✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈
✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕
❍❖⑨◆● ❚❍➚ ❍❾❯
P❍×❒◆● P❍⑩P ▲❆■ ●❍➆P ●■❷■
▼❐❚ ▲❰P ❇❻❚ ✣➃◆● ❚❍Ù❈ ❇■➌◆ P❍❹◆
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
❚❍⑩■ ◆●❯❨➊◆✱ ✺✴✷✵✶✾
✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈
✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕
❍❖⑨◆● ❚❍➚ ❍❾❯
P❍×❒◆● P❍⑩P ▲❆■ ●❍➆P ●■❷■
▼❐❚ ▲❰P ❇❻❚ ✣➃◆● ❚❍Ù❈ ❇■➌◆ P❍❹◆
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ö♥❣
▼➣ sè✿ ✽✹✻✵✶✶✷
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
●■⑩❖ ❱■➊◆ ❍×❰◆● ❉❼◆
P●❙✳❚❙✳ ◆●❯❨➍◆ ❚❍➚ ❚❍❯ ❚❍Õ❨
❚❍⑩■ ◆●❯❨➊◆✱ ✺✴✷✵✶✾
✐✐✐
▼ö❝ ❧ö❝
❇↔♥❣ ❦þ ❤✐➺✉
✶
▼ð ✤➛✉
✷
❈❤÷ì♥❣ ✶ ▼ët ❧î♣ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤æ♥❣
❣✐❛♥ ❇❛♥❛❝❤
✹
✶✳✶
✶✳✷
❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹
✶✳✶✳✶
❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕✱ ❧ç✐ ✤➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✶✳✶✳✷
P❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✼
✶✳✶✳✸
⑩♥❤ ①↕ ❧♦↕✐ ✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✽
❇➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❧♦↕✐ ✤ì♥ ✤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✶✳✷✳✶
❇➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✤ì♥ ✤✐➺✉
✶✳✷✳✷
❇➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ j ✲✤ì♥ ✤✐➺✉
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
❈❤÷ì♥❣ ✷ P❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ❣✐↔✐ ♠ët ❧î♣ ❜➜t ✤➥♥❣ t❤ù❝
❜✐➳♥ ♣❤➙♥
✶✺
✷✳✶
✷✳✷
P❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✷✳✶✳✶
❇➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✷✳✶✳✷
P❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
❙ü ❤ë✐ tö ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✷✳✷✳✶
❙ü ❤ë✐ tö ❝õ❛ P❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✳✷✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✷✳✷✳✷
❙ü ❤ë✐ tö ❝õ❛ P❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✳✸✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✷✳✷✳✸
❙ü ❤ë✐ tö ❝õ❛ P❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✳✹✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✷✳✷✳✹
❱➼ ❞ö ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✐✈
❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦
✸✻
✸✼
ỵ
H
ổ rt tỹ
E
ổ
E
ổ ố ừ E
SE
t ỡ ừ E
R
t số tỹ
R+
t số tỹ ổ
t rộ
x
ợ ồ x
D(F )
ừ F
R(F )
tr ừ F
F 1
ữủ ừ F
I
ỗ t
lp , 1 p <
ổ số tờ p
l
ổ số
Lp [a, b], 1 p <
ổ t p
tr [a, b]
d(x, C)
tứ tỷ x t ủ C
lim supn xn
ợ tr ừ số {xn }
lim inf n xn
ợ ữợ ừ số {xn }
xn x0
{xn } ở tử x0
xn
{xn } ở tử x0
x0
J
ố t
j
ố t ỡ tr
(T )
t t ở ừ T
✷
▼ð ✤➛✉
❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✈æ ❤↕♥ ❝❤✐➲✉
✤÷ñ❝ ♥❤➔ t♦→♥ ❤å❝ ♥❣÷í✐ ■t❛❧✐❛ ❧➔ ❙t❛♠♣❛❝❝❤✐❛ ✭①❡♠ ❬✶✷❪✮ ✈➔ ❝→❝ ✤ç♥❣
sü ✤÷❛ r❛ ❧➛♥ ✤➛✉ t✐➯♥ ✈➔♦ ♥❤ú♥❣ ♥➠♠ ✤➛✉ ❝õ❛ t❤➟♣ ♥✐➯♥ ✻✵ t❤➳ ❦➾ ❳❳
tr♦♥❣ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❜➔✐ t♦→♥ ❜✐➯♥ tü ❞♦✳ ❇➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥
❝â ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ♥❣❤✐➯♥ ❝ù✉ t♦→♥ ❤å❝ ❧þ t❤✉②➳t ✈➲ ❜➔✐ t♦→♥
tè✐ ÷✉✱ ❜➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥✱ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣✱ ❜➔✐ t♦→♥ ❜ò✱ ❜➔✐ t♦→♥ ❣✐→
trà ❜✐➯♥ ✈✳✈ ✳ ✳ ✳ ❉♦ ✤â✱ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➜t ✤➥♥❣
t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✤❛♥❣ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ✤➲ t➔✐ t❤✉ ❤ót ✤÷ñ❝ sü q✉❛♥
t➙♠ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ tr♦♥❣ ✈➔ ♥❣♦➔✐ ♥÷î❝ ✈➔ ✤➣ ♥❤➟♥
✤÷ñ❝ ♥❤✐➲✉ ❦➳t q✉↔ ❤❛②✱ s➙✉ s➢❝✳ ❇➯♥ ❝↕♥❤ ✤â✱ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥
❝á♥ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ ❝❤♦ ❝→❝ ❜➔✐ t♦→♥ t❤ü❝ t➳ ♥❤÷ ♠æ ❤➻♥❤ ❝➙♥ ❜➡♥❣
tr♦♥❣ ❦✐♥❤ t➳✱ ❣✐❛♦ t❤æ♥❣✱ ❜➔✐ t♦→♥ ❦❤æ✐ ♣❤ö❝ t➼♥ ❤✐➺✉✱ ❜➔✐ t♦→♥ ❝æ♥❣
♥❣❤➺ ❧å❝ ❦❤æ♥❣ ❣✐❛♥✱ ❜➔✐ t♦→♥ ♣❤➙♥ ♣❤è✐ ❜➠♥❣ t❤æ♥❣ ✈✳✈ ✳ ✳ ✳ ✭①❡♠ ❬✽❪✱
❬✶✵❪✱ ❬✶✶❪✮✳ ❈❤♦ ✤➳♥ ♥❛② ✈➝♥ ❝á♥ ♥❤✐➲✉ ✈➜♥ ✤➲ ♠î✐ ✈➔ ❦❤â ❝õ❛ ❜➜t ✤➥♥❣
t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❝➛♥ ✤÷ñ❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ✈î✐ ♥❤ú♥❣ ❝æ♥❣ ❝ö t♦→♥
❤å❝ ❤✐➺♥ ✤↕✐✳ ▼ët tr♦♥❣ ♥❤ú♥❣ ❤÷î♥❣ ♥❣❤✐➯♥ ❝ù✉ ✤❛♥❣ ✤÷ñ❝ q✉❛♥ t➙♠
❧➔ ①➙② ❞ü♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈î✐ t➟♣ r➔♥❣
❜✉ë❝ ❧➔ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✱ t➟♣
❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å →♥❤ ①↕ ❧♦↕✐ j ✲✤ì♥ ✤✐➺✉✱ t➟♣ ♥❣❤✐➺♠ ❝❤✉♥❣
❝õ❛ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣✱ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✱ ❜➔✐ t♦→♥ ✤✐➸♠
❜➜t ✤ë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳
▼ö❝ t✐➯✉ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣
❞è❝ ♥❤➜t ❣✐↔✐ ♠ët ❧î♣ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❜➔✐ ❜→♦ ❬✺❪✱ ❬✻❪ ✈➔
ừ ữớ ừ ổ
ố
ở ừ ữủ tr tr ữỡ ữỡ
ởt ợ t tự tr ổ ữỡ
ợ t ởt số tự ỡ ổ
ỗ ỡ j ỡ tr ổ ỗ
tớ tr t t tự ỡ j ỡ
tr ổ ữỡ Pữỡ
ởt ợ t tự ữỡ tr ữỡ
t t tự j ỡ tr
ổ ũ ỵ ở tử ừ ữỡ
P ố ừ ữỡ ởt ử ồ tr
ỵ ở tử ữủ tọ
ữủ t t rữớ ồ ồ ồ
r q tr ồ t tỹ rữớ
ồ ồ t ồ tốt t t ồ t
ự ữủ tọ ỏ t ỡ t
t ổ tr tr rữớ ồ ồ
ồ t t tọ ỏ t ỡ s s tợ
P ừ ữớ t t ữợ t
t
ụ ữủ ỷ ớ ỡ tợ trữớ P
t tốt t t ữủ t ồ t
ự
t
ữỡ
ởt ợ t tự
tr ổ
ữỡ ợ t ởt số tự ổ
t t tự ỡ j ỡ tr ổ
ử t ử tr ổ ỗ
tr tr ổ ỡ
j ỡ tr ổ ử tr
ử t t tự ỡ t tự
j ỡ tr ổ ở ừ ữỡ
ữủ t tr ỡ s tờ ủ tự tứ t
ổ
E ổ ỵ E ổ ố
ừ E r t sỷ ử ỵ . ừ
ổ E E ợ ộ x E x E t t x , x
x, x t ố t x (x) E = H ổ rt
t t ố t ổ ữợ ., . s tữỡ
ự .
ổ ỗ
ổ E ữủ ồ
ợ ồ tỷ x E ổ ủ tự ừ E
tỗ t tỷ x E s
x (x) = x (x ) x E .
ỵ E
ổ õ
s tữỡ ữỡ
(i) E ổ
(ii) ồ tr E õ ởt ở tử
ử ổ ỳ ổ
rt H ổ lp Lp [a, b], 1 < p < ổ
ỵ SE := {x E : x = 1} t ỡ ừ ổ
E
ổ E ữủ ồ
(i) ỗ t ợ ồ x, y SE x = y s r
(1 )x + y < 1 (0, 1);
(ii) ỗ ợ ồ (0, 2] t tự x 1, y 1
x y tọ t tỗ t = () > 0 s
(x + y)
1 ;
2
(iii) trỡ ợ
lim
t0
tỗ t ợ ồ x, y SE
ử
x + ty x
t
✻
(i) ❑❤æ♥❣ ❣✐❛♥ E = Rn ✈î✐ ❝❤✉➞♥ x
n
x
2
✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐
1/2
x2i
=
2
,
x = (x1 , x2 , . . . , xn ) ∈ Rn
i=1
❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ❝❤➦t✳ ❑❤æ♥❣ ❣✐❛♥ E = Rn , n ≥ 2 ✈î✐ ❝❤✉➞♥ x
1
①→❝
✤à♥❤ ❜ð✐
x
1
= |x1 | + |x2 | + . . . + |xn |, x = (x1 , x2 , . . . , xn ) ∈ Rn
❦❤æ♥❣ ♣❤↔✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ❝❤➦t✳
(ii) ❑❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤➲✉✳
▼➺♥❤ ✤➲ ✶✳✶✳✻ ✭①❡♠ ❬✷❪✮ ▼å✐ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❧ç✐ ✤➲✉ ❧➔ ❦❤æ♥❣ ❣✐❛♥
♣❤↔♥ ①↕ ✈➔ ❧ç✐ ❝❤➦t✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✼ ✭①❡♠ ❬✷❪✮
(i) ❈❤✉➞♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✤÷ñ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ●➙t❡❛✉① ♥➳✉ ✈î✐
♠é✐ y ∈ SE ❣✐î✐ ❤↕♥
lim
t→0
x + ty − x
t
tç♥ t↕✐ ✈î✐ x ∈ SE ✱ ❦þ ❤✐➺✉ y,
x ✳ ❑❤✐ ✤â
✭✶✳✶✮
x ✤÷ñ❝ ❣å✐ ❧➔ ✤↕♦
❤➔♠ ●➙t❡❛✉① ❝õ❛ ❝❤✉➞♥✳
(ii) ❈❤✉➞♥ ❝õ❛ E ✤÷ñ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉ ♥➳✉ ✈î✐ ♠é✐ y ∈ SE ✱
❣✐î✐ ❤↕♥ ✭✶✳✶✮ ✤↕t ✤÷ñ❝ ✤➲✉ ✈î✐ ♠å✐ x ∈ SE ✳
(iii) ❈❤✉➞♥ ❝õ❛ E ✤÷ñ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ♥➳✉ ✈î✐ ♠é✐ x ∈ SE ✱ ❣✐î✐
❤↕♥ ✭✶✳✶✮ tç♥ t↕✐ ✤➲✉ ✈î✐ ♠å✐ y ∈ SE ✳
(iv) ❈❤✉➞♥ ❝õ❛ E ✤÷ñ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ✤➲✉ ♥➳✉ ❣✐î✐ ❤↕♥ ✭✶✳✶✮ tç♥
t↕✐ ✤➲✉ ✈î✐ ♠å✐ x, y ∈ SE ✳
❱➼ ❞ö ✶✳✶✳✽ ❑❤æ♥❣
●➙t❡❛✉① ✈î✐
x
❣✐❛♥ ❍✐❧❜❡rt H ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝â ❝❤✉➞♥ ❦❤↔ ✈✐
= x/ x , x = 0✳ ❚❤➟t ✈➟②✱ ✈î✐ ♠é✐ x ∈ H ✈î✐
✼
x = 0✱ t❛ ❝â
x + ty 2 −
x + ty − x
= lim
lim
t→0 t( x + ty +
t→0
t
2t y, x + t2
= lim
t→0 t( x + ty +
❱➟② ❝❤✉➞♥ ❝õ❛ H ❧➔ ❦❤↔ ✈✐ ●➙t❡❛✉① ✈î✐
x 2
x )
y 2
=
x )
y,
x
x
.
x = x/ x , x = 0✳
✶✳✶✳✷ P❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
▼➺♥❤ ✤➲ ✶✳✶✳✾ ✭①❡♠ ❬✷❪✮ ❈❤♦ C ❧➔ t➟♣ ❝♦♥ ❧ç✐ ✤â♥❣ ❦❤→❝ ré♥❣ ❝õ❛
❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕ ✈➔ ❧ç✐ ❝❤➦t E ✳ ❑❤✐ ✤â✱ ✈î✐ ♠é✐ x ∈ E tç♥
t↕✐ ❞✉② ♥❤➜t ♠ët ✤✐➸♠ y ∈ C t❤ä❛ ♠➣♥
x − y = d(x, C),
✈î✐ d(x, C) = inf z∈C x − z ✳
✣✐➸♠ y ∈ C tr♦♥❣ ▼➺♥❤ ✤➲ ✶✳✶✳✾ ❝á♥ ✤÷ñ❝ ❣å✐ ❧➔ ①➜♣ ①➾ tèt ♥❤➜t ❝õ❛
x ∈ E ❜ð✐ C ✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✵ ✭①❡♠ ❬✷❪✮ ❈❤♦ C ❧➔ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ ❦❤æ♥❣
❣✐❛♥ ❇❛♥❛❝❤ E ✳ ⑩♥❤ ①↕ PC : E → 2C ①→❝ ✤à♥❤ ❜ð✐
PC (x) =
y ∈ C : x − y = d(x, C) ∀x ∈ E
✤÷ñ❝ ❣å✐ ❧➔ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tø E ❧➯♥ C ✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✶ ✭①❡♠ ❬✷❪✮ ❚➟♣ ❝♦♥ C
❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E
✤÷ñ❝ ❣å✐ ❧➔ t➟♣ ❈❤❡❜②s❤❡✈ tr♦♥❣ E ♥➳✉ ♠é✐ ✤✐➸♠ x ∈ E ❝â ❞✉② ♥❤➜t
♠ët ✤✐➸♠ y ∈ C ❧➔ ①➜♣ ①➾ tèt ♥❤➜t ❝õ❛ x✳
◆❤➟♥ ①➨t ✶✳✶✳✶✷
(i) ❚ø ▼➺♥❤ ✤➲ ✶✳✶✳✾ s✉② r❛✱ ♠å✐ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣✱ ❧ç✐✱ ✤â♥❣ ❝õ❛ ♠ët
❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕ ✈➔ ❧ç✐ ❝❤➦t ✤➲✉ ❧➔ t➟♣ ❈❤❡❜②s❤❡✈✳
✽
(ii) ❚ø ◆❤➟♥ ①➨t ✶✳✶✳✶✷(i) s✉② r❛ ♠å✐ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❧ç✐ ✤â♥❣ tr♦♥❣
❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✤➲✉ ❧➔ t➟♣ ❈❤❡❜②s❤❡✈✱ ✤✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔ ♣❤➨♣
❝❤✐➳✉ ♠➯tr✐❝ PC tø H ❧➯♥ ♠ët t➟♣ ❝♦♥ ❧ç✐ ✤â♥❣ C ❝õ❛ H ❧✉æ♥ ①→❝ ✤à♥❤
❞✉② ♥❤➜t✳
(iii) ❱î✐ ♠å✐ t➟♣ ❈❤❡❜②s❤❡✈ C ⊂ E ✱ t❛ ❝â
✭❛✮ PC (x) ❧➔ t➟♣ ❝❤➾ ❣ç♠ ♠ët ♣❤➛♥ tû✳
✭❜✮ x − PC (x) = d(x, C) ✈î✐ ♠å✐ x ∈ E ✳
❱➼ ❞ö ✶✳✶✳✶✸ ●✐↔ sû a, b ∈ Rn✱ a = 0✳ ❳➨t ♥û❛ ❦❤æ♥❣ ❣✐❛♥ D ⊂ Rn ✈➔
♠➦t ♣❤➥♥❣ Q ⊂ Rn ❝❤♦ ❜ð✐
D = {x ∈ Rn : a, x − b ≤ 0},
Q = {x ∈ Rn : a, x − b = 0}.
❑❤✐ ✤â t♦→♥ tû ❝❤✐➳✉ ❧➯♥ D ✈➔ Q ❧➛♥ ❧÷ñt ❝❤♦ ❜ð✐
PD (x) =
PQ (x) =
♥➳✉
x
x−
a,x−b
||a||2
♥➳✉
a
♥➳✉
x
x−
a,x−b
||a||2
a , x − b ≤ 0,
a , x − b > 0,
a , x − b = 0,
♥➳✉
a
a, x − b
= 0.
❱➼ ❞ö ✶✳✶✳✶✹ ●✐↔ sû a ∈ Rn✱ r > 0 ✈➔ ❤➻♥❤ ❝➛✉ B ①→❝ ✤à♥❤ ❜ð✐
B = {x ∈ Rn : x − a ≤ r}.
❑❤✐ ✤â t♦→♥ tû ❝❤✐➳✉ ❧➯♥ B ❝❤♦ ❜ð✐
PB (x) =
♥➳✉
x
a+
r
||x−a|| (x
− a) ♥➳✉
||x − a|| ≤ r,
||x − a|| > r.
✶✳✶✳✸ ⑩♥❤ ①↕ ❧♦↕✐ ✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✺ ✭①❡♠ ❬✸❪✮ ❈❤♦ C ❧➔ t➟♣ ❝♦♥ ❧ç✐ ✤â♥❣ ❦❤→❝ ré♥❣ ❝õ❛
❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✳ ⑩♥❤ ①↕ F : C → E ∗ ✤÷ñ❝ ❣å✐ ❧➔✿
✾
(i) ✤ì♥ ✤✐➺✉ tr➯♥ C ♥➳✉
✭✶✳✷✮
F (x) − F (y), x − y ≥ 0 ∀x, y ∈ C;
(ii) ✤ì♥ ✤✐➺✉ ❝❤➦t tr➯♥ C ♥➳✉ ❞➜✉ ” = ” tr♦♥❣ ✭✶✳✷✮ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾
❦❤✐ x = y ❀
(iii) ✤ì♥ ✤✐➺✉ ✤➲✉ tr➯♥ C ♥➳✉ tç♥ t↕✐ ♠ët ❤➔♠ ❧✐➯♥ tö❝ ✈➔ t➠♥❣ ♥❣➦t
α : [0, ∞) → [0, ∞) ✈î✐ α(0) = 0 ✈➔ α(t) → ∞ ❦❤✐ t → ∞ s❛♦ ❝❤♦
F (x) − F (y), x − y ≥ α( x − y ) x − y
∀x, y ∈ C;
(iv) η ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥ C ♥➳✉ tç♥ t↕✐ ❤➡♥❣ sè η > 0 s❛♦ ❝❤♦
F (x) − F (y), x − y ≥ η x − y
2
∀x, y ∈ C;
(v) ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr➯♥ C ♥➳✉ F ✤ì♥ ✤✐➺✉ tr➯♥ C ✈➔ ✤ç t❤à
G(F ) = (x, y) ∈ C × E ∗ : x ∈ C, y = F (x)
❝õ❛ F ❦❤æ♥❣ t❤ü❝ sü ❜à ❝❤ù❛ tr♦♥❣ ✤ç t❤à ❝õ❛ ♠ët →♥❤ ①↕ ✤ì♥ ✤✐➺✉ ♥➔♦
❦❤→❝✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✻ ✭①❡♠ ❬✸❪✮ ⑩♥❤ ①↕ Js : E → 2E , s > 1 ✭♥â✐ ❝❤✉♥❣
∗
❧➔ ✤❛ trà✮ ①→❝ ✤à♥❤ ❜ð✐
Js x = us ∈ E ∗ :
x, us = x
us ,
us = x
s−1
,
✤÷ñ❝ ❣å✐ ❧➔ →♥❤ ①↕ ✤è✐ ♥❣➝✉ tê♥❣ q✉→t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✳ ❑❤✐
s = 2✱ →♥❤ ①↕ J2 ✤÷ñ❝ ❦þ ❤✐➺✉ ❧➔ J ✈➔ ✤÷ñ❝ ❣å✐ ❧➔ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥
t➢❝ ❝õ❛ E ✳ ❚ù❝ ❧➔
Jx = {u ∈ E ∗ :
x, u = x u ,
u = x }.
◆❤➟♥ ①➨t ✶✳✶✳✶✼
(i) ⑩♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ tç♥ t↕✐ tr♦♥❣ ♠å✐ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳
(ii) ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✱ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ❧➔ →♥❤ ①↕
✤ì♥ ✈à I ✳
✶✵
▼➺♥❤ ✤➲ ✶✳✶✳✶✽ ✭①❡♠ ❬✷❪✮ ❈❤♦ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ ✈➔
J : E → 2E
∗
❧➔ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ❝õ❛ E ✳ ❑❤✐ ✤â✱
(i) J(0) = {0}✳
(ii) ❱î✐ ♠é✐ x ∈ E ✱ J(x) ❧➔ t➟♣ ❧ç✐ ✤â♥❣✱ ❜à ❝❤➦♥ ✈➔ ❦❤→❝ ré♥❣✳
(iii) J(λx) = λJ(x) ✈î✐ ♠å✐ x ∈ E ✈➔ λ ∈ R✳
(iv) ◆➳✉ E ∗ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ❝❤➦t t❤➻ J ❧➔ →♥❤ ①↕ ✤ì♥ trà✳
▼➺♥❤ ✤➲ ✶✳✶✳✶✾ ✭①❡♠ ❬✷❪✮ ❈❤♦ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ trì♥ tù❝ ❧➔
♥➳✉ ✈î✐ ♠é✐ ✤✐➸♠ x ♥➡♠ tr➯♥ ♠➦t ❝➛✉ ✤ì♥ ✈à SE tç♥ t↕✐ ❞✉② ♥❤➜t ♠ët
♣❤✐➳♠ ❤➔♠ gx ∈ E ∗ s❛♦ ❝❤♦ x, gx = x ✈➔ gx = 1. ❑❤✐ ✤â✱
x
2
+ 2 y, j(x) ≤ x + y
2
≤ x
2
+ 2 y, j(x + y)
✈î✐ ♠å✐ x, y ∈ E ✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✵ ✭①❡♠ ❬✶❪✮ ❈❤♦ C ❧➔ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ ❦❤æ♥❣
❣✐❛♥ ❇❛♥❛❝❤ E ✳
(i) ⑩♥❤ ①↕ T : C → E ✤÷ñ❝ ❣å✐ ❧➔ →♥❤ ①↕ L✲❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ ♥➳✉ tç♥
t↕✐ ❤➡♥❣ sè L ≥ 0 s❛♦ ❝❤♦
T (x) − T (y) ≤ L x − y
✭✶✳✸✮
∀x, y ∈ C.
(ii) ❚r♦♥❣ ✭✶✳✸✮✱ ♥➳✉ L ∈ [0, 1) t❤➻ T ✤÷ñ❝ ❣å✐ ❧➔ →♥❤ ①↕ ❝♦❀ ♥➳✉ L = 1
t❤➻ T ✤÷ñ❝ ❣å✐ ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✶ ✭①❡♠ ❬✷❪✮ ⑩♥❤ ①↕ T : C → E ✤÷ñ❝ ❣å✐ ❧➔ →♥❤ ①↕
γ ✲❣✐↔ ❝♦ ❝❤➦t ♥➳✉ tç♥ t↕✐ ❤➡♥❣ sè γ ∈ (0, 1) ✈➔ j(x − y) ∈ J(x − y) s❛♦
❝❤♦
T (x) − T (y), j(x − y) ≤ x−y 2 −γ (I−T )(x)−(I−T )(y)
2
∀x, y ∈ C,
✭✶✳✹✮
✈î✐ γ ❧➔ ❤➡♥❣ sè ❦❤æ♥❣ ➙♠ ❝è ✤à♥❤✳ ❚r♦♥❣ ✭✶✳✹✮✱ ♥➳✉ γ = 0 t❤➻ T ✤÷ñ❝
❣å✐ ❧➔ →♥❤ ①↕ ❣✐↔ ❝♦✳
✶✶
◆❤➟♥ ①➨t ✶✳✶✳✷✷ ✭①❡♠ ❬✷❪✮
(i) ◆➳✉ T : E → E ❧➔ →♥❤ ①↕ γ ✲❣✐↔ ❝♦ ❝❤➦t t❤➻ T ❧➔ →♥❤ ①↕ L✲❧✐➯♥ tö❝
▲✐♣s❝❤✐t③ ✈î✐ L = 1 + 1/γ ✳
(ii) ▼å✐ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✤➲✉ ❧➔ →♥❤ ①↕ ❣✐↔ ❝♦ ❧✐➯♥ tö❝✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✸ ✭①❡♠ ❬✸❪✮ ⑩♥❤ ①↕ F : E → E ✤÷ñ❝ ❣å✐ ❧➔
(i) η ✲j ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ♥➳✉ tç♥ t↕✐ ❤➡♥❣ sè η > 0 s❛♦ ❝❤♦ ✈î✐ ♠å✐ x, y
t❤✉ë❝ t➟♣ ①→❝ ✤à♥❤ D(F ) ❝õ❛ →♥❤ ①↕ F ✱ t❛ ❝â
F (x) − F (y), j(x − y) ≥ η x − y 2 , j(x − y) ∈ J(x − y);
(ii) α✲j ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ♥❣÷ñ❝ ✭❤❛② α✲✤ç♥❣ ❜ù❝ j ✲✤ì♥ ✤✐➺✉✮ ♥➳✉ tç♥ t↕✐
❤➡♥❣ sè α > 0 s❛♦ ❝❤♦ ✈î✐ ♠å✐ x, y ∈ D(F )✱ t❛ ❝â
F (x) − F (y), j(x − y) ≥ α F (x) − F (y) 2 , j(x − y) ∈ J(x − y);
(iii) j ✲✤ì♥ ✤✐➺✉ ♥➳✉ ✈î✐ ♠å✐ x, y ∈ D(F )✱ t❛ ❝â
F (x) − F (y), j(x − y) ≥ 0, j(x − y) ∈ J(x − y);
(iv) j ✲✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ♥➳✉ F ❧➔ →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ ✈➔ ✤ç t❤à G(F ) ❝õ❛
→♥❤ ①↕ F ❦❤æ♥❣ t❤ü❝ sü ❜à ❝❤ù❛ tr♦♥❣ ❜➜t ❦➻ ♠ët ✤ç t❤à ❝õ❛ ♠ët →♥❤
①↕ j ✲✤ì♥ ✤✐➺✉ ❦❤→❝❀
(v) m✲j ✲✤ì♥ ✤✐➺✉ ♥➳✉ F ❧➔ →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ ✈➔ R(F + I) = E ✱ ð ✤➙②
R(F ) ❧➔ ❦þ ❤✐➺✉ t➟♣ ❣✐→ trà ❝õ❛ →♥❤ ①↕ F ✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✹ ✭①❡♠ ❬✸❪✮ ⑩♥❤ ①↕ F :
E → E ∗ ✤÷ñ❝ ❣å✐ ❧➔ ❧✐➯♥ tö❝
t❤❡♦ t✐❛ t↕✐ ✤✐➸♠ x ∈ E ♥➳✉ F (x + th)
F (x)✱ ❦❤✐ t → 0 ✈➔ F ✤÷ñ❝
❣å✐ ❧➔ ❧✐➯♥ tö❝ t❤❡♦ t✐❛ tr➯♥ E ♥➳✉ ♥â ❧✐➯♥ tö❝ t❤❡♦ t✐❛ t↕✐ ♠å✐ x ∈ E ✳
◆❤➟♥ ①➨t ✶✳✶✳✷✺ ◆➳✉ F ❧➔ ♠ët →♥❤ ①↕ ❧✐➯♥ tö❝✱ t❤➻ F ❧➔ ❧✐➯♥ tö❝ t❤❡♦
t✐❛✳ ✣✐➲✉ ♥❣÷ñ❝ ❧↕✐ ❦❤æ♥❣ ✤ó♥❣✳ ◆❣♦➔✐ r❛ ♥➳✉ F : E → E ∗ ❧➔ →♥❤ ①↕
✤ì♥ ✤✐➺✉ ✈➔ ❧✐➯♥ tö❝ t❤❡♦ t✐❛ ✈î✐ D(F ) = E t❤➻ F ❧➔ →♥❤ ①↕ ✤ì♥ ✤✐➺✉
❝ü❝ ✤↕✐✳
t tự ỡ
t tự ỡ
C t rộ ỗ õ ừ ổ E
F : E E ổ ố ừ E t t tự
ợ F t r ở C ỵ (F, C)
ữủ t ữ s
tỷ x C tọ F (x ), x x 0
x C.
F ỡ t t tự ữủ ồ
t tự ỡ
ử f số tr [a, b] R x [a, b] s
f (x ) = min f (x).
[a,b]
x (a, b) t f (x ) = 0
x = a t f (x ) 0
x = b t f (x ) 0
r trữớ ủ t õ f (x )(x x ) 0 ởt t
tự
ử F ởt số tr t ỗ õ C ừ
ổ Rn x C tọ
F (x ) = min F (x).
xC
sỷ x ỹ t t x ởt tỷ tũ ỵ ừ C
C ởt t ỗ
(1 t)x + tx = x + t(x x ) = x + t(x x ) C
(t) = F (x + t(x x ) t [0; 1]
t [0; 1].
t ỹ t t t = 0 r (t) 0
(t) = F (x + t(x x )).(x x ).
ữ
(0) = F (x ), x x 0 x C.
ỹ tỗ t t ừ t tự tr
ổ E ữủ tr ỵ ữợ
ỵ C ởt t rộ ỗ õ ừ
ổ E F : C E ỡ tử
t t ợ C = D(F ) ự õ t ừ t
rộ r F ỡ t t ừ
t
t tự j ỡ
E ổ j : E E ố
t ỡ tr ừ E r t ổ sỷ F : E E
ỡ tr t t tự j ỡ ỵ
(F, C) ữủ t ữ s
x C tọ F (x ), j(x x ) 0
x C,
ợ j(x x ) J(x x )
ờ s ố ỳ j ỡ
ờ T : D(T ) E E ởt
õ T j ỡ I T ợ I
ỡ tr E
E ổ trỡ
F : E E j ỡ t ợ + > 1
õ
✶✹
(i) ⑩♥❤ ①↕ I − F ❧➔ →♥❤ ①↕ ❝♦ ✈î✐ ❤➺ sè ❝♦
(1 − η)/γ ✳
(ii) ❱î✐ ♠å✐ λ ∈ (0, 1)✱ I − λF ❧➔ →♥❤ ①↕ ❝♦ ✈î✐ ❤➺ sè ❝♦ 1 − λτ ✱ tr♦♥❣
✤â τ = 1 −
(1 − η)/γ ∈ (0, 1)✳
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✻ ✭①❡♠ ❬✷❪✮ ⑩♥❤ ①↕ QC : E → C ✤÷ñ❝ ❣å✐ ❧➔ ♣❤➨♣ ❝♦
rót ❦❤æ♥❣ ❣✐➣♥ t❤❡♦ t✐❛ tø E ❧➯♥ C ♥➳✉ QC t❤ä❛ ♠➣♥✿
(i) QC ❧➔ ♣❤➨♣ ❝♦ rót tr➯♥ C ✱ tù❝ ❧➔ Q2C = QC ❀
(ii) QC ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥❀
(iii) QC ❧➔ →♥❤ ①↕ t❤❡♦ t✐❛✱ tù❝ ❧➔ ✈î✐ ♠å✐ 0 < t < ∞
QC (QC (x) + t(x − QC (x))) = QC (x).
❚➟♣ C ✤÷ñ❝ ❣å✐ ❧➔ t➟♣ ❝♦ rót ❦❤æ♥❣ ❣✐➣♥ t❤❡♦ t✐❛ ♥➳✉ tç♥ t↕✐ ♣❤➨♣ ❝♦
rót ❦❤æ♥❣ ❣✐➣♥ t❤❡♦ t✐❛ QC tø E ❧➯♥ C ✳
❇ê ✤➲ ✶✳✷✳✼ ✭①❡♠ ❬✷❪✮ ❈❤♦ C ❧➔ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❧ç✐ ✤â♥❣ ❝õ❛ ❦❤æ♥❣
❣✐❛♥ ❇❛♥❛❝❤ trì♥ E ✈➔ QC : E → C ❧➔ ♣❤➨♣ ❝♦ rót tø E ❧➯♥ C ✳ ❑❤✐ ✤â✱
❝→❝ ♣❤→t ❜✐➸✉ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
(i) QC ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ t❤❡♦ t✐❛✳
✭✐✐✮ x − QC (x), j(y − QC (x)) ≤ 0 ∀x ∈ E, y ∈ C ✳
❈❤ó þ ✶✳✷✳✽ ❑❤✐ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H t❤➻ →♥❤ ①↕ QC ❝❤➼♥❤ ❧➔
♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ PC ❝❤✐➳✉ H ❧➯♥ C ✳
✶✺
❈❤÷ì♥❣ ✷
P❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ❣✐↔✐ ♠ët ❧î♣
❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥
❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣
❞è❝ ♥❤➜t ❣✐↔✐ ♠ët ❧î♣ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳
◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷ñ❝ ✈✐➳t tr➯♥ ❝ì sð ❝→❝ ❜➔✐ ❜→♦ ❬✺❪✱ ❬✻❪ ✈➔ ❬✾❪
❝æ♥❣ ❜è ♥➠♠ ✷✵✶✻✱ ✷✵✶✼ ✈➔ ✷✵✶✽✳
✷✳✶ P❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣ ❞è❝ ♥❤➜t
✷✳✶✳✶ ❇➔✐ t♦→♥
❈❤♦ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕ t❤ü❝✱ ❧ç✐ ❝❤➦t ✈➔ ❝â ❝❤✉➞♥ ❦❤↔
✈✐ ●➙t❡❛✉① ✤➲✉✱ F : E → E ❧➔ →♥❤ ①↕ η ✲j ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ γ ✲❣✐↔ ❝♦
❝❤➦t ✈î✐ η + γ > 1✳ ❈❤♦ {Ti }∞
i=1 ❧➔ ♠ët ❤å ✈æ ❤↕♥ ✤➳♠ ✤÷ñ❝ ❝→❝ →♥❤ ①↕
❦❤æ♥❣ ❣✐➣♥ tr➯♥ E ✈î✐
C := ∩∞
i=1 ❋✐①(Ti ) = ∅,
ð ✤➙② ❋✐①(Ti ) := x ∈ E : Ti (x) = x ❧➔ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕
Ti , i = 1, 2, ..✳ ▲î♣ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✤÷ñ❝ ♥❣❤✐➯♥ ❝ù✉
tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ❧➔✿
❚➻♠ ♣❤➛♥ tû x∗ ∈ C s❛♦ ❝❤♦✿ F (x∗ ), j(x − x∗ ) ≥ 0
∀x ∈ C,
✭✷✳✶✮
j ố t ỡ tr ừ E ỵ t
(F, C)
x C tọ ữủ ồ ừ t (F, C)
ừ t (F, C) ỵ S(F,C)
t
(i) t ữủ ỗ t t
t ỹ ởt ữỡ
t tr trữớ ủ E ổ ỗ
trỡ F j ỡ ữủ t ữủ sỹ ở
tử ừ r ụ r t t tự
õ ợ t t ở ừ t
t t ổ ừ t tỷ j ỡ
(ii) E ổ rt tỹ H t t tr t
t t tự ờ
tỷ x C s F (x ), x x 0
x C.
(iii) F : E E j ỡ t
õ ừ t t
t sỷ x , y S(F,C) õ
F (x ), j(y x ) 0
F (y ), j(x y ) 0.
ở ợ t tự tr ử (iii)
t ữủ
F (x ) F (y ), j(x y ) 0.
ứ t tự t ủ ợ tt F j ỡ
s r x y
2
0 õ x = y
Pữỡ
r ử t tr ữỡ ữớ
ố t t t tự ợ C t
t ở ừ ởt ồ ổ ữủ ổ
tr ổ tỹ ỗ t E
Pữỡ ợ t t x1 E tũ ỵ
{xk }
k=1 ữủ ỹ ữ s
xk+1 = (I k F )Sk (xk ),
k = 1, 2, . . .
tr õ Sk ữủ
k
Sk =
i=1
si i
T
si
ợ
T i = (1 i )I + i Ti ,
i = 1, 2, . . .
i k số tỹ ữỡ tở (0, 1) {si } sk
tọ
k
si > 0,
sk =
si = s < ,
si ,
i=1
i=1
{Ti }
i=1 ởt ồ ổ ữủ ổ tr E I
ỡ tr E
số tỹ {si } t ở tử 0 i t t
ữỡ s
Pữỡ ợ t t x1 E tũ ỵ
{xk }
k=1 ữủ ỹ ữ s
xk+1 = (I k F )Sk (xk ),
k = 1, 2, . . .
Sk ữủ
1
Sk =
s0 sk
k
(si1 si )T i
i=1
tr õ T i ữủ ữ tr i (0, 1) k (0, 1)
✶✽
P❤÷ì♥❣ ♣❤→♣ ✷✳✶✳✹ ✭①❡♠ ❬✾❪✮ ❱î✐ ①✉➜t ♣❤→t ❜❛♥ ✤➛✉ x1 ∈ E tò② þ✱
❞➣② ❧➦♣ {xk }∞
k=1 ✤÷ñ❝ ①➙② ❞ü♥❣ ♥❤÷ s❛✉
xk+1 = (I − λk F )S k (xk ),
k = 1, 2, . . .
✭✷✳✾✮
ð ✤➙② →♥❤ ①↕ S k ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐
k
k
S = αI + (1 − α)T
✈î✐ T =
k
k
i=1
si
Ti ,
s¯k
✭✷✳✶✵✮
tr♦♥❣ ✤â α ∈ (0, 1)✱ λk ∈ (0, 1)✱ si ✱ s¯k ✤÷ñ❝ ①→❝ ✤à♥❤ ✈➔ t❤ä❛ ♠➣♥ ✭✷✳✻✮✳
◆❤➟♥ ①➨t ✷✳✶✳✺ ✣è✐ ✈î✐ P❤÷ì♥❣ ♣❤→♣ ✷✳✶✳✷✱ ð ❜÷î❝ ❧➦♣ t❤ù k t❛ ❝➛♥
t❤ü❝ ❤✐➺♥ t➼♥❤ ❤❛✐ tê♥❣ t❤➔♥❤ ♣❤➛♥
k
i=1
(1 − αi )si
xk
s¯k
k
✈➔
i=1
αi si
Ti (xk ).
s¯k
❱î✐ ✈✐➺❝ sû ❞ö♥❣ →♥❤ ①↕ S k ♥❤÷ tr♦♥❣ ✭✷✳✶✵✮ t❤➻ ð ❜÷î❝ ❧➦♣ t❤ù k t❛
❦❤æ♥❣ ❝➛♥ t➼♥❤ tê♥❣ t❤➔♥❤ ♣❤➛♥ t❤ù ♥❤➜t✳ ❉♦ ✤â✱ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣
✭✷✳✾✮ ✤÷ñ❝ ❣✐↔♠ ❜ît sè t❤➔♥❤ ♣❤➛♥ ❝➛♥ t➼♥❤ t♦→♥ t↕✐ ♠é✐ ❜÷î❝ ❧➦♣ s♦
✈î✐ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✭✷✳✸✮✳
✷✳✷ ❙ü ❤ë✐ tö ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣
✷✳✷✳✶ ❙ü ❤ë✐ tö ❝õ❛ P❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✳✷✮
❙ü ❤ë✐ tö ♠↕♥❤ ❝õ❛ P❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✳✷✮ ✤÷ñ❝ ♣❤➙♥ t➼❝❤ ✈➔ ❝❤ù♥❣
♠✐♥❤ ❞ü❛ tr➯♥ ❝→❝ ❜ê ✤➲ s❛✉ ✤➙②✳
❇ê ✤➲ ✷✳✷✳✶ ✭①❡♠ ❬✶✸❪✮ ❈❤♦ {ak }∞k=1 ❧➔ ♠ët ❞➣② sè t❤ü❝ ❦❤æ♥❣ ➙♠ t❤ä❛
♠➣♥ ✤✐➲✉ ❦✐➺♥
ak+1 ≤ (1 − bk )ak + bk ck ,
∞
ð ✤➙② {bk }∞
k=1 ✈➔ {ck }k=1 ❧➔ ❞➣② ❝→❝ sè t❤ü❝ t❤ä❛ ♠➣♥
(i) bk ∈ [0, 1]❀
(ii)
∞
k=1 bk
= ∞❀
(iii) lim supk ck 0.
õ limk ak = 0
ờ sỷ {xk }k=1 {zk }k=1
tr ổ E tọ
xk+1 = hk xk + (1 hk )zk
ợ k 1,
{hk }
k=1 tọ
0 < lim inf hk lim sup hk < 1.
k
k
tt r
lim sup zk+1 zk xk+1 xk
0.
k
õ limk xk zk = 0.
ờ Q ởt t ỗ õ ừ ổ
ỗ t E {Ti }
i=1 ởt ồ ổ ữủ ổ
tr Q tọ
i=1 (Ti ) = sỷ {ti }i=1 ởt số tỹ
ữỡ tọ
i=1 ti
= 1 õ T tr Q
T (x) =
ti Ti (x)
i=1
ợ x Q t ỡ ỳ T ổ
(T ) =
i=1 (Ti )
ờ F ởt j ỡ
t ợ + > 1 T ởt ổ tr ổ
tỹ ỗ t E õ t ợ ộ
t (0, 1) ồ ởt số t (0, 1) tũ ỵ s t 0 t 0
sỷ {yt }
yt = (I t F )T (yt ).
õ {yt } ở tử tợ x ừ t t tự
ợ C = (T ) ữủ tt rộ t 0
ờ sỷ E, F Sk , S ữủ ữ tr ờ
tữỡ ự õ {xk }
k=1
t
lim sup F (p ), j(p xk ) 0.
k
ỵ E ổ tỹ ỗ
t õ t F : E E j ỡ
t ợ + > 1 {Ti }
i=1 ởt ồ ổ
ữủ ổ tr E ợ C :=
i=1 (Ti ) = sỷ
k (0, 1) tọ
limk k = 0
k=1 k =
si tọ õ {xk }
k=1
ở tử tợ t x ừ t t tự
(F, C) k
ự ữợ ự {xk }k=1
ứ ợ ồ p C t õ
xk+1 p = (I k F )Sk (xk ) p
(1 k ) xk p + (1 k F )Sk (p) Sk (p)
(1 k ) xk p + k F (p)
1
= (1 k ) xk p + k F (p)
1
max x1 p , F (p) .
õ {xk }
k=1
ữợ ự
xk+1 xk 0 k
{Sk (xk )}
k=1 {Sk+1 (xk )}k=1
{xk }
k=1
i
{F Sk (xk )}k=1 {T1 (xk )}
k=1 {T (xk )}k=1 ụ ợ i 1
ổ t t tờ qt t tt
✷✶
❤➡♥❣ sè ❞÷ì♥❣ M1 ✳ ▼➦t ❦❤→❝ tø ✭✷✳✹✮ ✈➔ ✭✷✳✺✮✱ t❛ ❝â t❤➸ ✈✐➳t
xk+1 = λk (I − F )S¯k (xk ) + (1 − λk )S¯k (xk ),
= λk (I − F )S¯k (xk ) + (1 − λk )
s1
(1 − α1 )xk
s¯k
s1 α1
s¯k − s1 ¯
+
T1 (xk ) +
Sk,−1 (xk )
s¯k
s¯k
= hk xk + (1 − hk )zk ,
ð ✤➙②
S¯k,−1 =
1
s˜k − s1
k
si T i , hk =
i=2
(1 − λk )(1 − α1 )s1
,
s˜k
λk (I − F )S¯k (xk ) (1 − λk )s1 α1 T1 (xk )
zk =
+
1 − hk
s¯k (1 − hk )
(1 − λk )(¯
sk − s1 )S¯k,−1 (xk )
+
.
s¯k (1 − hk )
❱➻
λk+1 (I − F )S¯k+1 (xk+1 ) λk (I − F )S¯k (xk )
−
C1 :=
1 − hk+1
1 − hk
λk+1
=
(I − F )S¯k+1 (xk+1 ) − (I − F )S¯k+1 (xk )
1 − hk+1
λk+1
(I − F )S¯k+1 (xk ) − (I − F )S¯k (xk )
+
1 − hk+1
λk
λk+1
+
−
×(I − F )S¯k (xk ),
1 − hk+1 1 − hk
(1 − λk+1 )T1 (xk+1 ) (1 − λk )T1 (xk )
−
s¯k+1 (1 − hk+1 )
s¯k (1 − hk )
1 − λk+1
=
T1 (xk+1 ) − T1 (xk )
s¯k+1 (1 − hk+1 )
1 − λk+1
1 − λk
+
−
T1 (xk ),
s¯k+1 (1 − hk+1 ) s¯k (1 − hk )
C2 : =
✈➔
(1 − λk+1 )(¯
sk+1 − s1 )S¯k+1,−1 (xk+1 )
C3 :=
s¯k+1 (1 − hk+1 )
(1 − λk )(¯
sk − 1)S¯k,−1 (xk )
−
s¯k (1 − hk )
✭✷✳✶✷✮