Sự tồn tại và tính liên thông của tập nghiệm đối với bài toán tựa cân bằng véctơ suy rộng

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

◆❣✉②➵♥ ▼✐♥❤ ❍✐➲♥

❙Ü ❚➬◆ ❚❸■ ❱⑨ ❚➑◆❍ ▲■➊◆ ❚❍➷◆● ❈Õ❆ ❚❾P ◆●❍■➏▼
✣➮■ ❱❰■ ❇⑨■ ❚❖⑩◆ ❚Ü❆ ❈❹◆ ❇➀◆● ❱➆❈❚❒ ❙❯❨ ❘❐◆●

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✾


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

◆❣✉②➵♥ ▼✐♥❤ ❍✐➲♥
❙Ü ❚➬◆ ❚❸■ ❱⑨ ❚➑◆❍ ▲■➊◆ ❚❍➷◆● ❈Õ❆ ❚❾P ◆●❍■➏▼
✣➮■ ❱❰■ ❇⑨■ ❚❖⑩◆ ❚Ü❆ ❈❹◆ ❇➀◆● ❱➆❈❚❒ ❙❯❨ ❘❐◆●
◆❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t
số


ữớ ữợ ❦❤♦❛ ❤å❝
❚❙✳ ❇Ò■ ❚❍➌ ❍Ò◆●

❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✾

▲í✐ ❝❛♠ ✤♦❛♥
❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣ t❤ü❝
✈➔ ❦❤ỉ♥❣ trị♥❣ ợ t ỗ t sỷ ử
t ỗ t ❈→❝ t❤ỉ♥❣ t✐♥✱ t➔✐ ❧✐➺✉ tr♦♥❣ ❧✉➟♥ ✈➠♥
♥➔② ✤➣ ✤÷đ❝ ró ỗ ố

t

ữớ t ❧✉➟♥ ✈➠♥

◆❣✉②➵♥ ▼✐♥❤ ❍✐➲♥
❳→❝ ♥❤➟♥
❝õ❛ ❦❤♦❛ ❝❤✉②➯♥ ♠ỉ♥

❳→❝ ♥❤➟♥
❝õ❛ ♥❣÷í✐ ữợ
ũ ũ




ớ ỡ
rữợ tr ở ừ ❧✉➟♥ ✈➠♥✱ tỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t
ì♥ s➙✉ s➢❝ tợ s ũ ũ ữớ trỹ t ữợ
ú ù t t t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ❣✐ó♣ tỉ✐ ❤♦➔♥
t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳
❚ỉ✐ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉✱ ❦❤♦❛ ❚♦→♥ ❝ị♥❣ t♦➔♥ t❤➸ ❝→❝
t❤➛② ❝ỉ ❣✐→♦ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❱✐➺♥ ❚♦→♥
❤å❝ ✈➔ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✤➣ tr✉②➲♥ t❤ư ❝❤♦ tỉ✐ ♥❤ú♥❣ ❦✐➳♥
t❤ù❝ q✉❛♥ trå♥❣✱ t↕♦ ✤✐➲✉ ❦✐➺♥ t ủ tổ ỳ ỵ õ õ

qỵ ❜→✉ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳
❈✉è✐ ❝ị♥❣✱ tỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ q✉❛♥ t➙♠ ❣✐ó♣ ✤ï✱
✤ë♥❣ ✈✐➯♥ tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❧➔♠ ❧✉➟♥ ✈➠♥✳
❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦

❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✾

◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥

◆❣✉②➵♥ ▼✐♥❤ ❍✐➲♥

✐✐


▼ư❝ ❧ư❝
▲í✐ ❝❛♠ ✤♦❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
▲í✐ ❝↔♠ ì♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
ử ỵ ỳ t t➢t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳







ỗ ởt số t t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

ổ ỗ ữỡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✶✳✸✳ ❑❤→✐ ♥✐➺♠ →♥❤ ①↕ ✤❛ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

✶✳✹✳ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ →♥❤ ①↕ ✤❛ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✹✳✶✳ ◆â♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✶✳✹✳✷✳ ❚➼♥❤ ❧✐➯♥ tö❝ t❤❡♦ ♥â♥ ❝õ❛ →♥❤ ①↕ ✤❛ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



ỗ t õ ❝õ❛ →♥❤ ①↕ ✤❛ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹

❈❤÷ì♥❣ ✷✳ ỹ tỗ t t tổ ừ t ✤è✐ ✈ỵ✐
❜➔✐ t♦→♥ tü❛ ❝➙♥ ❜➡♥❣ ✈➨❝tì s✉② rë♥❣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✷✳✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ♠ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

ỹ tỗ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✸

✷✳✸✳ ❚➼♥❤ ❧✐➯♥ t❤æ♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✽

❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✐✐✐


ử ỵ ỳ t
tt
R

t số t❤ü❝

R+

t➟♣ sè t❤ü❝ ❦❤ỉ♥❣ ➙♠

R−

t➟♣ sè t❤ü❝ ❦❤ỉ♥❣ ❞÷ì♥❣

Rn

❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì ❊✉❝❧✐❞❡ n− ❝❤✐➲✉

Rn+

t➟♣ ❝→❝ ✈➨❝tì ❦❤ỉ♥❣ ➙♠ ❝õ❛ Rn

Rn−

t➟♣ ❝→❝ ✈➨❝tì ❦❤ỉ♥❣ ❞÷ì♥❣ ❝õ❛ Rn

f :X→Y

→♥❤ ①↕ tø t➟♣ X ✈➔♦ t➟♣ Y

A := B

A ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜➡♥❣ B

∅

t➟♣ ré♥❣

A⊆B

A ❧➔ t➟♣ ❝♦♥ ❝õ❛ B

A⊆B

A ❦❤æ♥❣ ❧➔ t➟♣ ❝♦♥ ❝õ❛ B

A∪B

❤ñ♣ ❝õ❛ ❤❛✐ t➟♣ ❤ñ♣ A ✈➔ B

dom F

ừ tr F

gph F

ỗ t❤à ❝õ❛ →♥❤ ①↕ ✤❛ trà F

✐✈


A∩B

❣✐❛♦ ❝õ❛ ❤❛✐ t➟♣ ❤ñ♣ A ✈➔ B

A\B

❤✐➺✉ ❝õ❛ ❤❛✐ t➟♣ ❤đ♣ A ✈➔ B

B

t➼❝❤ ❉❡s❝❛rt❡s ❝õ❛ ❤❛✐ t➟♣ ❤đ♣ A ✈➔ B

cl A

❜❛♦ ✤â♥❣ tỉ♣ỉ ❝õ❛ t➟♣ ❤đ♣ A

co A

ỗ ừ t ủ A

int A

tr tổổ ừ t ủ A

conv A

ỗ ừ t ủ A



t tú ❝❤ù♥❣ ♠✐♥❤

✈


▼ð ✤➛✉
❇➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ✈➨❝tì ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ q trồ tr t ỵ
t ồ ỵ tt sỹ tỗ t ừ t ố ợ t♦→♥ ❝➙♥
❜➡♥❣ ✈➨❝tì ✤➣ ✤÷đ❝ r➜t ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ ự ữợ tt
ử t tỹ ỗ tỹ ỗ t õ
ự sỹ tỗ t↕✐ ✤è✐ ✈ỵ✐ t➟♣ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉
②➳✉ ✈➔ t➟♣ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❝õ❛ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ✈➨❝tì s✉② rë♥❣ ợ
tt ử t ỗ t õ ❝→❝ t→❝ ❣✐↔ ✤➣ ♠ð rë♥❣ ❦➳t q✉↔
tr➯♥ ❝❤♦ ❧ỵ♣ ❜➔✐ t♦→♥ tü❛ ❝➙♥ ❜➡♥❣ ✈➨❝tì s✉② rë♥❣ ❬✻❪✳ ◆❣♦➔✐
ự sỹ tỗ t ừ t ❜➡♥❣ ✈➨❝tì ♥❣÷í✐ t❛ ❝á♥ q✉❛♥ t➙♠

♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❝❤➜t ❝õ❛ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ♥➔②✳ ❚r♦♥❣ sè ❝→❝ t➼♥❤
❝❤➜t ❝õ❛ t➟♣ ♥❣❤✐➺♠ t❤➻ t➼♥❤ ❧✐➯♥ t❤æ♥❣ ❝â ✈❛✐ trá r➜t q✉❛♥ trå♥❣✱ ✈➻ ♥â
✤÷đ❝ ❜↔♦ t♦➔♥ ❦❤✐ ❝❤✉②➸♥ q✉❛ →♥❤ ①↕ ❧✐➯♥ tư❝✳ ❇❛♥ ✤➛✉✱ ♥❣÷í✐ t❛ ♥❣❤✐➯♥
❝ù✉ t➼♥❤ ❧✐➯♥ t❤æ♥❣ ❝õ❛ t➟♣ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ❧✐➯♥ q✉❛♥ ✤➳♥ →♥❤ ①↕ ✤ì♥
trà tø ❦❤ỉ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ♥➔② s❛♥❣ ❦❤æ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ❦❤→❝
❬✹❪✳ ❙❛✉ õ t ữủ rở ợ ổ ❣✐❛♥ ❝â sè ❝❤✐➲✉ ✈æ
❤↕♥ ❬✽❪✳ ◆➠♠ ✷✵✶✻✱ ❍❛♥ ✈➔ ❍✉❛♥❣ ❬✻❪ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❧✐➯♥ t❤æ♥❣ ❝õ❛ t➟♣
♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ✈➨❝tì s✉② rë♥❣✳ ❙❛✉ ✤â ❝→❝ t→❝ ❣✐↔ ✤➣ ♠ð
rë♥❣ ❦➳t q✉↔ tr➯♥ ❝❤♦ ❧ỵ♣ ❜➔✐ t♦→♥ tü❛ ❝➙♥ ❜➡♥❣ ✈➨❝tì s✉② rë♥❣ ❬✻❪✳
▼ư❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥❤➡♠ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ❤➺ t❤è♥❣ ❝→❝ t q
tr ổ tr sỹ tỗ t t➼♥❤ ❧✐➯♥ t❤ỉ♥❣ ❝õ❛ t➟♣ ♥❣❤✐➺♠ ✤è✐ ✈ỵ✐
❜➔✐ t♦→♥ tü❛ tỡ s rở
ỗ ❤❛✐ ❝❤÷ì♥❣ ♥ë✐ ❞✉♥❣✱ ♣❤➛♥ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉
t❤❛♠ ❦❤↔♦✳
✶


❈❤÷ì♥❣ ✶ ❝õ❛ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ tự t ỗ ổ
ỗ tr ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ →♥❤ ①↕ ✤❛ trà✳
❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ♠ët sè ✤✐➲✉ ❦✐➺♥ ✤õ ❝❤♦ sü tỗ t ỳ
ỳ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❝õ❛ ❜➔✐ t♦→♥ tü❛ ❝➙♥ ❜➡♥❣
✈➨❝tì s✉② rë♥❣✳ ❍ì♥ ♥ú❛ t➼♥❤ ❧✐➯♥ t❤ỉ♥❣ ❝õ❛ t➟♣ ♥❣❤✐➺♠ ✤è✐ ợ t
tỹ tỡ s rở ụ ữủ tr➻♥❤ ❜➔② tr♦♥❣ ❝❤÷ì♥❣ ♥➔②✳

✷


❈❤÷ì♥❣ ✶
❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
●✐↔✐ t➼❝❤ ✤❛ trà ✤÷đ❝ ❤➻♥❤ t❤➔♥❤ tø ♥❤ú♥❣ ♥➠♠ ✸✵ ❝õ❛ t❤➳ ❦✛ ✷✵ ❞♦ ❝❤➼♥❤

♥❤✉ ❝➛✉ ❝õ❛ ❝→❝ ✈➜♥ ✤➲ ♥↔② s✐♥❤ tø t❤ü❝ t✐➵♥ ✈➔ ❝✉ë❝ sè♥❣✳ ❚ø ❦❤♦↔♥❣ ✶✵
♥➠♠ trð ❧↕✐ ✤➙② ✈ỵ✐ ❝ỉ♥❣ ❝ư ❣✐↔✐ t➼❝❤ ✤❛ trà✱ ❝→❝ ♥❣➔♥❤ t♦→♥ ồ ữ ỵ
tt ữỡ tr ữỡ tr ❤➔♠ r✐➯♥❣✱ ❜➜t ✤➥♥❣ t❤ù❝
❜✐➳♥ ♣❤➙♥ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ s✉② rở ỵ tt tố ữ ỵ tt
tố ữ ử t ồ q ỵ t t➳✱ ✳✳✳ ♣❤→t tr✐➸♥ ♠ët ❝→❝❤
♠↕♥❤ ♠➩ ✈➔ ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ s➙✉ s➢❝✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤
❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ✈➔ ❦➳t q✉↔ q✉❡♥ ❜✐➳t ✈➲ ❣✐↔✐ t➼❝❤ ✤❛ trà ✤÷đ❝ tr➼❝❤ r❛
tø ❝✉è♥ s→❝❤ ❝❤✉②➯♥ ❦❤↔♦ ✈➲ ❣✐↔✐ t➼❝❤ ✤❛ trà ❬✶❪✳ ❈→❝ ❦➳t q✉↔ ❝õ❛ ❝❤÷ì♥❣
♥➔② ❧➔ ❝ì sð ❝❤♦ ✈✐➺❝ tr➻♥❤ ❜➔② ❦➳t q ừ ữỡ

ỗ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ●✐↔ sû X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤✳ ❚➟♣ A ⊆ X ✤÷đ❝
❣å✐ ỗ ợ ồ x1 , x2 A t❛ ❧✉ỉ♥ ❝â

λx1 + (1 − λ)x2 ∈ A ✈ỵ✐ ồ [0, 1].

ữợ rộ t ỗ
sỷ A X t ỗ ợ ồ I ợ I ❧➔ t➟♣
❝❤➾ sè ❜➜t ❦➻✳ ❑❤✐ ✤â t➟♣ A = A ỗ
I




❈❤ù♥❣ ♠✐♥❤✳ ▲➜② x, y ∈ A✳ ❑❤✐ ✤â x, y ∈ Aα , ✈ỵ✐ ♠å✐ α ∈ I ✳ A ỗ
ợ ồ I x + (1 − λ)y ∈ Aα , ✈ỵ✐ ♠å✐ λ ∈ [0, 1], α ∈ I. ❉♦ ✤â

λx + (1 )y A. A t ỗ

sỷ Ai X t ỗ λi ∈ R (i = 1, 2, . . . , m)✳ ❑❤✐
✤â λ1 A1 + λ2 A2 + · Ã Ã + m Am t ỗ
ự t A = λ1 A1 + λ2 A2 + · · · + λm Am . ▲➜② x, y ∈ A, õ tỗ
t xi Ai , yi Ai , i = 1, 2, . . . , m s❛♦ ❝❤♦ x = λ1 x1 + λ2 x2 + · · · + λm xm ,

y = λ1 y1 + λ2 y2 + · · · + λm ym .
❚❛ ❝â

λx + (1 − λ)y = λ(λ1 x1 + · · · + λm xm ) + (1 − λ)(λ1 y1 + · · · + λm ym )
= λ1 [λx1 + (1 − λ)y1 ] + · · · + λm [λxm + (1 − λ)ym ].
❉♦ Ai t ỗ xi +(1)yi Ai , ợ ♠å✐ λ ∈ [0, 1], i ∈ {1, 2}, . . . , m.
❙✉② r❛ λx + (1 − λ)y ∈ A, ✈ỵ✐ ♠å✐ λ ∈ [0, 1]. ❱➟② A t ỗ

sỷ X ổ ❣✐❛♥ t✉②➳♥ t➼♥❤✱ A ❧➔ ♠ët t➟♣ ❝♦♥
❝õ❛ X ✳ õ ừ tt t ỗ ự A ữủ ồ ỗ ừ
t A co A.

t ỗ ự tt tờ ủ ỗ ừ
tỷ tr

ỵ sỷ A ❧➔ t➟♣ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ X ✳ ❑❤✐ ✤â
co A trị♥❣ ✈ỵ✐ t➟♣ t➜t ❝↔ ❝→❝ tê ủ ỗ ừ t A tự
n

n

i xi : xi ∈ A, αi ≥ 0,

co A =
i=1

αi = 1 .
i=1

❈❤ù♥❣ õ co A t ỗ A ⊂ co A ♥➯♥ co A ❝❤ù❛ t➜t ❝↔ ❝→❝
tê ủ ỗ ừ A ỡ ỳ t tt tờ ủ ỗ ừ A ỗ ự A
✤â ♥â ❝❤ù❛ co A ✭✈➻ co A ❧➔ t➟♣ ỗ ọ t ự co A trũ ợ
t tt tờ ủ ỗ ừ A



ổ ỗ ữỡ

sỷ X ❧➔ ♠ët t➟♣ ❤ñ♣ ❦❤→❝ ré♥❣✳ ❍å τ ♥❤ú♥❣ t➟♣
❝♦♥ ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ ♠ët tỉ♣ỉ tr➯♥ X ♥➳✉
✭✐✮ ❍❛✐ t➟♣ ∅, X ✤➲✉ t❤✉ë❝ ❤å τ ❀
✭✐✐✮ τ ❦➼♥ ✤è✐ ✈ỵ✐ ♣❤➨♣ ❣✐❛♦ ❤ú✉ ❤↕♥✱ tù❝ ❧➔ ❣✐❛♦ ❝õ❛ ♠ët sè ❤ú✉ ❤↕♥ t➟♣
t❤✉ë❝ ❤å τ t❤➻ ❝ơ♥❣ t❤✉ë❝ ❤å τ ❀
✭✐✐✐✮ τ ❦➼♥ ✤è✐ ✈ỵ✐ ♣❤➨♣ ❤ñ♣ ❜➜t ❦➻✱ tù❝ ❧➔ ❤ñ♣ ❝õ❛ ♠ët sè ❤ú✉ ❤↕♥ ❤❛②
✈ỉ ❤↕♥ t➟♣ t❤✉ë❝ ❤å τ t❤➻ ❝ơ♥❣ t❤✉ë❝ ❤å τ ✳
❈➦♣ (X, τ ) ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ tæ♣æ✳ ❈→❝ ♣❤➛♥ tû t❤✉ë❝ X t❛ ❣å✐ ❧➔
✤✐➸♠ ✈➔ ❝→❝ t➟♣ t❤✉ë❝ ❤å τ ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ♠ð✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✷✳ ●✐↔ sû τ, τ

❧➔ ❝→❝ tæ♣æ tr➯♥ X ✳ ◆➳✉ τ ⊆ τ ✱ t❛ ♥â✐

tæ♣æ τ ②➳✉ ❤ì♥ ✭t❤ỉ ❤ì♥✮ tỉ♣ỉ τ ❤❛② tỉ♣ỉ τ ♠↕♥❤ ❤ì♥ ✭♠à♥ ❤ì♥✮ tỉ♣ỉ τ ✳
❚r÷í♥❣ ❤đ♣ ❦❤ỉ♥❣ ❝â q✉❛♥ ❤➺ ✤â✱ t❛ ♥â✐ ❤❛✐ tỉ♣ỉ ❦❤ỉ♥❣ s♦ s→♥❤ ✤÷đ❝✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✸✳ ❈❤♦ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ (X, τ ) ✈➔ A ⊆ X ✳
✭✐✮ ❚➟♣ ❝♦♥ U ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ X ✤÷đ❝ ❣å✐ ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ A ♥➳✉ U ❧➔ ❜❛♦
❤➔♠ t➟♣ ♠ð ❝❤ù❛ A❀
✭✐✐✮ ▲➙♥ ❝➟♥ ❝õ❛ ♣❤➛♥ tû x ∈ X ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ t➟♣ ❝♦♥ {x}✳ ❍å t➜t ❝↔
❝→❝ ❧➙♥ ❝➟♥ ❝õ❛ ♠ët ✤✐➸♠ ❣å✐ ❧➔ ❤➺ ❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ ✤â✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✹✳ ❑❤æ♥❣ ❣✐❛♥ tỉ♣ỉ (X, τ ) ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ s
r ố ợ tũ ỵ x, y X ổ tỗ t
U ❝õ❛ x, V ❝õ❛ y s❛♦ ❝❤♦ U ∩ V = ∅✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✺✳ ❈❤♦ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì tr➯♥ tr÷í♥❣ K✳
✭✐✮ ▼ët tỉ♣ỉ τ tr➯♥ X ✤÷đ❝ ❣å✐ tữỡ t ợ trú số ừ X
t ở ổ ữợ →♥❤ ①↕ ❧✐➯♥ tư❝✳
✭✐✐✮ ▼ët ❦❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ t✉②➳♥ t➼♥❤ ❤❛② ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì tỉ♣ỉ tr➯♥
tr÷í♥❣ K ❧➔ ♠ët ❝➦♣ (X, τ )✱ tr♦♥❣ ✤â X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì tr trữớ K
ởt tổổ tữỡ t ợ ❝➜✉ tró❝ ✤↕✐ sè ❝õ❛ X ✳

✺


✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✻✳ ❑❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ t✉②➳♥ t➼♥❤ X ✤÷đ❝ ồ ổ
ỗ ữỡ tổổ ừ õ tổổ ỗ ữỡ tr X õ ởt
ỡ s ừ ố ỗ t t ỗ ỡ ổ ỗ
ữỡ X ỗ tớ ổ sr t X ữủ ồ ổ
ỗ ✤à❛ ♣❤÷ì♥❣ ❍❛✉s❞♦r❢❢✳

❱➼ ❞ư ✶✳✷✳✼✳ ❑❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ❦❤ỉ♥❣ rt ổ
ỗ ữỡ sr

♥✐➺♠ →♥❤ ①↕ ✤❛ trà

●✐↔ sû X ✈➔ Y ❧➔ ❤❛✐ t➟♣ ❤ñ♣✳ ❑➼ ❤✐➺✉ 2X ❧➔ t➟♣ t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ ❝õ❛

X✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✳ ▼ët →♥❤ ①↕ ✤❛ trà F tø X ✈➔♦ Y

♠➔ ù♥❣ ✈ỵ✐ ♠é✐ ♣❤➛♥

tû x ∈ X ❝❤♦ ♠ët t➟♣ ❝♦♥ ❝õ❛ Y ✱ ữủ ỵ F : X 2Y
ỹ t ♠é✐ →♥❤ ①↕ ✤❛ trà F : X → 2Y ữủ trữ ởt t
ừ X ì Y ✱ ❦➼ ❤✐➺✉ ❧➔ gph F ✈➔ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐

gph F := (x, y) ∈ X × Y : y ∈ F (x) .
❚➟♣ ❤đ♣ gph F ✤÷đ❝ ❣å✐ ỗ t ừ F

ừ F ✱ ❦➼ ❤✐➺✉ dom F ✱ ①→❝ ✤à♥❤ ❜ð✐

dom F := x ∈ X : F (x) = ∅ .

❱➼ ử t ữỡ tr tự ợ số t❤ü❝
xn + a1 xn−1 + ... + an−1 x + an = 0,
◗✉② t➢❝ ❝❤♦ ù♥❣ ♠é✐ ✈➨❝tì a = (a1 , a2 , ..., an ) ∈ Rn ✈ỵ✐ t➟♣ ♥❣❤✐➺♠ ❝õ❛
♣❤÷ì♥❣ tr➻♥❤ tr➯♥✱ ❦➼ ❤✐➺✉ ❜ð✐ F (a)✱ ❝❤♦ t❛ ♠ët →♥❤ ①↕ ✤❛ trà

F : Rn → 2C
tø ❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ Rn ✈➔♦ ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝ C✳
✻


✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✸✳ ❈❤♦ X, Y

❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✈➔ →♥❤ ①↕ ✤❛

trà F : X → 2Y õ r
F õ tr ỗ F (x) t ỗ tr Y ợ ồ x X;
F ỗ gph F t ỗ tr X ì Y.

❈❤♦ X, Y

❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ ✈➔ F : X → 2Y ❧➔

→♥❤ ①↕ ✤❛ trà✳ ❚❛ ♥â✐ r➡♥❣
✭✐✮ F ❝â ❣✐→ trà ✤â♥❣ ♥➳✉ F (x) ❧➔ t➟♣ ✤â♥❣ tr♦♥❣ Y ✱ ✈ỵ✐ ♠å✐ x ∈ X ❀
✭✐✐✮ F ❧➔ →♥❤ ①↕ ✤â♥❣ ♥➳✉ gph F ❧➔ t➟♣ ✤â♥❣ tr♦♥❣ X × Y ❀
✭✐✐✮ F ❧➔ →♥❤ ①↕ ♠ð ♥➳✉ gph F ❧➔ t➟♣ ♠ð tr♦♥❣ X × Y ❀
✭✐✐✐✮ F ❧➔ →♥❤ ①↕ ❝♦♠♣❛❝t ♥➳✉ F (X) ❧➔ t➟♣ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ tr♦♥❣ Y ✳

▼➺♥❤ ✤➲ ✶✳✸✳✺✳ ●✐↔ sû X, Y

❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ t✉②➳♥ t➼♥❤ ✈➔ →♥❤ ①↕

✤❛ trà F : X → 2Y ✳ ❑❤✐ ✤â
✭✐✮ ◆➳✉ F ❧➔ →♥❤ ①↕ ✤â♥❣ t❤➻ F ❝â ❣✐→ trà ✤â♥❣❀
✭✐✐✮ ◆➳✉ F ❧➔ →♥❤ ①↕ ♠ð t❤➻ F ❝â ❣✐→ trà ♠ð❀
✭✐✐✐✮ ◆➳✉ F ❧➔ →♥❤ ỗ t F õ tr ỗ
F ỗ

(1 t)F (x) + tF (x ) ⊆ F ((1 − t)x + tx ) ✈ỵ✐ ♠å✐ x, x ∈ X ✈➔ t [0, 1]

ử ữợ r r tr õ tr ỗ ữ

ỗ tr õ trà ✤â♥❣ ❝❤÷❛ ❝❤➢❝ ❧➔ →♥❤ ①↕ ✤â♥❣✳

❱➼ ❞ư ✶✳✸✳✻✳ ❈❤♦ →♥❤ ①↕ ✤❛ trà F : N∗ → 2R ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉

 co 1, 2, ..., n − 1 , ♥➳✉ n ≥ 2,
F (n) =
 {0}, ♥➳✉ ♥❂✶.
❍✐➸♥ ♥❤✐➯♥ F ❧➔ →♥❤ ①↕ ✤❛ trà ✈ỵ✐ ❣✐→ tr ỗ F ổ
ỗ




❱➼ ❞ö ✶✳✸✳✼✳ ❳➨t →♥❤ ①↕ ✤❛ trà F : R → 2R ①→❝ ✤à♥❤ ❜ð✐

 [0, 1], ♥➳✉ x = 0,
F (x) =
 R, tr♦♥❣ tr÷í♥❣ ❤đ♣ ❝á♥ ❧↕✐.
❍✐➸♥ ♥❤✐➯♥ →♥❤ ①↕ F ❝â ❣✐→ trà ✤â♥❣✳ ▼➦t ❦❤→❝ t❛ ❝â

gph F = (x, y) ∈ R2 : y ∈ F (x) = ({0} × [0, 1]) ∪ (R\{0} × R)
❧➔ t➟♣ ❦❤ỉ♥❣ ✤â♥❣ tr♦♥❣ R2 ✈➔ ♥❤÷ ✈➟② F ❦❤æ♥❣ ❧➔ →♥❤ ①↕ ✤â♥❣✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✽✳ ❈❤♦ X, Y ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ✳ ⑩♥❤ ①↕ ❜❛♦ ✤â♥❣ ❝õ❛
F ❧➔ →♥❤ ①↕ ✤❛ trà cl F : X 2Y ỗ t ừ õ õ ừ ỗ t
ừ F tự

gph(cl F ) = cl(gph F ).

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✾✳ ●✐↔ sû F : X → 2Y

❧➔ →♥❤ ①↕ ✤❛ trà tø X ✈➔♦ Y ✳ ❚❛

❣å✐ →♥❤ ①↕ ♥❣÷đ❝ ❝õ❛ F ✱ ỵ F 1 : Y 2X ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐

F −1 (y) = x ∈ X : y ∈ F (x) , ✈ỵ✐ y ∈ Y.
❚❛ ♥â✐ F −1 ❧➔ ↔♥❤ ♥❣÷đ❝ ❝õ❛ F ✳
▼å✐ →♥❤ ①↕ ✤❛ trà ✤➲✉ ❝â →♥❤ ①↕ ♥❣÷đ❝✱ ✤✐➲✉ ♥➔② ❦❤ỉ♥❣ ✤ó♥❣ ✤è✐ ✈ỵ✐ →♥❤
①↕ ✤ì♥ trà✳ ❚❛ ❝ơ♥❣ ❞➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷đ❝ ♠å✐ →♥❤ ①↕ ✤❛ trà ❝â ↔♥❤ ♥❣÷đ❝
t↕✐ ♠é✐ ✤✐➸♠ ❧➔ ♠ð ✤➲✉ ❧➔ →♥❤ ①↕ ỷ tử ữợ ữủ ổ
ú

ởt sè t➼♥❤ ❝❤➜t ❝õ❛ →♥❤ ①↕ ✤❛ trà
❚r♦♥❣ ♣❤➛♥ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② t➼♥❤ ❝❤➜t ❧✐➯♥ tư❝ t❤❡♦ ♥â♥ ừ
tr t ỗ t õ ừ →♥❤ ①↕ ✤❛ trà✳ ❈→❝ ❦❤→✐ ♥✐➺♠ tr♦♥❣ ♣❤➛♥
♥➔② ❧➔ sü ♠ð rë♥❣ ❝õ❛ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ t➼♥❤ ❧✐➯♥ tử t ỗ ừ
tr rữợ t t tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ♥â♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤✳

✽


✶✳✹✳✶✳ ◆â♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✶✳ ❈❤♦ Y

❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✈➔ C ❧➔ ♠ët t➟♣ ❝♦♥

❦❤æ♥❣ ré♥❣ tr♦♥❣ Y ✳ ❚❛ ♥â✐ r➡♥❣ C ❧➔ ♥â♥ ❝â ✤➾♥❤ t↕✐ ❣è❝ tr♦♥❣ Y ♥➳✉

tc ∈ C ✱ ✈ỵ✐ ♠å✐ c ∈ C ✈➔ t > 0✳

◆➳✉ C ❧➔ ♥â♥ ❝â ✤➾♥❤ t↕✐ ❣è❝ t❤➻ C + x0 ❧➔ ♥â♥ ❝â ✤➾♥❤ t↕✐ x0 ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✷✳ ❈❤♦ C ❧➔ ♥â♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ Y õ
r
C õ ỗ C t ỗ
C õ ồ l(C) = {0} tr♦♥❣ ✤â l(C) = C ∩ (−C)✳
◆â♥ C ❣å✐ ❧➔ ✤â♥❣ ♥➳✉ C ❧➔ t➟♣ ✤â♥❣ tr♦♥❣ Y ✳ ❚❛ õ C õ ỗ õ

ồ C õ ỗ õ ồ
ữợ ởt số ử ✈➲ ♥â♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤✳

❱➼ ❞ư ✶✳✹✳✸✳ ✶✳ ❈❤♦ Y

❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤✳ ❑❤✐ ✤â 0 ✱ Y ❧➔ ❝→❝ ♥â♥

tr♦♥❣ Y ✈➔ t❛ ❣å✐ ❝❤ó♥❣ ❧➔ ❝→❝ ♥â♥ t➛♠ t❤÷í♥❣ tr♦♥❣ Y ✳
✷✳ ❈❤♦ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ Rn ✳ ❑❤✐ ✤â t➟♣

Rn+ = x = (x1 , x2 , ..., xn ) ∈ Rn : xi ≥ 0, i ∈ {1, 2, ..., n}
❧➔ ♥â♥ ỗ õ ồ tr Rn t ồ õ ❖rt❤❛♥t ❦❤æ♥❣ ➙♠ tr♦♥❣ Rn ✳
✸✳ ●å✐ C[0, 1] ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ❝→❝ ❤➔♠ sè ①→❝ ✤à♥❤ ✈➔ ❧✐➯♥ tư❝
tr➯♥ ✤♦↕♥ [0, 1] ✈ỵ✐ ❝→❝ ♣❤➨♣ t♦→♥ ❝ë♥❣ ổ ữợ

(x + y)(t) = x(t) + y(t),
(x)(t) = λx(t).
❑❤✐ ✤â t➟♣

C+ [0, 1] = x ∈ C[0, 1] : x(t) ≥ 0 ✈ỵ✐ ♠å✐ t ∈ [0, 1]
õ ỗ õ ồ tr C[0, 1]

✶✳✹✳✷✳ ❚➼♥❤ ❧✐➯♥ tö❝ t❤❡♦ ♥â♥ ❝õ❛ →♥❤ ①↕ ✤❛ tr
rữợ t t tử ừ →♥❤ ①↕ ✤ì♥ trà ❣✐ú❛ ❝→❝ ❦❤ỉ♥❣
❣✐❛♥ tỉ♣ỉ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✹✳ ▼ët →♥❤ ①↕ ✤ì♥ trà f

: X → Y tø ❦❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ

X ✈➔♦ ❦❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ Y ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tư❝ t↕✐ x0 ∈ X ♥➳✉ ✈ỵ✐ ♠å✐ t
V tr Y ự f (x0 ) tỗ t ❧➙♥ ❝➟♥ ♠ð U tr♦♥❣ X ❝❤ù❛ x0 s❛♦ ❝❤♦

f (U ) ⊆ V ✳
❚r♦♥❣ tr÷í♥❣ ❤đ♣ F : X → 2Y ❧➔ →♥❤ ①↕ ✤❛ trà tø ❦❤æ♥❣ ❣✐❛♥ tỉ♣ỉ X
✈➔♦ ❦❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ Y ✱ ❇❡r❣❡ ✤➣ ✤÷❛ r❛ ❦❤→✐ ♥✐➺♠ ✈➲ t➼♥❤ ♥û❛ ❧✐➯♥ tö❝ tr➯♥
✈➔ ♥û❛ tử ữợ ừ tr

⑩♥❤ ①↕ ✤❛ trà F : X → 2Y ✤÷đ❝ ồ ỷ tử tr
ữợ t x0 ợ ♠é✐ t➟♣ ♠ð V tr♦♥❣ Y t❤ä❛ ♠➣♥ F (x0 ) ⊆ V ✭t÷ì♥❣
ù♥❣✱ F (x0 ) ∩ V = tỗ t U ừ x0 tr X s❛♦ ❝❤♦ F (x) ⊆ V
✭t÷ì♥❣ ù♥❣✱ F (x) ∩ V = ∅✮ ✈ỵ✐ ♠å✐ x ∈ U ✳
●✐↔ sû X, Y ❧➔ ❤❛✐ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ t✉②➳♥ t➼♥❤ ✈➔ C ❧➔ ♥â♥ tr➯♥ ❨✳ ❚❛
♥❤➢❝ ❧↕✐ ❦❤→✐ ♥✐➺♠ ❧✐➯♥ tö❝ t❤❡♦ ♥â♥ ❝õ❛ →♥❤ ①↕ ✤❛ trà✳ ❑❤→✐ ♥✐➺♠ ♥➔② ❧➔
♠ð rë♥❣ ❦❤→✐ ♥✐➺♠ ❝õ❛ ❇❡r❣❡ ✈➲ t➼♥❤ ♥û❛ tử tr ỷ tử ữợ
ừ ✤❛ trà✳

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✻✳ ❈❤♦ →♥❤ ①↕ ✤❛ trà F : X → 2Y ✳
✭✐✮ F ✤÷đ❝ ❣å✐ ❧➔ C tử tr ữợ t x
dom F ♥➳✉ ✈ỵ✐ ♠é✐ ❧➙♥

❝➟♥ V ❝õ❛ ❣è❝ tr♦♥❣ Y ✱ tỗ t U ừ x
tr X s ❝❤♦

F (x) ⊆ F (¯
x) + V + C
(F (¯
x) F (x) + V C, tữỡ ự)
ợ ồ x ∈ U ∩ dom F ✳
✭✐✐✮ ◆➳✉ F ❧➔ C ✲ ❧✐➯♥ tö❝ tr➯♥ ✈➔ C ✲ ❧✐➯♥ tö❝ ữợ t x
ỗ tớ t t
õ F C ✲ ❧✐➯♥ tö❝ t↕✐ x
¯✳
✶✵


✭✐✐✐✮ ◆➳✉ F ❧➔ C ✲ ❧✐➯♥ tö❝ tr➯♥✱ C tử ữợ C tử t ♠å✐
✤✐➸♠ tr♦♥❣ dom F ✱ t❛ ♥â✐ F ❧➔ C tử tr C tử ữợ C ✲ ❧✐➯♥
tö❝ tr♦♥❣ X ✳
❈→❝ ❦❤→✐ ♥✐➺♠ ♥û❛ ❧✐➯♥ tử tr ỷ tử ữợ t r
❤♦➔♥ t♦➔♥ ❦❤→❝ ♥❤❛✉✳ ❉♦ ✤â ❦❤→✐ ♥✐➺♠ ❧✐➯♥ tö❝ tr t õ tử
ữợ t õ ụ t ử ữợ ồ ❝❤♦
✤✐➲✉ ❦❤➥♥❣ ✤à♥❤ ✤â✳

❱➼ ❞ö ✶✳✹✳✼✳ ❈❤♦ →♥❤ ①↕ ✤❛ tràF : R → 2R ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝
 R, ♥➳✉ x = 0,
F (x) =
 {0}, ♥➳✉ x = 0.

❑❤✐ ✤â ❞➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷đ❝ →♥❤ ①↕ ✤❛ trà F ❧➔ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ t↕✐ x0 = 0
ữ F ổ ỷ tử ữợ t x0 = 0✳

❱➼ ❞ö ✶✳✹✳✽✳ ❈❤♦ →♥❤ ①↕✤❛ trà F : R → 2R ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝
 {0}, ♥➳✉ x = 0,
F (x) =
 R, tr♦♥❣ tr÷í♥❣ ❤đ♣ ❝á♥ ❧↕✐.

❑❤✐ ✤â ❞➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷đ❝ →♥❤ ①↕ ✤❛ tr F ỷ tử ữợ t

x0 = 0 ♥❤÷♥❣ F ❦❤ỉ♥❣ ♥û❛ ❧✐➯♥ tư❝ tr➯♥ t↕✐ x0 = 0✳
▼➺♥❤ ✤➲ s❛✉ ✤÷❛ r❛ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ →♥❤ ①↕ ✤❛ trà ❧✐➯♥ tö❝ t❤❡♦
♥â♥✳

▼➺♥❤ ✤➲ ✶✳✹✳✾✳ ●✐↔ sû X ❧➔ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ✱ Y

❧➔ ❦❤æ♥❣ ❣✐❛♥ tổổ t

t ợ tự tỹ s õ ỗ C ✈➔ →♥❤ ①↕ ✤❛ trà F : X → 2Y ✈ỵ✐ F (x0 )
❧➔ t➟♣ ❝♦♠♣❛❝t tr♦♥❣ Y ✳ ❑❤✐ ✤â
✭✐✮ F ❧➔ C ✲ ❧✐➯♥ tö❝ tr➯♥ t↕✐ x0 ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✈ỵ✐ ♠å✐ t➟♣ ♠ð V ✱ F (x0 )

V + C tỗ t ❝➟♥ U ❝õ❛ x0 s❛♦ ❝❤♦ F (x) ⊆ V + C, ✈ỵ✐ ♠å✐
x ∈ U ∩ dom F.
✭✐✐✮ F C tử ữợ t x0 ❝❤➾ ♥➳✉ ✈ỵ✐ ♠é✐ y ∈ F (x0 ) ✈➔

V ừ y tỗ t U ❝õ❛ x0 s❛♦ ❝❤♦ F (x) ∩ (V + C) = ∅ ✈ỵ✐ ♠å✐

x ∈ U ∩ dom F ✳
✶✶

F C tử ữợ t x0 ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✈ỵ✐ ♠å✐ t➟♣ ♠ð G

t❤ä❛ ♠➣♥ F (x0 ) ∩ (G + C) = ∅✱ ❧✉æ♥ tỗ t U ừ x0 s

F (x) ∩ (G + C) = ∅ ✈ỵ✐ ♠å✐ x ∈ U ∩ dom F.
❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ●✐↔ sû F ❧➔ C ✲ ❧✐➯♥ tö❝ tr➯♥ t↕✐ x0 ✳ ▲➜② V ❧➔ t➟♣ ♠ð tr♦♥❣

Y s❛♦ ❝❤♦ F (x0 ) ⊆ V + C ✳ ❱➻ F (x0 ) ❝♦♠♣❛❝t ♥➯♥ tỗ t V0 ừ 0
s F (x0 ) + V0 ⊆ V + C. ❱➻ F ❧➔ C tử tr t x0 tỗ t ❧➙♥
❝➟♥ U ❝õ❛ x0 s❛♦ ❝❤♦

F (x) ⊆ F (x0 ) + V0 + C ✈ỵ✐ ♠å✐ x ∈ U ∩ dom F.
❚ø ✤â s✉② r❛

F (x) ⊆ V + C + C = V + C ✈ỵ✐ ♠å✐ x ∈ U ∩ dom F.
◆❣÷đ❝ ❧↕✐✱ ❧➜② W ❧➔ ❧➙♥ ❝➟♥ ♠ð ❜➜t ❦ý ❝õ❛ 0 tr♦♥❣ Y ✳ ✣➦t V = F (x0 ) + W ✳
❑❤✐ ✤â V ❧➔ t➟♣ ♠ð t❤ä❛ ♠➣♥ F (x0 ) ⊆ V + C tt tỗ t
U ❝õ❛ x0 tr♦♥❣ X s❛♦ ❝❤♦ F (x) ⊆ V + C ✈ỵ✐ ♠å✐ x ∈ U ∩ dom F. ❚ø
✤â s✉② r❛

F (x) ⊆ F (x0 ) + W + C ✈ỵ✐ ♠å✐ x ∈ U ∩ dom F.
✣✐➲✉ ♥➔② ❝❤ù♥❣ tä F ❧➔ C ✲ ❧✐➯♥ tö❝ tr➯♥ t↕✐ x0 .
✭✐✐✮ ●✐↔ sû F ❧➔ C ✲ ❧✐➯♥ tử ữợ t x0 y F (x0 ) tũ ỵ V
ừ y t ❦ý✳ ✣➦t W = y − V ✳ ❑❤✐ ✤â W ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ 0 tr♦♥❣ Y ✳ ❱➻

F C tử ữợ t x0 tỗ t↕✐ ❧➙♥ ❝➟♥ U ❝õ❛ x0 tr♦♥❣ X s❛♦ ❝❤♦
F (x0 ) ⊆ F (x) + W − C ✈ỵ✐ ♠å✐ x ∈ U ∩ dom F.
❱➻ y ∈ F (x0 ) ♥➯♥ y ∈ F (x) + W − C ✳ ❚❛ ❝â t❤➸ ✈✐➳t y = y ∗ + w − c✱ ð
✤➙② y ∗ ∈ F (x), w ∈ W ✈➔ c ∈ C ✳ ❚ø ✤â ❦➨♦ t❤❡♦ y ∗ = y − w + c ∈ V + C ✳
✣✐➲✉ ♥➔② ❝❤ù♥❣ tä y ∗ ∈ F (x) ∩ (V + C)✳ ❱➟② F (x) ∩ (V + C) = ∅ ✈ỵ✐ ♠å✐

x ∈ U ∩ dom F.
◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû V ❧➔ ❧➙♥ ❝➟♥ ♠ð ❝õ❛ 0 tr♦♥❣ Y ✳ ❚❛ ❝â

F (x0 ) ⊆

(y + V ).
y∈F (x0 )

✶✷


F (x0 ) t tỗ t y1 , y2 , ..., yn ∈ F (x0 ) s❛♦ ❝❤♦
n

F (x0 ) ⊆

(yi + V ).
i=1

❱ỵ✐ i ∈ {1, 2, ..., n}✱ ✈➻ yi + V ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ yi F (x0 ) tỗ t

Ui ❝õ❛ x0 s❛♦ ❝❤♦
F (x) ∩ (yi + V + C) = ∅ ✈ỵ✐ ♠å✐ x ∈ Ui ∩ dom F.
✣➦t U = ∩ni=1 Ui ✳ ❑❤✐ ✤â U ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ x0 ✈➔

F (x) ∩ (yi + V + C) = ∅ ✈ỵ✐ ♠å✐ x ∈ U ∩ dom F ✈➔ i ∈ {1, 2, ..., n}.
❚❛ ❝❤ù♥❣ ♠✐♥❤

F (x0 ) ⊆ F (x) + V − C ✈ỵ✐ ♠å✐ x ∈ U ∩ dom F.

❚❤➟t ✈➟②✱ ❧➜② y0 F (x0 ) tũ ỵ ứ õ s r
n

y0

(yi + V ).
i=1

tỗ t i0 {1, 2, ..., n} ✈➔ v ∈ V s❛♦ ❝❤♦ y0 = yi0 + v ✳ ▼➦t ❦❤→❝
❜ð✐ F (x) ∩ (yi0 + V + C) = ∅ ✈ỵ✐ ♠å✐ x U dom F tỗ t y ∈ F (x)
✈➔ v ∈ V, c ∈ C s❛♦ ❝❤♦ y = yi0 + v + c✳ ❚ø ✤â s✉② r❛ y0 = y + v − v − c ∈

F (x) + V − C ✳ ❱➟② F (x0 ) ⊆ F (x) + V − C ✈ỵ✐ ♠å✐ x ∈ U ∩ dom F ✳ ❈❤ù♥❣
tä F C tử ữợ t x0
sỷ F C tử ữợ t x0 ✳ ▲➜② G ❧➔ t➟♣ ♠ð ❜➜t ❦ý t❤ä❛ ♠➣♥

F (x0 ) ∩ (G + C) = ∅✳ ❚ø ✤â s r tỗ t y0 F (x0 ) s ❝❤♦ y0 = g + c✱
ð ✤➙② g ∈ G ✈➔ c ∈ C ✳ ❱➻ G ❧➔ ♠ð ♥➯♥ tỗ t V ừ 0 s

g + V ⊆ G✳ ❚ø ✤â s✉② r❛ g + c + V ⊆ G + C ❤❛② y0 + V ⊆ G + C ✳ ▼➦t
❦❤→❝ y0 + V ừ y0 t tỗ t ❧➙♥ ❝➟♥ U ❝õ❛ x0 s❛♦
❝❤♦ F (x) ∩ (y0 + V + C) = ∅ ✈ỵ✐ ♠å✐ x ∈ U ∩ dom F ✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦

F (x) ∩ (G + C) = ∅ ✈ỵ✐ ♠å✐ x ∈ U ∩ dom F.
✶✸


◆❣÷đ❝ ❧↕✐✱ ❧➜② y0 ∈ F (x0 ) ✈➔ V ❧➔ ❧➙♥ ❝➟♥ ♠ð ❝õ❛ y0 ✳ ❚ø ✤â s✉② r❛ y0 ∈

F (x0 ) ∩ (V + C)✳ ❚❤❡♦ tt tỗ t U ừ x0 s ❝❤♦ F (x) ∩

(V + C) = ∅ ✈ỵ✐ ♠å✐ x ∈ U ∩ dom F. ❚❤❡♦ ✭✐✐✮ t❛ s✉② r F C tử
ữợ t x0

①➨t ✶✳✹✳✶✵✳ ✭✐✮ ◆➳✉ C = {0} ✈➔ F (x0) ❧➔ t➟♣ ❝♦♠♣❛❝t t❤➻ ✣à♥❤ ♥❣❤➽❛
✶✳✹✳✻ ♣❤➛♥ ✭✐✮ ð tr➯♥ ỗ t ợ t ỷ tử tr
ỷ tử ữợ ừ r r trữớ ủ Y ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥
✈➔ F ✈ø❛ {0}✲ ❧✐➯♥ tử tr ứ {0} tử ữợ t x0 t❤➻ F ❧✐➯♥ tö❝
t↕✐ x0 t❤❡♦ ❦❤♦↔♥❣ ❝→❝❤ ❍❛✉s❞♦r❢❢✳
✭✐✐✮ ◆➳✉ F ❧➔ →♥❤ ①↕ ✤ì♥ trà t❤➻ tø ✤à♥❤ ♥❣❤➽❛ t❛ t❤➜② t➼♥❤ C ✲ ❧✐➯♥ tö❝
tr➯♥ ✈➔ C ✲ tử ữợ trũ õ t õ F ❧➔ C ✲ ❧✐➯♥ tư❝✳
✭✐✐✐✮ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ Y = R✱ C = R+ ✈➔ ♥➳✉ →♥❤ ①↕ ✤ì♥ trà F ❧➔ C ✲
❧✐➯♥ tö❝ t↕✐ x0 t❤➻ F ỷ tử ữợ t x0 t tổ tữớ ◆➳✉
❧➜② C = R− ✈➔ F ❧➔ C ✲ ❧✐➯♥ tö❝ t↕✐ x0 t❤➻ F ♥û❛ ❧✐➯♥ tö❝ tr➯♥ t↕✐ x0 ✳
✭✐✈✮ ❚ø ♠➺♥❤ ✤➲ tr➯♥ t❛ ❝â t❤➸ ♥â✐ r➡♥❣ ♠ët →♥❤ ①↕ ✤❛ trà F ❧➔ C ✲ ❧✐➯♥
tư❝ tr➯♥ t↕✐ x0 ♥➳✉ F (x) ❦❤ỉ♥❣ ❣✐➣♥ r❛ q✉→ s♦ ✈ỵ✐ F (x0 ) + C ❦❤✐ x x0
F C tử ữợ t x0 ♥➳✉ F (x) ❦❤æ♥❣ ❜à t❤✉ ❧↕✐ q✉→ ♥❤ä s♦ ✈ỵ✐

F (x0 ) + C ❦❤✐ x ❣➛♥ x0

ỗ t õ ừ tr

D t rộ ỗ tr ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤
❝❤✉➞♥ X ✱ C ⊆ Y ❧➔ t➟♣ õ ỗ õ ồ ợ tr rộ
①↕ ✤❛ trà Φ : D → 2Y ✤÷đ❝ ❣å✐ ❧➔
✭✐✮ C ✲ ❧ã♠ ♥➳✉ ✈ỵ✐ x1 , x2 ❜➜t ❦ý t❤✉ë❝ D ✈➔ t ∈ [0, 1] t❛ ❝â

Φ(tx1 + (1 − t)x2 ) ⊆ tΦ(x1 ) + (1 − t)Φ(x2 ) + C.
✭✐✐✮ C ✲ ❧ã♠ ♥❣❤✐➯♠ ♥❣➦t ♥➳✉ ✈ỵ✐ x1 , x2 ❜➜t ❦ý t❤✉ë❝ D ♠➔ x1 = x2 ✈➔

t ∈ [0, 1] t❛ ❝â

Φ(tx1 + (1 − t)x2 ) ⊆ tΦ(x1 ) + (1 − t)Φ(x2 ) + int C.
✶✹


C ỗ ợ x1 , x2 t ❦ý t❤✉ë❝ D ✈➔ t ∈ [0, 1] t❛ ❝â

tΦ(x1 ) + (1 − t)Φ(x2 ) ⊆ Φ(tx1 + (1 t)x2 ) + C.
C ố ữ ỗ ♥➳✉ ✈ỵ✐ x1 , x2 ❜➜t ❦ý t❤✉ë❝ D ✈➔ t [0, 1] tỗ t

x3 D s
t(x1 ) + (1 − t)Φ(x2 ) ⊆ Φ(x3 ) + C.
C tỹ ỗ tữớ ợ x1 , x2 ❜➜t ❦ý t❤✉ë❝ D ✈➔ t ∈ [0, 1]
t❛ ❝â
❤♦➦❝ Φ(x1 ) ⊆ Φ(tx1 + (1 − t)x2 ) + C,
❤♦➦❝ Φ(x2 ) ⊆ Φ(tx1 + (1 − t)x2 ) + C.
C tỹ ỗ tỹ ♥➳✉ ✈ỵ✐ x1 , x2 ❜➜t ❦ý t❤✉ë❝ D ✈➔ t [0, 1] tỗ
t [0, 1] s ❝❤♦

λΦ(x1 ) + (1 − λ)Φ(x2 ) ⊆ Φ(tx1 + (1 − t)x2 ) + C.

◆❤➟♥ ①➨t ✶✳✹✳✶✷✳ ◆➳✉ Φ C ỗ t C ố ữ ỗ
t C ỗ t C tỹ ỗ tỹ
C tỹ ỗ tữớ t C tỹ ỗ tỹ ợ
C tỹ ỗ tỹ rở ỡ ợ C ỗ C tỹ ỗ
tữớ

ử sỷ Y = R2 C = R2+ = {(x1, x2) ∈ R2 : x1 ≥ 0, x2 ≥ 0}✱
π
X = R ✈➔ A = [0, ]✳ ⑩♥❤ ①↕ ✤❛ trà Φ : A → 2Y ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐
2

Φ(x) = (sin x, 3 − sin x) + BY .
❑❤✐ ✤â t❛ t❤➜② Φ ❦❤æ♥❣ ❧➔ C ỗ C tỹ ỗ tữớ C
tỹ ỗ tỹ




ữỡ
ỹ tỗ t t tổ ừ t
ố ợ t tỹ
tỡ s rở
r ữỡ ú tổ tr ởt số ỵ sỹ tỗ t
ừ t tỹ tỡ s✉② rë♥❣ ✈➔ t➼♥❤ ❧✐➯♥ t❤ỉ♥❣ ❝õ❛ t➟♣ ♥❣❤✐➺♠
✤è✐ ✈ỵ✐ ❜➔✐ t♦→♥ ♥➔②✳ ❈→❝ ❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ ❝❤ó♥❣ tỉ✐ tr➼❝❤
❞➝♥ tø ❝ỉ♥❣ tr➻♥❤ ❬✻❪✳

✷✳✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ♠ð ✤➛✉
❚r♦♥❣ ♣❤➛♥ ♥➔②✱ ♥➳✉ ❦❤ỉ♥❣ ❝â tr÷í♥❣ ❤ñ♣ ✤➦❝ ❜✐➺t✱ t❛ ❦➼ ❤✐➺✉ Λ✱ W ✱ ∆✱

X Y ổ ỗ ữỡ sr sỷ C Y t
õ ỗ õ ♥❤å♥ ✈ỵ✐ ♣❤➛♥ tr♦♥❣ ❦❤→❝ ré♥❣✱ R+ := {x ∈ R : x ≥ 0}
✈➔ P ⊆ ∆ ❧➔ ♠ët ỗ õ ồ ồ BY õ tr♦♥❣ Y ✱ Y ∗ ❧➔
❦❤æ♥❣ ❣✐❛♥ tæ ♣æ ✤è✐ ♥❣➝✉ ❝õ❛ Y ✈➔ C ∗ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐

C ∗ := {f ∈ Y ∗ : f (c) ≥ 0, ✈ỵ✐ ♠å✐ c ∈ C}.
❑➼ ❤✐➺✉ tü❛ ♣❤➛♥ tr♦♥❣ ❝õ❛ C ∗ ❧➔ C # ✱ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐

C # := {f ∈ Y ∗ : f (c) > 0, ✈ỵ✐ ♠å✐ c ∈ C\{0}}.
✶✻

●å✐ D ❧➔ t➟♣ ❦❤→❝ ré♥❣ ❝õ❛ Y ✱ ❦❤✐ ✤â ❜❛♦ ♥â♥ ❝õ❛ D ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐

cone D := {td : t ≥ 0, d ∈ D}.
▼ët t➟♣ ỗ rộ B ừ õ ỗ C ữủ ❣å✐ ❧➔ ❝ì sð ❝õ❛ C ♥➳✉

C = cone B ✈➔ 0 ∈ cl B ✳ ❍✐➸♥ ♥❤✐➯♥✱ C # = ∅ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ C ❝â ❝ì sð✳
❱ỵ✐ e ∈ int C ✱ t❛ ❦➼ ❤✐➺✉

B ∗ := {f ∈ C ∗ : f (e) = 1},
✈➔

B # := {f ∈ C # : f (e) = 1}.
❍✐➸♥ ♥❤✐➯♥✱ B ∗ ❧➔ ❝ì sð ❝♦♠♣❛❝t ②➳✉ ✯ ❝õ❛ C ∗ ✱ B # ❧➔ ❝ì sð ❝õ❛ C # ✈➔ B ∗
❂ cl B # ✳

◆❤➟♥ ①➨t ✷✳✶✳✶✳ ❈❤♦ f ∈ C #✳ ◆➳✉ f (z) = 0 t❤➻ z ∈ −C\{0}✳
●✐↔ sû K ❧➔ ♠ët t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ X ✱ S : X → 2∆ ✈➔ F : X ×∆×X →

2Y ❧➔ ❝→❝ →♥❤ ①↕ ✤❛ trà✳ ❚❛ ①➨t ❜➔✐ t♦→♥ (GV QEP )✿ ❚➻♠ x0 ∈ K s❛♦ ❝❤♦
F (x0 , u, y) ∩ (−Ω) = ∅ ✈ỵ✐ ♠å✐ u ∈ S(x0 ) ✈➔ y ∈ K,
tr♦♥❣ ✤â Ω ∪ {0} ❧➔ t➟♣ ♥â♥ tr♦♥❣ Y ✳
◆➳✉ t❤❛② F (x, u, y) ❜ð✐ F (x, y) t❤➻ ❜➔✐ t♦→♥ (GV QEP ) trð t❤➔♥❤ ❜➔✐
t♦→♥✿ ❚➻♠ x0 ∈ K s❛♦ ❝❤♦

F (x0 , y) ∩ (−Ω) = ∅ ✈ỵ✐ ♠å✐ y ∈ K.
❇➔✐ t♦→♥ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ✈➨❝tì s✉② rë♥❣ ✭GV EP ✮✳ ❍ì♥
♥ú❛✱ ♥➳✉ t❤❛② F (x, y) ❜ð✐ f (y) − f (x)✱ tr♦♥❣ ✤â f : X → Y ❧➔ →♥❤ ①↕✱ t❤➻
❜➔✐ t♦→♥ (GV EP ) trð t❤➔♥❤ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈➨❝tì ✭V OP ✮✿ ❚➻♠ x0 ∈ K s❛♦
❝❤♦

f (y) − f (x) ∈
/ −Ω ✈ỵ✐ ♠å✐ y ∈ K.
❚❛ ❦➼ ❤✐➺✉ G(F, S, K) ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ♠↕♥❤ ❝õ❛ ❜➔✐ t♦→♥

(GV QEP )✱ tù❝ ❧➔
G(F, S, K) = {x ∈ K : F (x, u, y) ⊆ C, ✈ỵ✐ ♠å✐ u ∈ S(x) ✈➔ y ∈ K},
✶✼


W (F, S, K) ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ❝õ❛ ❜➔✐ t♦→♥ (GV QEP )✱
tù❝ ❧➔

W (F, S, K) = {x ∈ K : F (x, u, y)∩(− int C) = ∅, ✈ỵ✐ ♠å✐ u ∈ S(x) ✈➔ y ∈ K},
✈➔ E(F, S, K) ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❝õ❛ ❜➔✐ t♦→♥ (GV QEP )✱
tù❝ ❧➔

E(F, S, K) = {x ∈ K : F (x, u, y)∩(−C\{0}) = ∅, ✈ỵ✐ ♠å✐ u ∈ S(x) ✈➔ y ∈ K}.
❑➼ ❤✐➺✉ WV (f, K) ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ❝õ❛ ❜➔✐ t♦→♥

(V OP )✱ tù❝ ❧➔
WV (f, K) = {x ∈ K : f (y) − f (x) ∈
/ − int C, ✈ỵ✐ ♠å✐ y ∈ K},
✈➔ EV (f, K) ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❝õ❛ ❜➔✐ t♦→♥ (V OP )✱ tù❝ ❧➔

EV (f, K) = {x ∈ K : f (y) − f (x) ∈
/ −C\{0}, ✈ỵ✐ ♠å✐ y ∈ K}.
❱ỵ✐ f ∈ C ∗ ✱ ❦➼ ❤✐➺✉ Q(f ) ❧➔ t➟♣ t➜t ❝↔ ❝→❝ f ✲ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ (GV QEP )✱
tù❝ ❧➔

Q(f ) = {x ∈ K : f (F (x, u, y)) ⊆ R+ , ✈ỵ✐ ♠å✐ u ∈ S(x) ✈➔ y ∈ K}.

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✷✳ ❈❤♦ K ❧➔ t➟♣ ❦❤→❝ ré♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤
X ✳ ⑩♥❤ ①↕ ✤❛ trà F : K → 2X ✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ KKM ♥➳✉ ✈ỵ✐ ♠é✐ t➟♣
❤ú✉ ❤↕♥ {y1 , y2 , ..., yn } tr♦♥❣ K t❛ ❝â
n

conv({y1 , y2 , ..., yn }) ⊆

F (yi ).
i=1

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✸✳ ⑩♥❤ ①↕ ✤❛ trà Φ : ∆ → 2Y

✤÷đ❝ ❣å✐ ❧➔ P ✲C ✲ t➠♥❣ ♥➳✉

✈ỵ✐ u1 , u2 ∈ ∆ ♠➔ u1 − u2 ∈ P t❛ ❝â

Φ(u1 ) ⊆ Φ(u2 ) + C.

◆❤➟♥ ①➨t ✷✳✶✳✹✳ ❚r÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t✱ ❤➔♠ f : R R ữủ ồ R+R+
t ợ u1 , u2 ∈ R ♠➔ u1 ≥ u2 t❛ ❝â f (u1 ) ≥ f (u2 )✳
✶✽