞n=1 t❤ä❛ ♠➣♥ 0 < a ≤ Cn < 1
✈ỵ✐ ♠å✐ n ≥ 1✳ ❑❤✐ ✤â {xn}∞n=0 ❤ë✐ tö ②➳✉ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ f ✳
❈❤ù♥❣ ♠✐♥❤✳ ❱➻ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❖♣✐❛❧ ✈➔ f ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✱ ♥➯♥ (I −f )
♥û❛ ✤â♥❣✳ ◆❣♦➔✐ r❛✱ t ỵ f t q ❉♦ ✤â✱ t❤❡♦
❇r♦✇❞❡r ✈➔ P❡tr②s❤②♥ ✭✶✾✻✻✮✱ ✈ỵ✐ ❜➜t ❦ý ✤✐➸♠ tö ②➳✉ ❝õ❛ {xn }∞
n=0 ⊆ K ✤➲✉
❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ f ✳ ❚❛ s➩ ❝❤➾ r❛ {xn }∞
n=0 ⊆ K ❝â ❞✉② ♥❤➜t ✤✐➸♠
tö ②➳✉✳ ❚❤➟t sỷ tỗ t tử ❜✐➺t ❝õ❛ {xn }∞
n=0 ✱ ❧➔
∞
∞
q1 ✈➔ q2 ✱ ✈➔ ❤❛✐ ❞➣② ❝♦♥ {xni }∞
i=1 ❱➔ {xnj }j=1 s❛♦ ❝❤♦ {xni }i=1 ❤ë✐ tư ②➳✉ tỵ✐

q1 ✈➔ {xnj }∞
j=1 ❤ë✐ tư ②➳✉ tỵ✐ q2 ✳ ❳➨t p ∈ ❋✐①(f ) ✈ỵ✐ ❋✐①(f ) ❧➔ t➟♣ ❝→❝ ✤✐➸♠
❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ f ✳ ❑❤✐ ✤â t❛ ❞➵ ❞➔♥❣ ❝â xn+1 − p ≤ xn − p ✈ỵ✐ ♠é✐

n ≥ 0 s lim xn p tỗ t ợ ồ p ∈ ❋✐①(f )✳ ❱➻ ✈➟② ✭X ❧➔ ❦❤æ♥❣
n→∞

❣✐❛♥ ❖♣✐❛❧✮ t❛ ❝â

lim xn − q1 = lim xni − q1 < lim xni − q2 = lim xn − q2

n→∞

i→∞

i→∞

n→∞

✈➔

lim xn − q2 = lim xnj − q2 < lim xnj − q1 = lim xn − q1 ,

n→∞

i→∞

i→∞

n→∞

✤✐➲✉ ♥➔② t ợ tt s r tỗ t ♥❤➜t ♠ët ✤✐➸♠ tö ②➳✉
∞
q ❝õ❛ ❞➣② {xn }∞
n=0 ⊆ K ✳ ❚❤❡♦ t➼♥❤ ❝♦♠♣❛❝t ②➳✉ ❝õ❛ K ✱ t❛ õ {xn }n=0 ở tử
tợ q

ỵ K t ỗ õ ừ ổ ❇❛♥❛❝❤ ♣❤↔♥ ①↕

X✱

✈➔ T : K → X ❧✐➯♥ tö❝ t❤ä❛ ♠➣♥

(i) ❋✐①(T ) = ∅❀