✣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➼❝❤ ▼➣ sè✿ ữớ ữợ ❦❤♦❛ ❤å❝ ❚❙✳ ❇Ò■ ❚❍➌ ❍Ò◆● ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✾ ▲í✐ ❝❛♠ ✤♦❛♥ ❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ổ trũ ợ t ỗ t sỷ t ỗ t➔✐ ❧✐➺✉ ♠ð✳ ❈→❝ t❤æ♥❣ t✐♥✱ t➔✐ ❧✐➺✉ tr♦♥❣ ❧✉➟♥ ữủ ró ỗ ố t❤→♥❣ ✹ ♥➠♠ ✷✵✶✾ ◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥ ◆❣✉②➵♥ ▼✐♥❤ ❍✐➲♥ ❳→❝ ♥❤➟♥ ❝õ❛ ❦❤♦❛ ❝❤✉②➯♥ ♠æ♥ ❳→❝ ♥❤➟♥ ❝õ❛ ữớ ữợ ũ ũ ỡ rữợ tr ✈➠♥✱ 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♥ ❇❛♥ ●✐→♠ ❤✐➺✉✱ ❦❤♦❛ ❚♦→♥ ❝ò♥❣ 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❤✉➟♥ ❧ñ✐ ✈➔ ❝❤♦ tổ ỳ ỵ õ õ qỵ tr sốt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❈✉è✐ ❝ò♥❣✱ 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♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ỳ t tt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỗ ✈➔ ♠ët sè 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 →♥❤ ①↕ ✤❛ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ❈❤÷ì♥❣ ỹ tỗ t t tổ t ♥❣❤✐➺♠ ✤è✐ ✈ỵ✐ ❜➔✐ t♦→♥ tü❛ ❝➙♥ ❜➡♥❣ ✈➨❝tì s✉② rë♥❣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ♠ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ỹ tỗ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✸✳ ❚➼♥❤ ❧✐➯♥ t❤æ♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ❝→❝ ❝❤ú ✈✐➳t t➢t R t➟♣ ❝→❝ sè t❤ü❝ R+ t➟♣ sè t❤ü❝ ❦❤æ♥❣ ➙♠ R− t➟♣ sè t❤ü❝ ❦❤æ♥❣ ❞÷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 ♠✐➲♥ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ →♥❤ ①↕ ✤❛ trà F gph F ỗ t tr F ✐✈ A∩B ❣✐❛♦ ❝õ❛ ❤❛✐ t➟♣ ❤ñ♣ A ✈➔ B A\B ❤✐➺✉ ❝õ❛ ❤❛✐ t➟♣ ❤ñ♣ A ✈➔ B B t➼❝❤ ❉❡s❝❛rt❡s ❝õ❛ ❤❛✐ t➟♣ ❤đ♣ A ✈➔ B cl A ❜❛♦ ✤â♥❣ tỉ♣ỉ ❝õ❛ t➟♣ ❤đ♣ A co A ỗ t ủ A int A tr tổổ t ủ A conv A ỗ ❝õ❛ t➟♣ ❤đ♣ A ✷ ❦➳t t❤ó❝ ❝❤ù♥❣ ♠✐♥❤ ✈ ▼ð ✤➛✉ ❇➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ✈➨❝tì ❝â ♥❤✐➲✉ ù♥❣ q trồ tr t ỵ t ỵ tt sỹ tỗ t t ố ợ t tỡ ữủ rt t ự ữợ tt t tỹ ỗ tỹ ỗ t õ ❬✸❪✱ ❬✹❪✮✳ ◆➠♠ ✷✵✶✺✱ ❍❛♥ ✈➔ ❍✉❛♥❣ ❬✹❪ ♥❣❤✐➯♥ ❝ù✉ sỹ tỗ t ố ợ t ❤ú✉ ❤✐➺✉ ②➳✉ ✈➔ t➟♣ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❝õ❛ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ✈➨❝tì s✉② rë♥❣ ✈ỵ✐ ❣✐↔ t❤✐➳t ❤➔♠ t ỗ t õ t ✤➣ ♠ð rë♥❣ ❦➳t q✉↔ tr➯♥ ❝❤♦ ❧ỵ♣ ❜➔✐ t♦→♥ tü❛ ❝➙♥ ❜➡♥❣ ✈➨❝tì s✉② rë♥❣ ❬✻❪✳ ◆❣♦➔✐ ✈✐➺❝ ♥❣❤✐➯♥ ự sỹ tỗ t 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 ❦❤æ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ♥➔② s❛♥❣ ❦❤æ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ❦❤→❝ ❬✹❪✳ ❙❛✉ ✤â✱ ❝→❝ ❜➔✐ t♦→♥ ♥➔② ữủ rở ợ ổ õ số ổ ❤↕♥ ❬✽❪✳ ◆➠♠ ✷✵✶✻✱ ❍❛♥ ✈➔ ❍✉❛♥❣ ❬✻❪ ♥❣❤✐➯♥ ❝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♦→♥ tỹ tỡ s rở ỗ ♠ð ✤➛✉✱ ❤❛✐ ❝❤÷ì♥❣ ♥ë✐ ❞✉♥❣✱ ♣❤➛♥ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ✶ ❈❤÷ì♥❣ ✶ ❝õ❛ ❧✉➟♥ ✈➠♥ tr ởt số tự t ỗ ổ ỗ tr ởt số t ❝❤➜t ❝õ❛ →♥❤ ①↕ ✤❛ trà✳ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ởt số sỹ tỗ t ❤ú✉ ❤✐➺✉ ♠↕♥❤✱ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✈➔ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❝õ❛ ❜➔✐ t♦→♥ tü❛ ❝➙♥ ❜➡♥❣ ✈➨❝tì s✉② rë♥❣✳ ❍ì♥ ♥ú❛ t➼♥❤ ❧✐➯♥ t❤ỉ♥❣ ❝õ❛ t➟♣ ♥❣❤✐➺♠ ✤è✐ ✈ỵ✐ ❜➔✐ t♦→♥ tü❛ ❝➙♥ ❜➡♥❣ ✈➨❝tì s✉② rë♥❣ ❝ơ♥❣ ✤÷đ❝ 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♦→♥ ữ ỵ tt ữỡ tr ữỡ tr ✤↕♦ ❤➔♠ r✐➯♥❣✱ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔ ♣❤÷ì♥❣ tr s rở ỵ tt tố ữ ỵ tt tố ữ t 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✉↔ ❝❤➼♥❤ ❝õ❛ ữỡ ỗ ởt sè t➼♥❤ ❝❤➜t ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ●✐↔ sû X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤✳ ❚➟♣ A ⊆ X ✤÷đ❝ ❣å✐ ỗ ợ x1 , x2 A t❛ ❧✉æ♥ ❝â λx1 + (1 − λ)x2 ∈ A ợ [0, 1] ữợ rộ t ỗ sỷ A X t ỗ ợ I ✱ ✈ỵ✐ I ❧➔ t➟♣ ❝❤➾ sè ❜➜t ❦➻✳ ❑❤✐ õ t A = A ỗ I F : X ì ì X 2Y ữủ ①→❝ ✤à♥❤ ♥❤÷ s❛✉ F (x, u, y) = (f1 (x, u, y), f2 (x, u, y)) + BY , ợ (x, u, y) X ì ì X, tr♦♥❣ ✤â f1 (x, u, y) = sin x + u2 − 2u − sin y + , 12 f2 (x, u, y) = x + u2 − 2u − sin y + ❚❛ t❤➜② ỵ ữủ tọ ❍ì♥ ♥ú❛✱ ❜➡♥❣ ❦✐➸♠ π tr❛ trü❝ t✐➳♣ t❛ ❝â ∈ G(F, S, K)✳ ❱➟② G(F, S, K) = ∅✳ sỷ K t rộ ỗ t tr ổ ỵ ỗ ữỡ ❍❛✉s❞♦r❢❢ X ✳ ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥ ✭✐✮ ❱ỵ✐ ♠é✐ x ∈ K ✈➔ u ∈ S(x)✱ F (x, u, x) ∩ (− int C) = ∅❀ ✭✐✐✮ ❱ỵ✐ ♠é✐ y ∈ K ✱ {x ∈ K : F (x, u, y) ∩ (− int C) = ∅ ✈ỵ✐ ♠å✐ u ∈ S(x)} ❧➔ t➟♣ ✤â♥❣❀ ✭✐✐✐✮ ❱ỵ✐ ♠é✐ x ∈ K ✱ S(x) ❧➔ t ỗ F (x, ã, ã) C tỹ ỗ tỹ tr S(x) ì K ✤â W (F, S, K) = ∅✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠é✐ y ∈ K ✱ ✤➦t Q(y) = {x ∈ K : F (x, u, y) ∩ (− int C) = ∅ ✈ỵ✐ ♠å✐ u ∈ S(x)} ❍✐➸♥ ♥❤✐➯♥ y ∈ Q(y)✳ ❚ø ❣✐↔ t❤✐➳t t❛ t❤➜② Q(y) ❧➔ t➟♣ ❝♦♥ ✤â♥❣✱ ❦❤→❝ ré♥❣ ❝õ❛ K ✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ Q : K → 2K ❧➔ →♥❤ ①↕ KKM ✳ t sỷ ữủ tỗ t t {y1 , , yn } ⊆ K ✈➔ y0 ∈ conv({y1 , , yn }) s❛♦ ❝❤♦ y0 ∈ / Q(yi ), i {1, 2, , n} õ tỗ t↕✐ ui ∈ S(y0 ) (i ∈ {1, 2, , n}) s❛♦ ❝❤♦ F (y0 , ui , yi ) ∩ (− int C) = ∅, i ∈ {1, 2, , n} ứ õ t t tỗ t zi F (y0 , ui , yi ), i ∈ {1, 2, , n} ✷✺ ✭✷✳✺✮ s❛♦ ❝❤♦ ✭✷✳✻✮ zi ∈ − int C, i ∈ {1, 2, , n} n y0 conv({y1 , , yn }) tỗ t↕✐ λi ≥ (i ∈ {1, 2, , n}) ✈ỵ✐ n s❛♦ ❝❤♦ y0 = i=1 n λi yi S(y0 ) t ỗ õ i=1 λi = λi ui ∈ S(y0 )✳ i=1 ❚ø ❣✐↔ t❤✐➳t F (y0 , ·, ·) ❧➔ C ✲ tỹ ỗ tỹ tr S(y0 ) ì K tỗ t ti n (i {1, 2, , n}) ✈ỵ✐ ti = s❛♦ ❝❤♦ i=1 n n n ti F (y0 , ui , yi ) ⊆ F (y0 , t=1 n λi ui , i=1 λi yi ) + C = F (y0 , i=1 λi ui , y0 ) + C i=1 ✭✷✳✼✮ ❚ø t õ tỗ t n z0 F (y0 , λi ui , y0 ), i=1 ✈➔ c0 ∈ C s❛♦ ❝❤♦ n ti zi = z0 + c0 ✭✷✳✽✮ i=1 ❚ø ✭✷✳✻✮ ✈➔ ✭✷✳✽✮ t❛ ❝â z0 ∈ − int C ✳ ✣✐➲✉ ♥➔② ♠➙✉ t ợ tt (i) ỵ Q(y) = ∅✱ ❞♦ ✈➟② W (F, S, K) = ∅✳ y∈K ◆❤➟♥ ①➨t ✷✳✷✳✺✳ ●✐↔ sû S(·) ❧➔ ♥û❛ ❧✐➯♥ tử ữợ tr K ợ ộ y K, F (ã, ã, y) ỷ tử ữợ tr K × S(K)✳ ❑❤✐ ✤â {x ∈ K : F (x, u, y) ∩ (− int C) = ∅ ✈ỵ✐ ♠å✐ u ∈ S(x)} ❧➔ t➟♣ ✤â♥❣✳ ❍➺ q✉↔ ✷✳✷✳✻✳ sỷ K t rộ ỗ t tr ổ ỗ ữỡ sr X ①↕ ✤❛ trà G : X × ∆ × X → 2R ✳ ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ①↔② r❛ ✭✐✮ ❱ỵ✐ ♠é✐ x ∈ K ✈➔ u ∈ S(K)✱ G(x, u, x) ⊆ R+ ❀ ✷✻ ✭✐✐✮ ❱ỵ✐ ♠é✐ y ∈ K, {x ∈ K : G(x, u, y) ⊆ R+ ✈ỵ✐ ♠å✐ u ∈ S(x)} ❧➔ t➟♣ õ ợ ộ x K S(x) ỗ G(x, ã, ã) R+ tỹ ỗ tỹ tr S(x) ì K õ tỗ t x0 ∈ K s❛♦ ❝❤♦ G(x0 , u, y) ⊆ R+ ✈ỵ✐ ♠å✐ u ∈ S(x0 ) ✈➔ y ∈ K ỵ sỷ K t rộ ỗ t tr ổ ỗ ữỡ ❍❛✉s❞♦r❢❢ X ✱ ✈➔ C # = ∅✳ ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ①↔② r❛ ✭✐✮ ❱ỵ✐ ♠é✐ x ∈ K ✈➔ u ∈ S(x)✱ F (x, u, x) ⊆ C ❀ ✭✐✐✮ ❱ỵ✐ ♠é✐ y ∈ K ✱ {x ∈ K : F (x, u, y) ⊆ C ✈ỵ✐ ♠å✐ u ∈ S(x)} ❧➔ t➟♣ ✤â♥❣❀ ✭✐✐✐✮ ❱ỵ✐ ộ x K S(x) t ỗ F (x, ã, ã) C tỹ ỗ tỹ ♥❤✐➯♥ tr➯♥ S(x) × K ✳ ❑❤✐ ✤â E(F, S, K) = ∅ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ ϕ ∈ C # ✳ ⑩♥❤ ①↕ ✤❛ trà H : X × ∆ × X → 2R ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉ H(x, u, y) = ϕ(F (x, u, y)) = {ϕ(z)} ✈ỵ✐ ♠å✐ (x, u, y) ∈ X×∆×X z∈F (x,u,y) ❉➵ t❤➜② H t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❍➺ q✉↔ ✷✳✷✳✻ tỗ t x0 K s H(x0 , u, y) ⊆ R+ ✈ỵ✐ ♠å✐ u ∈ S(x0 ) ✈➔ y ∈ K ❚ø ◆❤➟♥ ①➨t ✷✳✶✳✶ t❛ t❤➜② F (x0 , u, y) ∩ (−C \ {0}) = ∅ ✈ỵ✐ ♠å✐ u ∈ S(x0 ) ✈➔ y ∈ K ❚ø ✤â s✉② r❛ x0 ∈ E(F, S, K) E(F, S, K) = ữợ ỵ ỵ ❱➼ ❞ö ✷✳✷✳✽✳ ●✐↔ sû Y = R2 ✱ C = R2+ = {(x1 , x2 ) ∈ R2 : x1 ≥ 0, x2 ≥ 0}✱ X = R ✈➔ K = [0, π]✳ ⑩♥❤ ①↕ S : X → 2Y ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ S(x) = {u ∈ R : cos x − ≤ u ≤ sin x + 1}, x ∈ X ✷✼ ✈➔ F : X × ∆ × X → Y ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉ F (x, u, y) = (f1 (x, u, y), f2 (x, u, y)) + BY ✈ỵ✐ ♠å✐ (x, u, y) ∈ X × ∆ × X, tr♦♥❣ ✤â✱ f1 (x, u, y) = sin x + u2 − sin y + , 2 2 f2 (x, u, y) = −2x + u + u + 2y + y + ❉➵ t❤➜② ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ỵ ữủ tọ ỡ ỳ ❦✐➸♠ tr❛ trü❝ t✐➳♣ t❛ ❝â t❤❡♦ W (F, S, K) = ∅ π ∈ E(F, S, K)✳ ❱➟② E(F, S, K) = ∅✱ ❦➨♦ ✷✳✸✳ ❚➼♥❤ ❧✐➯♥ t❤æ♥❣ ❚ø ❍➺ q✉↔ ✷✳✷✳✻ t❛ s✉② r❛ ❜ê ✤➲ s❛✉✳ ❇ê ✤➲ ✷✳✸✳✶✳ ●✐↔ sû K ❧➔ t➟♣ ❦❤→❝ ré♥❣✱ ỗ t tr ổ ỗ ữỡ sr X ✳ ●✐↔ sû f ∈ C ∗ \ {0Y ∗ } ✈➔ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ①↔② r❛ ✭✐✮ ❱ỵ✐ ♠é✐ x ∈ K ✈➔ u ∈ S(x)✱ F (x, u, x) ⊆ C ❀ ✭✐✐✮ ❱ỵ✐ ♠é✐ y ∈ K ✱ {x ∈ K : F (x, u, y) ⊆ C ✈ỵ✐ ♠å✐ u ∈ S(x)} ❧➔ t➟♣ ✤â♥❣❀ ✭✐✐✐✮ ❱ỵ✐ ♠é✐ x ∈ K ✱ S(x) ❧➔ t ỗ F (x, ã, ã) C tỹ ỗ tỹ tr S(x) ì K ✤â Q(f ) = ∅ ❇ê ✤➲ ✷✳✸✳✷✳ ●✐↔ sû K t rộ ỗ tr ổ ỗ ✤à❛ ♣❤÷ì♥❣ ❍❛✉s❞♦r❢❢ X ✈➔ f ∈ C ∗ \ {0Y ∗ }✳ ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ①↔② r❛ ✭✐✮ S(·) ❧➔ P ✲ ❧ã♠ tr➯♥ K ❀ ợ ộ (x, y) K ì K F (x, ·, y) ❧➔ P ✲C ✲ t➠♥❣❀ ✭✐✐✐✮ ❱ỵ✐ ♠é✐ y ∈ K ✱ F (·, ·, y) ❧➔ C ✲ ❧ã♠ tr➯♥ K × S(K)✳ ❑❤✐ ✤â Q(f ) t ỗ ỡ ỳ G(F, S, K) ụ t ỗ ự x1 , x2 ∈ Q(f ) ✈➔ t ∈ [0, 1]✳ ❑❤✐ ✤â f (F (xi , u, y)) ⊆ R+ ✈ỵ✐ ♠å✐ u ∈ S(xi ) ✈➔ y ∈ K, i ∈ {1, 2} ✭✷✳✾✮ ❱➻ S(·) ❧➔ P ✲ ❧ã♠ tr➯♥ K ♥➯♥ S(tx1 + (1 − t)x2 ) ⊆ tS(x1 ) + (1 − t)S(x2 ) + P ❱ỵ✐ ộ ut S(tx1 + (1 t)x2 ) tỗ t↕✐ u1 ∈ S(x1 ), u2 ∈ S(x2 ) ✈➔ p0 ∈ P s❛♦ ❝❤♦ ut = tu1 + (1 − t)u2 + p0 ✳ ❉♦ F (tx1 + (1 − t)x − 2, ·, y) ❧➔ P ✲C ✲ t➠♥❣ ♥➯♥ t❛ ❝â F (tx1 +(1−t)x2 , tu1 +(1−t)u2 , y) ⊆ F (tx1 +(1−t)x2 , tu1 +(1−t)u2 , y)+C ✭✷✳✶✵✮ ❍ì♥ ♥ú❛✱ ❞♦ F (·, ·, y) ❧➔ C ✲ ❧ã♠ tr➯♥ K × S(K) ♥➯♥ F (tx1 +(1−t)x2 , tu1 +(1−t)u2 , y) ⊆ tF (x1 , u1 , y)+(1−t)F (x2 , u2 , y)+C ✭✷✳✶✶✮ ❑➳t ❤ñ♣ ✭✷✳✾✮✱ ✭✷✳✶✵✮✱ ✭✷✳✶✶✮ ✈➔ f ∈ C ∗ t❛ ❝â f (F (tx1 + (1 − t)x2 , ut , y)) ⊆ R+ ✈ỵ✐ ♠å✐ ut ∈ S(tx1 + (1 − t)x2 ) ✈➔ y ∈ K ❑➨♦ t❤❡♦ tx1 + (1 − t)x2 ∈ Q(f )✳ ❱➟② G(F, S, K) t ỗ t G(F, S, K) t ỗ t G(F, S, K) t➟♣ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣✳ ❇ê ✤➲ ✷✳✸✳✹✳ ●✐↔ sû K t rộ tr ổ ỗ ữỡ sr X S(ã) ỷ tử ữợ tr K ✈➔ ✈ỵ✐ ♠é✐ y ∈ K ✱ F (·, ã, y) ỷ tử ữợ tr K ì S(K)✳ ❑❤✐ ✤â ✭✐✮ ◆➳✉ K ❧➔ t➟♣ ✤â♥❣ t❤➻ Q(f ) ❧➔ t➟♣ ✤â♥❣❀ ✭✐✐✮ ◆➳✉ K ❧➔ t➟♣ ❝♦♠♣❛❝t t❤➻ Q(·) ❧➔ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ C ∗ \ {0Y ∗ }✱ tr♦♥❣ ✤â tæ♣æ tr➯♥ C ∗ \ {0Y ∗ } ❧➔ tæ♣æ ②➳✉ ✯✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ▲➜② ❞➣② {xn } ⊆ Q(f ) ✈ỵ✐ xn → x0 ✳ ❑❤✐ ✤â t❛ ❝â f (F (xn , u, y)) ⊆ R+ ✈ỵ✐ ♠å✐ u ∈ S(xn ) ✈➔ y ∈ K ✷✾ ✭✷✳✶✷✮ ❚❤❡♦ ❣✐↔ t❤✐➳t K ❧➔ t➟♣ ✤â♥❣ ♥➯♥ x0 ∈ K ✳ ❱ỵ✐ u S(x0 ) tũ ỵ S(ã) ỷ tử ữợ t x0 t tỗ t un S(xn ) s un → u✳ ❱ỵ✐ ♠é✐ z ∈ F (x0 , u, y)✱ ❞♦ F (·, ·, y) ❧➔ ♥û❛ ❧✐➯♥ tö❝ ữợ t (x0 , u) t tỗ t zn F (xn , un , y) s❛♦ ❝❤♦ zn → z ✳ ❚ø ✭✷✳✶✷✮ t❛ ❝â f (zn ) ≥ 0✳ ❱➻ f ❧✐➯♥ tö❝ ♥➯♥ f (zn ) → f (z)✳ ❚ø ✤â s✉② r❛ f (z) ≥ 0✳ ❉♦ ✈➟② f (F (x0 , u, y)) ⊆ R+ ✈ỵ✐ ♠å✐ u ∈ S(x0 ) ✈➔ y ∈ K ❙✉② r❛ x0 ∈ Q(f )✳ ❱➟② Q(f ) ❧➔ t➟♣ ✤â♥❣✳ ✭✐✐✮ ●✐↔ sû ♥❣÷đ❝ ❧↕✐ Q(·) ❦❤ỉ♥❣ ❧➔ ♥û❛ ❧✐➯♥ tư❝ tr➯♥ t↕✐ f0 C \ {0Y } õ tỗ t↕✐ ❧➙♥ ❝➟♥ W0 ❝õ❛ Q(f0 ) ✈➔ ❞➣② {fn } ✈ỵ✐ fn ❤ë✐ tư ②➳✉ ✈➲ f0 s❛♦ ❝❤♦ Q(fn ) W0 ✳ ❚ø ✤â s✉② r❛ xn ∈ Q(fn ), n = 1, 2, ✭✷✳✶✸✮ xn ∈ / W0 ✈ỵ✐ ♠å✐ n ∈ N ✭✷✳✶✹✮ s❛♦ ❝❤♦ ❉➵ t❤➜② r➡♥❣ xn ∈ K ✳ ❉♦ K ❧➔ t➟♣ ❝♦♠♣❛❝t✱ ❦❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ ❣✐↔ sû xn → x0 ∈ K ✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ x0 ∈ Q(f0 ) t sỷ ữủ tỗ t u0 ∈ S(x0 ) ✈➔ y0 ∈ K s❛♦ ❝❤♦ f0 (F (x0 , u0 , y0 )) R+ õ tỗ t z0 F (x0 , u0 , y0 ) s❛♦ ❝❤♦ f0 (z0 ) < S(ã) ỷ tử ữợ t x0 t tỗ t un S(xn ) s❛♦ ❝❤♦ un → u0 ✳ ▼➦t ❦❤→❝ F (ã, ã, y0 ) ỷ tử ữợ t (x0 , u0 ) ♥➯♥ ❝ô♥❣ t❤❡♦ ❇ê ✤➲ ✷✳✶✳✾✱ tỗ t zn F (xn , un , y0 ) s❛♦ ❝❤♦ zn → z0 ✳ ❙✉② r❛ fn ❤ë✐ tö ②➳✉ ✈➲ f0 ✳ ❚ø ✤â s✉② r❛ fn (zn ) → f0 (z0 )✳ ✣✐➲✉ ♥➔② ❦➳t ❤đ♣ ✈ỵ✐ ✭✷✳✶✺✮ t❛ ❝â fn (zn ) < ✈ỵ✐ n ✤õ ❧ỵ♥✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✭✷✳✶✸✮✳ ❉♦ ✤â x0 ∈ Q(f0 )✳ ❙✉② r❛ xn → x0 ∈ W0 ✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✭✷✳✶✹✮✳ ❱➟② Q(·) ❧➔ ♥û❛ ❧✐➯♥ tö❝ tr➯♥✳ ✸✵ ❇ê ✤➲ sỷ K t rộ ỗ t tr ổ ỗ ữỡ sr X ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ①↔② r❛ ✭✐✮ S(·) ỷ tử ữợ P ó tr K ✈ỵ✐ ❣✐→ trà ❦❤→❝ ré♥❣✱ ❝♦♠♣❛❝t❀ ✭✐✐✮ ❱ỵ✐ ♠é✐ (x, y) ∈ K × K, F (x, ·, y) ❧➔ P ✲C ✲ t➠♥❣❀ ✭✐✐✐✮ ❱ỵ✐ ♠é✐ y ∈ K, F (·, ·, y) ❧➔ C ✲ ❧ã♠ ♥❣❤✐➯♠ ♥❣➦t tr➯♥ K × S(K)❀ ✭✐✈✮ F (·, ·, ·) tử tr K ì S(K) ì K ợ trà ❦❤→❝ ré♥❣✱ ❝♦♠♣❛❝t✳ ❑❤✐ ✤â Q(·) ❧➔ ❧✐➯♥ tö❝ ữợ tr C \ {0Y } tr õ tæ♣æ tr➯♥ C ∗ \ {0Y ∗ } ❧➔ tæ♣æ ②➳✉ ✯✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ♥❣÷đ❝ ❧↕✐ Q(·) ❦❤ỉ♥❣ ỷ tử ữợ t f0 C \ {0Y } õ tỗ t x0 Q(f0 )✱ ♠ët ❧➙♥ ❝➟♥ W0 ❝õ❛ ∈ X ✈➔ ❞➣② {fn } ✈ỵ✐ fn ❤ë✐ tư ②➳✉ ✈➲ f0 s❛♦ ❝❤♦ (x0 + W0 ) ∩ Q(fn ) = ∅ ✈ỵ✐ ♠å✐ n ∈ N ✭✷✳✶✻✮ ❚❛ ①➨t ✷ tr÷í♥❣ ❤đ♣ ❚r÷í♥❣ ❤đ♣ ✶✱ Q(f0 ) ❧➔ t➟♣ ♠ët ♣❤➛♥ tû✳ ❚❛ ❝❤å♥ xn ∈ Q(fn ) ✈ỵ✐ ♠å✐ n ∈ N ✭✷✳✶✼✮ ❍✐➸♥ ♥❤✐➯♥✱ xn ∈ K ✳ ❱➻ K ❧➔ t➟♣ ❝♦♠♣❛❝t✱ ❦❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ t❛ ❣✐↔ sû xn → x ∈ K ✳ ❚÷ì♥❣ tü ♥❤÷ ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ ❇ê ✤➲ ✷✳✸✳✹ t❛ ❝â x ∈ Q(f0 )✳ ❱➻ Q(f0 ) ❧➔ t➟♣ ♠ët ♣❤➛♥ tû ♥➯♥ x = x0 ✈➔ ❞♦ ✈➟② xn → x0 ✳ ✣✐➲✉ ♥➔② ❦➳t ❤đ♣ ✈ỵ✐ ✭✷✳✶✼✮ t❛ ✤÷đ❝ xn ∈ (x0 + W0 ) ∩ Q(fn ) ✈ỵ✐ n ✤õ ❧ỵ♥✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✭✷✳✶✻✮✳ ❚r÷í♥❣ ❤đ♣ ✷✱ Q(f0 ) ❧➔ t➟♣ ❝❤ù❛ ➼t t tỷ õ tỗ t x Q(f0 ) ♠➔ x = x0 ✳ ❱➻ x , x0 ∈ Q(f0 ) ♥➯♥ f0 (F (x , u, y)) ⊆ R+ ✈ỵ✐ ♠å✐ u ∈ S(x ) ✈➔ y ∈ K ✭✷✳✶✽✮ f0 (F (x0 , u, y)) ⊆ R+ ✈ỵ✐ ♠å✐ u ∈ S(x0 ) ✈➔ y ∈ K ✭✷✳✶✾✮ ✈➔ ❉♦ S(·) ❧➔ P ✲ ❧ã♠ tr➯♥ K ✱ ✈ỵ✐ ♠é✐ t ∈ [0, 1] t❛ ❝â S(tx + (1 − t)x0 ) ⊆ tS(x + (1 − t)S(x0 ) + P ✸✶ ❱ỵ✐ ♠é✐ ut S(tx + (1 t)x0 ) tỗ t u ∈ S(x ), u0 ∈ S(x0 ) ✈➔ p0 ∈ P s❛♦ ❝❤♦ ut = tu + (1 − t)u0 + p0 ✳ ❚❤❡♦ ❣✐↔ t❤✐➳t F (tx ) + (1 − t)x0 , ·, y) ❧➔ P ✲C ✲ t➠♥❣ ♥➯♥ t❛ ❝â F (tx + (1 − t)x0 , ut , y) ⊆ F (tx + (1 − t)x0 , tu + (1 − t)u0 , y) + C ✭✷✳✷✵✮ ✣➦t x(t) := tx + (1 − t)x0 ✳ ❑❤✐ ✤â x(t) ∈ K ✳ ❚❤❡♦ ❣✐↔ t❤✐➳t F (·, ·, y) ❧➔ C ✲ ❧ã♠ ♥❣❤✐➯♠ ♥❣➦t tr➯♥ K × S(K) ♥➯♥ F (x(t), tu +(1−t)u0 , y) ⊆ tF (x , u , y)+(1−t)F (x0 , u0 , y)+int C t tỗ t t0 ∈ [0, 1] s❛♦ ❝❤♦ x(t0 ) ∈ x0 + W0 ✳ ❚ø ✭✷✳✶✻✮ s✉② r❛ x(t0 ) ∈ / Q(fn ) õ tỗ t un S(x(t0 )) ✈➔ yn ∈ K s❛♦ ❝❤♦ fn (F (x(t0 ), un , yn ) R+ õ tỗ t zn ∈ F (x(t0 ), un , yn ) s❛♦ ❝❤♦ fn (zn ) < ✈ỵ✐ ♠å✐ n ∈ N ✭✷✳✷✷✮ ❚❤❡♦ ❣✐↔ t❤✐➳t S(x(t0 )) ✈➔ K ❧➔ ❝♦♠♣❛❝t✱ ❦❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ t❛ ❣✐↔ sû un → u ∈ S(x(t0 )) ✈➔ yn → y0 ∈ K ứ tỗ t z0 F (x(t0 ), u, y0 ) ✈➔ ❞➣② ❝♦♥ {znk } ❝õ❛ ❞➣② {zn } s❛♦ ❝❤♦ znk → z0 ✳ ❑❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ t❛ ❣✐↔ sû zn → z0 ✳ ❚ø ✤â s✉② r❛ fn (zn ) → f0 (z0 )✳ ❇ð✐ ✭✷✳✷✷✮✱ t❛ ❝â ✭✷✳✷✸✮ f0 (z0 ) ≤ ▼➦t ❦❤→❝✱ tø ✭✷✳✶✽✮✲✭✷✳✷✶✮ t❛ ❝â f0 (z0 ) > 0✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ỵ ữủ ự ●✐↔ sû ✈ỵ✐ ♠é✐ x ∈ K, F (x, ·, ã) C ố ữ ỗ tr S(x) ì K ✳ ❑❤✐ ✤â W (F, S, K) = Q(f ) f ∈B ∗ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠é✐ x ∈ Q(f ) tỗ t f0 B s x ∈ Q(f0 )✳ f ∈B ∗ ❉♦ ✤â f0 (F (x, u, y)) ⊆ R+ ✈ỵ✐ ♠å✐ u ∈ S(x) ✈➔ y ∈ K ✸✷ ✭✷✳✷✹✮ ●✐↔ sû x / W (F, S, K) õ tỗ t u0 ∈ S(x)✱ y0 ∈ K s❛♦ ❝❤♦ F (x, u0 , y0 ) ∩ (− int C) = ∅ ứ õ tỗ t z0 F (x, u0 , y0 ) s❛♦ ❝❤♦ f0 (z0 ) < 0✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✭✷✳✷✹✮✳ ❉♦ ✤â x ∈ W (F, S, K) ❱➟② Q(f ) ⊆ W (F, S, K) f ∈B ∗ Q(f ) ❚❤➟t ✈➟②✱ ❧➜② x ∈ W (F, S, K) tò② ❚❛ ❝❤ù♥❣ ♠✐♥❤ W (F, S, K) f B ỵ õ F (x, u, y) ∩ (− int C) = ∅ ✈ỵ✐ ♠å✐ u ∈ S(x) ✈➔ y ∈ K ❉➵ t❤➜② F (x, S(x), K) + C) ∩ (− int C) = ∅ ❱ỵ✐ ♠é✐ x ∈ K ✱ ❞♦ F (x, ã, ã) C ố ữ ỗ tr➯♥ S(x) × K ♥➯♥ F (x, S(x), K) + C t ỗ ỵ t t ỗ tỗ t g Y \ {0} s inf{g(z+c) : u ∈ S(x), y ∈ K, z ∈ F (x, u, y), c ∈ C} ≥ sup{g(c ) : c ∈ −C} ❉♦ ✤â g ∈ C ∗ ✈➔ g(F (x, u, y)) ⊆ R+ ✈ỵ✐ ♠å✐ u ∈ S(x) ✈➔ y ∈ K ❱➻ e ∈ int C ✈➔ g ∈ C ∗ \ {0} ♥➯♥ g(e) > 0✳ ✣➦t ψ = g ✳ ❑❤✐ ✤â ψ ∈ B ∗ g(e) ✈➔ ψ(F (x, u, y)) ⊆ R+ ✈ỵ✐ ♠å✐ u ∈ S(x) ✈➔ y ∈ K ❉♦ ✈➟② x ∈ Q(ψ) tø ✤â s✉② r❛ x ∈ f ∈B ∗ Q(f ) ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ Q(f ) ❱➟② W (F, S, K) = W (F, S, K) ⊆ f ∈B ∗ Q(f ) f ∈B ∗ ❇ê ✤➲ ✷✳✸✳✼✳ ●✐↔ sû K ❧➔ t➟♣ ❦❤→❝ rộ ỗ t tr ổ ỗ ữỡ ❍❛✉s❞♦r❢❢ X ✳ ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ①↔② r S(ã) ỷ tử ữợ P ✲ ❧ã♠ tr➯♥ K ✈ỵ✐ ❣✐→ trà ❦❤→❝ ré♥❣✱ ❝♦♠♣❛❝t❀ ợ ộ (x, y) K ì K, F (x, ·, y) ❧➔ P ✲C ✲ t➠♥❣❀ ✭✐✐✐✮ ❱ỵ✐ ♠é✐ y ∈ K, F (·, ·, y) ❧➔ C ✲ ❧ã♠ ♥❣❤✐➯♠ ♥❣➦t tr➯♥ K × S(K)❀ ✭✐✈✮ F (·, ·, ·) ❧✐➯♥ tư❝ tr➯♥ K × S(K) × K ✈ỵ✐ ❣✐→ trà ❦❤→❝ ré♥❣✱ ❝♦♠♣❛❝t❀ ✭✈✮ ❱ỵ✐ ♠é✐ x ∈ K, F (x, ·, ·) ❧➔ C ố ữ ỗ tr S(K) ì K ✤â Q(f ) ⊆ cl Q(f ) ⊆ E(F, S, K) ⊆ W (F, S, K) = f ∈B ∗ f ∈B # Q(f ) f ∈B # ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❇ê ✤➲ ✷✳✸✳✻✱ t❛ ❝â Q(f ) ⊆ E(F, S, K) ⊆ W (F, S, K) = Q(f ) f ∈B # f ∈B # ❇ð✐ ❇ê ✤➲ ✷✳✸✳✺✱ s r Q(ã) ỷ tử ữợ tr B ∗ = cl B # ✳ ✣✐➲✉ ♥➔② ✈➔ ❇ê ✤➲ ✷✳✶✳✶✷ ❦➨♦ t❤❡♦ Q(f ) ⊆ cl f ∈B ∗ Q(f ) f ∈B # ✈➔ ❞♦ ✈➟② Q(f ) ⊆ E(F, S, K) ⊆ W (F, S, K) = Q(f ) ⊆ cl f ∈B ∗ f ∈B # Q(f ) f ∈B # ❚ø ◆❤➟♥ ①➨t ✷✳✷✳✷ ✈➔ ❝→❝ ❇ê ✤➲ ✷✳✶✳✶✸✱ ✷✳✸✳✶✲✷✳✸✳✹ ✈➔ ✷✳✸✳✻ t❛ ❝â ỵ ữợ ỵ sỷ K t rộ ỗ t tr ổ ỗ ữỡ sr X sỷ ❦✐➺♥ s❛✉ ①↔② r❛ ✭✐✮ ❱ỵ✐ ♠é✐ x ∈ K ✱ u ∈ S(x) t❛ ❝â F (x, u, x) C S(ã) ỷ tử ữợ ✈➔ P ✲ ❧ã♠ tr➯♥ K ✈ỵ✐ ❣✐→ trà ❦❤→❝ rộ t ợ ộ (x, y) K ì K, F (x, ·, y) ❧➔ P ✲C ✲ t➠♥❣❀ ✭✐✈✮ ❱ỵ✐ ♠é✐ y ∈ K, F (·, ·, y) C ó ỷ tử ữợ tr K ì S(K) ợ ộ x K F (x, ã, ã) C tỹ ỗ tỹ C ố ữ ỗ tr S(x) × K ✳ ❑❤✐ ✤â W (F, S, K) ❧➔ t➟♣ ❧✐➯♥ t❤æ♥❣✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✸✳✷ t õ Q(f ) t ỗ Q(f ) ❧➔ t➟♣ ❧✐➯♥ t❤æ♥❣✳ ❚ø ❇ê ✤➲ ✷✳✸✳✶ ✈➔ ◆❤➟♥ ①➨t ✷✳✷✳✷ t❛ t❤➜② Q(f ) = ∅✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✸✳✹ s✉② r❛ Q(·) ❧➔ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ tr➯♥ C ∗ \ {0Y ∗ }✳ ❑➳t ❤ñ♣ ❇ê ✤➲ ✷✳✶✳✶✸ Q(f ) ❧➔ t➟♣ ❧✐➯♥ t❤æ♥❣✳ ✈➔ ❇ê ✤➲ ✷✳✸✳✻ t❛ ❝â W (F, S, K) = f ∈B ∗ ◆❤➟♥ ①➨t ✷✳✸✳✾✳ ❚ø ◆❤➟♥ ①➨t ✶✳✹✳✶✷ ✈➔ ✶✳✹✳✶✸ t❛ t❤➜②✱ ✈ỵ✐ ♠é✐ x ∈ K ✱ F (x, ã, ã) C ỗ tr S(x) × K t❤➻ F (x, ·, ·) ❧➔ C tỹ ỗ tỹ C ố ữ ỗ tr S(x) ì K ứ ỵ t s r q ữợ q sỷ K t rộ ỗ t tr ổ ỗ ữỡ sr X f tử C ỗ tr K ✳ ❑❤✐ ✤â WV (f, K) ❧➔ t➟♣ ❧✐➯♥ t❤æ♥❣✳ ỵ sỷ K t rộ ỗ t tr ổ ỗ ữỡ sr X ✳ ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ①↔② r❛ ✭✐✮ ❱ỵ✐ ♠é✐ x ∈ K ✈➔ u ∈ S(x) t❛ ❝â F (x, u, x) ⊆ C ❀ ✭✐✐✮ S(ã) tử ữợ P ó tr K ✈ỵ✐ ❣✐→ trà ❦❤→❝ ré♥❣✱ ❝♦♠♣❛❝t❀ ✭✐✐✐✮ ❱ỵ✐ ♠é✐ (x, y) ∈ K × K, F (x, ·, y) ❧➔ P ✲C ✲ t➠♥❣❀ ✭✐✈✮ ❱ỵ✐ ♠é✐ y ∈ K ✱ F (·, ·, y) ❧➔ C ✲ ❧ã♠ ♥❣❤✐➯♠ ♥❣➦t tr➯♥ K × S(K)❀ ✭✈✮ F (·, ·, ·) tử tr K ì S(K) ì K ợ trà ❦❤→❝ ré♥❣✱ ❝♦♠♣❛❝t❀ ✭✈✐✮ ❱ỵ✐ ♠é✐ x ∈ K, F (x, ã, ã) C tỹ ỗ tỹ C ố ữ ỗ tr S(x) ì K ✳ ❑❤✐ ✤â E(F, S, K) ❧➔ t➟♣ ❧✐➯♥ t❤ỉ♥❣✳ ❍ì♥ ♥ú❛✱ W (F, S, K) ❧➔ t➟♣ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚÷ì♥❣ tü ♥❤÷ ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ ỵ t ự Q(f ) ❧➔ t➟♣ ❧✐➯♥ t❤æ♥❣✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✸✳✼✱ E(F, S, K) ❧➔ t➟♣ ❧✐➯♥ f ∈B # ✸✺ t❤æ♥❣✳ ❚ø ❇ê ✤➲ ✷✳✸✳✺✱ t❛ ❝â Q(·) ❧➔ ♥û❛ ❧✐➯♥ tö❝ ữợ tr C \ {0Y } tr õ tæ♣æ tr➯♥ C ∗ \ {0Y ∗ } ❧➔ tæ♣æ ②➳✉ ✯✳ ❉♦ fn → f0 ❦➨♦ t❤❡♦ fn ❤ë✐ tö ②➳✉ ✈➲ f0 ✳ ❱➻ Q(·) ❧➔ ♥û❛ ❧✐➯♥ tử ữợ tr C \ {0Y } tr ✤â tæ♣æ tr➯♥ C ∗ \ {0Y ∗ } ❧➔ tổổ Q(ã) ỷ tử ữợ tr➯♥ C ∗ \ {0Y ∗ }✱ tr♦♥❣ ✤â tæ♣æ tr➯♥ C ∗ \ {0Y ∗ } ✤è✐ ✈ỵ✐ tỉ♣ỉ s✐♥❤ ❜ð✐ ❝❤✉➞♥✳ ❍✐➸♥ ♥❤✐➯♥ B ∗ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✈➔ ❞♦ ✈➟② B ∗ ❧➔ t✐➲♥ ❝♦♠♣❛❝t✳ ❚ø B t ỗ B t❤ỉ♥❣ ✤÷í♥❣✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✸✳✻ t❛ ❝â Q(f ) W (F, S, K) = f ∈B ∗ ❑➳t ❤ñ♣ ❇ê ✤➲ ✷✳✸✳✶✲✷✳✸✳✹✱ ✈ỵ✐ ♠é✐ f ∈ B ∗ , Q(f ) t rộ ỗ õ ✤➲ ✷✳✸✳✼ s✉② r❛ W (F, S, K) ❧➔ t➟♣ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣✳ ❍➺ q✉↔ ✷✳✸✳✶✷✳ ●✐↔ sû K ❧➔ t rộ ỗ t tr ổ ỗ ✤à❛ ♣❤÷ì♥❣ ❍❛✉s❞♦r❢❢ X ✱ f ❧➔ ❤➔♠ ❧✐➯♥ tư❝ C ỗ t tr K ✤â EV (f, K) ❧➔ t➟♣ ❧✐➯♥ t❤æ♥❣ ✈➔ WV (f, K) t tổ ữớ ữợ ỵ ❈❤♦ Y = R2 , C = R2+ = {(x1 , x2 ) ∈ R2 : x1 ≥ 0, x2 ≥ 0}✱ X = R✱ P = R+ = {x ∈ R : x ≥ 0} ✈➔ K = [0, π]✳ ⑩♥❤ ①↕ ✤❛ trà S : X → ∆ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ S(x) = {u ∈ R : −2 ≤ u ≤ sin x}, ✈ỵ✐ ♠å✐ x ∈ X, ✈➔ →♥❤ ①↕ F : X × ∆ × X → Y ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉ F (x, u, y) = (f1 (x, u, y), f2 (x, u, y)) + BY , ✈ỵ✐ ♠å✐ (x, u, y) ∈ X × ∆ × X, tr♦♥❣ ✤â f1 (x, u, y) = −x2 + u + y2 + y + 4, 11 f2 (x, u, y) = sin x + 2u − sin y + ❉➵ t❤➜② ❝→❝ ỵ ✤÷đ❝ t❤ä❛ ♠➣♥✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✸✳✶ t❛ ❝â Q(f ) = ✈ỵ✐ ♠é✐ f ∈ C ∗ \{0Y ∗ }✱ ❦➨♦ t❤❡♦ W (F, S, K) = ∅✳ ỡ ỳ tứ ỵ s r W (F, S, K) ❧➔ t➟♣ ❧✐➯♥ t❤ỉ♥❣✳ ✸✻ ❱➼ ❞ư t✐➳♣ t ỵ ❈❤♦ Y = R2 , C = R2+ = {(x1 , x2 ) ∈ R2 : x1 ≥ 0, x2 ≥ 0}✱ X = R✱ P = R+ = {x ∈ R : x ≥ 0} ✈➔ K = [0, π]✳ ⑩♥❤ ①↕ ✤❛ trà S : X → ∆ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ S(x) = {u ∈ R : ≤ u ≤ + sin x}, ✈ỵ✐ ♠å✐ x ∈ X, ✈➔ →♥❤ ①↕ F : X × ì X Y ữủ ữ s F (x, u, y) = (f1 (x, u, y), f2 (x, u, y)) + BY , ✈ỵ✐ ♠å✐ (x, u, y) ∈ X × ∆ × X, tr♦♥❣ ✤â 1 x + y + 2u + 2y + 6, 4 f1 (x, u, y) = −x2 + sin f2 (x, u, y) = −x2 + x + 1 x + y + u + y + 4 ❉➵ t❤➜② ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❇ê ✤➲ ỵ ữủ tọ ✤➲ ✷✳✸✳✷ t❛ ❝â Q(f ) ✈ỵ✐ ♠é✐ f ∈ C ∗ \{0Y ∗ }✱ ✤✐➲✉ ♥➔② ❦➨♦ t❤❡♦ E(F, S, K) = ỡ ỳ t ỵ t❛ ❝â E(F, S, K) ❧➔ t➟♣ ❧✐➯♥ t❤æ♥❣ ✈➔ W (F, S, K) ❧➔ t➟♣ ❧✐➯♥ t❤ỉ♥❣ ✤÷í♥❣✳ ✸✼ ❑➳t ❧✉➟♥ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ✤➣ tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ❝❤➼♥❤ s❛✉✿ ✶✳ ❚r➻♥❤ ❜➔② ởt số ỵ sỹ tỗ t ỳ ỵ ỳ ỵ ỳ ỵ ❜➔✐ t♦→♥ tü❛ ❝➙♥ ❜➡♥❣ ✈➨❝tì s✉② rë♥❣✳ ✷✳ ❚r➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✈➲ t➼♥❤ ❧✐➯♥ t❤æ♥❣ ❝õ❛ t➟♣ ♥❣❤✐➺♠ ✤è✐ ✈ỵ✐ ❜➔✐ t♦→♥ tü❛ ❝➙♥ ❜➡♥❣ ✈➨❝tì s rở ỵ ỵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ ✣æ♥❣ ❨➯♥✱ ●✐↔✐ t➼❝❤ ✤❛ trà✱ ◆❤➔ ①✉➜t ❜↔♥ ❣✐→♦ ❞ö❝ ✭✷✵✵✼✮✳ ❚✐➳♥❣ ❆♥❤ ❬✷❪ ❆✉❜✐♥✱ ❏✳P✳✱ ❊❦❡❧❛♥❞✱ ■✳✱ ❆♣♣❧✐❡❞ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s✳ ❲✐❧❡②✱ ◆❡✇ ❨♦r❦ ✭✶✾✽✹✮✳ ❬✸❪ ●♦♥❣✱ ❳✳❍✳✿ ❖♥ t❤❡ ❝♦♥tr❛❝t✐❜✐❧✐t② ❛♥❞ ❝♦♥♥❡❝t❡❞♥❡ss ♦❢ ❛♥ ❡❢❢✐❝✐❡♥t ♣♦✐♥t s❡t✳ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✷✻✹✱ ✹✻✺✲✹✼✽ ✭✷✵✵✶✮✳ ❬✹❪ ●♦♣❢❡rt✱ ❆✳✱ ❘✐❛❤✐✱ ❍✳✱ ❚❛♠♠❡r✱ ❈✳✱ ❩❛❧✐♥❡s❝✉✱ ❈✳✱ ❱❛r✐❛t✐♦♥❛❧ ▼❡t❤♦❞s ✐♥ P❛rt✐❛❧❧② ❖r❞❡r❡❞ ❙♣❛❝❡s✳ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥ ✭✷✵✵✸✮✳ ❬✺❪ ❍❛♥✱ ❨✳✱ ❍✉❛♥❣✱ ◆✳❏✳✱ ❙t❛❜✐❧✐t② ♦❢ ❡❢❢✐❝✐❡♥t s♦❧✉t✐♦♥s t♦ ♣❛r❛♠❡tr✐❝ ❣❡♥✲ ❡r❛❧✐③❡❞ ✈❡❝t♦r ❡q✉✐❧✐❜r✐✉♠ ♣r♦❜❧❡♠s ✭✐♥ ❈❤✐♥❡s❡✮✳ ❙❝✐✳ ❙✐♥✳ ▼❛t❤✳ ✹✻✱ ✶✕✶✷ ✭✷✵✶✻✮✳ ❞♦✐✿✶✵✳✶✸✻✵✴✵✶✷✵✶✻✲✶✷✳ ❬✻❪ ❍❛♥✱ ❨✳✱ ❍✉❛♥❣✱ ◆✳❏✳✱ ❊①✐st❡♥❝❡ ❛♥❞ ❈♦♥♥❡❝t❡❞♥❡ss ♦❢ ❙♦❧✉t✐♦♥s ❢♦r ●❡♥❡r❛❧✐③❡❞ ❱❡❝t♦r ◗✉❛s✐ ✕ ❊q✉✐❧✐❜r✐✉♠ Pr♦❜❧❡♠s✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✶✼✾✱ ✻✺✲✽✺ ✭✷✵✶✻✮✳ ❬✼❪ ❍✐r✐❛rt✲❯rr✉t②✱ ❏✳❇✳✱ ■♠❛❣❡s ♦❢ ❝♦♥♥❡❝t❡❞ s❡ts ❜② s❡♠✐❝♦♥t✐♥✉♦✉s ♠✉❧✲ t✐❢✉♥❝t✐♦♥s✳ ❏✳▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✶✶✶✱ ✹✵✼✕✹✷✷ ✭✶✾✽✺✮✳ ❬✽❪ ❍✉✱ ❨✳❉✳✱ ▲✐♥❣✱ ❈✳✱ ❈♦♥♥❡❝t❡❞♥❡ss ♦❢ ❝♦♥❡ s✉♣❡r❡❢❢✐❝✐❡♥t ♣♦✐♥t s❡ts ✐♥ ❧♦❝❛❧❧② ❝♦♥✈❡① t♦♣♦❧♦❣✐❝❛❧ ✈❡❝t♦r s♣❛❝❡s✳ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✶✵✼✱ ✹✸✸✲✹✹✻ ✭✷✵✵✵✮✳ ✸✾