THÔNG TIN TÀI LIỆU
❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆ ❍❯Ý◆❍ ❱■➏❚ ❚❘❯◆● ▼❐❚ ❙➮ ❱❻◆ ✣➋ ❱➋ ❚➑◆❍ ❈❍➑◆❍ ◗❯❨ ❚❍❊❖ ❍×❰◆● ❈Õ❆ ⑩◆❍ ❳❸ ✣❆ ❚❘➚ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❇➻♥❤ ✣à♥❤ ✲ ◆➠♠ ✷✵✶✾ download by : skknchat@gmail.com ❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆ ❍❯Ý◆❍ ❱■➏❚ ❚❘❯◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ▼❐❚ ❙➮ ❱❻◆ ✣➋ ❱➋ ❚➑◆❍ ❈❍➑◆❍ ◗❯❨ ❚❍❊❖ ❍×❰◆● ❈Õ❆ ⑩◆❍ ❳❸ ✣❆ ❚❘➚ ❈❤✉②➯♥ t số ữớ ữợ ❚❙✳ ◆●❯❨➍◆ ❍Ú❯ ❚❘➴◆ download by : skknchat@gmail.com ✐ ▼ö❝ ❧ö❝ ▼Ð ✣❺❯ ✶ ✶ ✸ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ⑩♥❤ ①↕ ✤❛ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❚➼♥❤ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ữợ õ t t ữợ ố q tr r t ữợ ❦❤→✐ ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✷ ❈→❝ ✤➦❝ tr÷♥❣ ❝õ❛ t q r t ữợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✷✳✶ ✣➦❝ tr÷♥❣ q✉❛ ✤ë ❞è❝ ♠↕♥❤ ❝❤♦ t➼♥❤ ❝❤➼♥❤ q✉② ♠❡tr✐❝ r t ữợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✷✳✷ ❚➼♥❤ ❝❤➼♥❤ q✉② ♠❡tr✐❝ t❤❡♦ ữợ tr ✳ ✳ ✷✶ ✷✳✷✳✸ ❚➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ❝❤➼♥❤ q✉② tr r t ữợ ữợ t t✉②➳♥ ✲ ✤è✐ ✤↕♦ ❤➔♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸ Ù♥❣ ❞ö♥❣ ❝õ❛ t➼♥❤ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ❍☎♦❧❞❡r ✸✺ ❑➌❚ ▲❯❾◆ ✹✶ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ download by : skknchat@gmail.com ✹✷ ✶ ▼Ð ✣❺❯ ❚r♦♥❣ s✉èt ❜❛ t❤➟♣ ❦➾ ❝✉è✐ ❝õ❛ t❤➳ ❦➾ ❳❳✱ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ✤➣ ♣❤→t tr✐➸♥ ✈➔ trð t❤➔♥❤ ♠ët tr♦♥❣ ❝→❝ ❦❤→✐ ♥✐➺♠ tr✉♥❣ t➙♠ ❝õ❛ ●✐↔✐ t➼❝❤ ❜✐➳♥ ♣❤➙♥ ❤✐➺♥ ✤↕✐✳ ❚❤✉➟t ♥❣ú ✧❝❤➼♥❤ q✉② ♠❡tr✐❝✧ ✤÷đ❝ ✤➦t ❜ð✐ ❇♦r✇❡✐♥✱ ♥❤÷♥❣ ỗ ố t ỗ tứ õ ỵ ♥â ✤÷đ❝ tê♥❣ q✉→t ❝❤♦ →♥❤ ①↕ ♣❤✐ t✉②➳♥ ✤÷đ❝ t ữ ỵ str rs ỵ tt ❝❤➼♥❤ q✉② ♠❡tr✐❝ ✤÷đ❝ ❞ị♥❣ ❝❤♦ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❞→♥❣ ✤✐➺✉ ♥❣❤✐➺♠ ❝õ❛ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ♣❤✐ t✉②➳♥ tr➯♥ ♠ët ❞ú ❧✐➺✉ ♥❤✐➵✉✱ ❤♦➦❝ tê♥❣ q✉→t ❤ì♥ ❧➔ ❞→♥❣ ✤✐➺✉ ❝❤♦ t➟♣ ❤đ♣ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ tê♥❣ q✉→t ❧✐➯♥ ❦➳t ✈ỵ✐ ♠ët →♥❤ ①↕ ✤❛ trà✳ ❑➳t q✉↔ ❧➔✱ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ✤â♥❣ ♠ët ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ♥❤✐➲✉ ♥❤→♥❤ ❝õ❛ t♦→♥ ❤å❝ ♥❤÷ tè✐ ÷✉✱ ❝→❝ t ỵ tt ữỡ ♣❤→♣ sè ✈➔ tr♦♥❣ ♥❤✐➲✉ ❜➔✐ t♦→♥ ●✐↔✐ t➼❝❤✳ ❍✐➺♥ ♥❛②✱ ❦❤→✐ ♥✐➺♠ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ✤❛♥❣ ✤÷đ❝ sü q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ♥❤✐➲✉ ❝❤✉②➯♥ ❣✐❛ tr♦♥❣ ❧➽♥❤ ✈ü❝✳ ◆❤✐➲✉ ❜✐➳♥ t❤➸ ❝õ❛ t➼♥❤ ❝❤➜t ♥➔② ❝ơ♥❣ ✤÷đ❝ ❦❤↔♦ st ự ữợ ởt ữợ ự tr ỹ ự ữợ t t ✈➔ ✤÷đ❝ ❣å✐ ❧➔ t➼♥❤ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ❝➜♣ ❝❛♦ t ữợ ụ t ự tr ❚è✐ ÷✉ ✈➔ ●✐↔✐ t➼❝❤ ❜✐➳♥ ♣❤➙♥✳ ❱➻ ✈➟② ✈✐➺❝ ự t q tr t ữợ ❧➔ ♠ët ✈➜♥ ✤➲ t❤í✐ sü✳ ❚r♦♥❣ ❦❤✉ỉ♥ ❦❤ê ❝õ❛ ♠ët ❧✉➟♥ ✈➠♥ t❤↕❝ sÿ ❝❤✉②➯♥ ♥❣➔♥❤ ❚♦→♥ ●✐↔✐ t➼❝❤✱ ♠ư❝ ✤➼❝❤ ❝õ❛ ❝❤ó♥❣ tỉ✐ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr➻♥❤ ❜➔② ♣❤✐➯♥ ❜↔♥ t➼♥❤ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ❍☎ ♦❧❞❡r t ữợ t trữ t q tr t ữợ q ởt số ổ tr ❣➛♥ ✤➙② ✈➲ ❧➽♥❤ ✈ü❝ ♥➔②✳ ▲✉➟♥ ✈➠♥ s➩ ❤➺ t❤è♥❣ ❤â❛ ✈➔ ❝❤✐ t✐➳t ❤â❛ ♥❤ú♥❣ ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ ♥❤ú♥❣ ❦➳t q✉↔ q✉❛♥ trå♥❣ ✈➲ t➼♥❤ ❝❤➼♥❤ q✉② ♠❡tr✐❝ t ữợ ỳ ự t q tr♦♥❣ ✈✐➺❝ ❣✐↔✐ ❝→❝ ❇➔✐ t♦→♥ ❚è✐ ÷✉✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ✈✐➳t ❞ü❛ tr➯♥ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤➼♥❤ ❧➔ ❬✸❪ ✈➔ ❬✹❪ ✳ ❈➜✉ download by : skknchat@gmail.com trú ỗ ữỡ ữỡ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❈❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ✤➸ ❝õ♥❣ ❝è ❦✐➳♥ t❤ù❝ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì sð ❧✐➯♥ q✉❛♥ ✤➳♥ t➼♥❤ ❝❤➼♥❤ q✉② ♠❡tr✐❝✳ ❈❤÷ì♥❣ ✷✳ q tr r t ữợ ữỡ ỗ r ❜➔② ❦❤→✐ ♥✐➺♠ ✈➲ t➼♥❤ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ❍☎ ♦❧❞❡r t ữợ ởt số r ❝→❝ ✤➦❝ tr÷♥❣ ❝õ❛ t➼♥❤ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ❍☎ ♦❧❞❡r t ữợ ữỡ t q tr r t ữợ ú tổ tr ự t q tr r t ữợ tr ố ÷✉ ❜➡♥❣ ✈✐➺❝ ✤÷❛ r❛ ♣❤÷ì♥❣ ♣❤→♣ ✤➸ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐ t♦→♥ ❚è✐ ÷✉✳ ▼➦❝ ❞ị r➜t ❝è ❣➢♥❣ ♥❤÷♥❣ ❞♦ ❤↕♥ ❝❤➳ ✈➲ t❤í✐ ❣✐❛♥ ✈➔ tr➻♥❤ ✤ë ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳ ✈➔ t sõt t ữủ ỳ õ ỵ t qỵ t ổ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ◗✉② ◆❤ì♥✱ ♥❣➔② ✷ t❤→♥❣ ✽ ♥➠♠ ✷✵✶✾ ❍å❝ ✈✐➯♥ ❍✉ý♥❤ ❱✐➺t ❚r✉♥❣ download by : skknchat@gmail.com ✸ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ✈➲ →♥❤ ①↕ ✤❛ trà✱ ❦❤→✐ ♥✐➺♠ ✈➲ t➼♥❤ ❝❤➼♥❤ q✉② ✭①❡♠ ■♦❢❢❡ ❬✸❪✮ ✈➔ ❝→❝ ❦✐➳♥ t❤ù❝ ❞ò♥❣ ✤➸ ❜ê trđ ❝❤♦ ❝→❝ ✤➦❝ tr÷♥❣ ❝õ❛ t➼♥❤ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ✭①❡♠ ❍✉②♥❤ ❱❛♥ ◆❣❛✐✱ ◆❣✉②❡♥ ❍✉✉ ❚r♦♥✱ ▼✐❝❤❡❧ ❚❤❡r❛ ❬✹❪✮✳ ✶✳✶ ⑩♥❤ ①↕ ✤❛ trà ❈❤♦ X ✈➔ Y ổ tr ợ tr ữủ ❜ð✐ d(., )❀ ❝→❝ ❤➻♥❤ ¯ (x, r)✳ ❝➛✉ ♠ð ✈➔ ❤➻♥❤ ❝➛✉ ✤â♥❣ ✈ỵ✐ t➙♠ x ✈➔ ❜→♥ ❦➼♥❤ r > ❧➛♥ ❧÷đt ❧➔ ❇(x, r) ✈➔ ❇ ❈❤♦ t➟♣ ❤ñ♣ C ⊂ X ✱ ♣❤➛♥ tr♦♥❣ ❝õ❛ C ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ ✐♥t ❈✳ ▼ët →♥❤ ①↕ ✤❛ trà ❧➔ ♠ët →♥❤ ①↕ F : X ⇒ Y q✉② t➢❝ ✈ỵ✐ ♠é✐ x ∈ X ✱ ♠ët t➟♣ ✭❝â t❤➸ ❜➡♥❣ ré♥❣✮ F (x) ⊂ Y ✳ ❚❛ ❞ò♥❣ ❦➼ ❤✐➺✉ ❣♣❤ F := {(x, y) ∈ X ì Y : y F (x)} ỗ t❤à ❝õ❛ ❤➔♠ ❋✳ ❱ỵ✐ ♠é✐ →♥❤ ①↕ F : X ⇒ Y ✱ t❛ ✤à♥❤ ♥❣❤➽❛ →♥❤ ①↕ ♥❣÷đ❝ ❝õ❛ ❋ ❜➡♥❣ F −1 : Y ⇒ X ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ F −1 (y) := {x ∈ X : y ∈ F (y)}, y ∈ Y ✈➔ t❤ä❛ ♠➣♥ (x, y) ∈ ❣♣❤ F ⇐⇒ (y, x) ∈ ❣♣❤ F −1 ❚❛ ❞ò♥❣ ❦➼ ❤✐➺✉ d(x, C) ✤➸ ✤à♥❤ ♥❣❤➽❛ ❦❤♦↔♥❣ ❝→❝❤ tø ✤✐➸♠ x ✤➳♥ ♠ët t➟♣ ❈❀ ❦❤♦↔♥❣ ❝→❝❤ ♥➔② ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ❝ỉ♥❣ t❤ù❝ d(x, C) = inf d(x, z) z∈C ✈➔ q✉② ÷ỵ❝ d(x, S) = +∞ ♥➳✉ ❙ ❧➔ t➟♣ ré♥❣✳ download by : skknchat@gmail.com ✹ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❈❤♦ f : Rn → R ∪ {+∞} ❧➔ ♠ët ❤➔♠ ♥❤➟♥ ❣✐→ trà t❤ü❝ ♠ð rë♥❣✳ ✭✐✮ ▼✐➲♥ ❤ú✉ ❤✐➺✉ ❝õ❛ ❤➔♠ f ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ❞♦♠ f = {x ∈ Rn : f (x) < +∞} ❍➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ❝❤➼♥❤ t❤÷í♥❣ ♥➳✉ ♠✐➲♥ ❤ú✉ ❤✐➺✉ ❝õ❛ f ❦❤æ♥❣ ré♥❣ ✐✳❡✳ ❞♦♠ f = ∅✳ ✭✐✐✮ ❈❤♦ f : Rn → R∪{+∞} ❧➔ ❤➔♠ ❝❤➼♥❤ t❤÷í♥❣✱ f ✤÷đ❝ ỷ tử ữợ s t x0 ợ < f (x0 ) t tỗ t r > s❛♦ ❝❤♦ ♠å✐ x ∈ ❇(x0 , r), λ < f (x)✳ f ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ ✭✉s❝✮ t↕✐ x0 ♥➳✉ −f ❧➔ ❧s❝✳ f ữủ ỷ tử ữợ f tử ữợ t t❤✉ë❝ ❞♦♠ f ✈➔ f ❧➔ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f ❧✐➯♥ tö❝ tr➯♥ t↕✐ ♠å✐ ✤✐➸♠ t❤✉ë❝ ❞♦♠ f ✳ ✶✳✷ ❚➼♥❤ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ❈❤♦ →♥❤ ①↕ ✤❛ trà F : X ⇒ Y ✈➔ ♠ët ❝➦♣ (x, y) ∈ ❣♣❤ F ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ✭❈→❝ t➼♥❤ ❝❤➜t t➼♥❤ ❝❤➼♥❤ q✉② ✤à❛ ♣❤÷ì♥❣✮✳ ❚❛ ♥â✐ r➡♥❣ F ❧➔ • P❤õ ❤♦➦❝ ♠ð t➾ ❧➺ t✉②➳♥ t➼♥❤ t↕✐ ♠ët ✤✐➸♠ ❣➛♥ (¯ x, y¯) tỗ t r > 0, > s ❝❤♦ ❇(y, rt) ∩ ❇(¯ y , ε) ⊂ F (❇(x, t)), ∀(x, y) ∈ ❣♣❤ F, d(x, x¯) < ε, t ≥ • ❈❤➼♥❤ q✉② ♠❡tr✐❝ ❣➛♥ (¯ x, y) F tỗ t K > 0, ε > s❛♦ ❝❤♦ d(x, F −1 (y)) ≤ Kd(y, F (x)), ♥➳✉ d(x, x ¯) < ε, d(y, y¯) < ε ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ✭❚➼♥❤ ❝❤➼♥❤ q✉② tr ỗ t F ữủ q tr ỗ t ( x, y) F tỗ t số K > 0, > s❛♦ ❝❤♦ ❜➜t ✤➥♥❣ t❤ù❝ d(x, F −1 (y)) ≤ Kd (x, y), ❣♣❤ F download by : skknchat@gmail.com ✺ ✤ó♥❣✱ ✈ỵ✐ d(x, x ¯) < ε, d(y, y¯) < ε ▼➺♥❤ ✤➲ ✶✳✹✳ ✭❚➼♥❤ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ✈➔ q ỗ t F : X Y, ✈➔ (¯x, y¯) ∈ ❣♣❤ F ❑❤✐ ✤â F ❧➔ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ F ❧➔ ❝❤➼♥❤ q ỗ t ỵ ỵ r t ♥➠♠ 1974 ❧➔ ♠ët ❝ỉ♥❣ ❝ư ♠↕♥❤ tr♦♥❣ ●✐↔✐ t➼❝❤ ♣❤✐ t✉②➳♥✱ ●✐↔✐ t➼❝❤ ❦❤ỉ♥❣ trì♥✱ ●✐↔✐ t➼❝❤ ✤❛ trà✱ t tr ữợ tr ỵ ❊❦❡❧❛♥❞✮ ❈❤♦ ✭❳✱❞✮ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✤õ ✈➔ ❤➔♠ f : X → R ∪ {∞} ❧➔ ♥û❛ tử ữợ ữợ sỷ > ✈➔ xε ∈ X t❤ä❛ ♠➣♥✿ f (xε ) < inf f + ε X ❑❤✐ ✤â✱ ✈ỵ✐ λ > t t tỗ t x X s❛♦ ❝❤♦ ✭✐✮ d(¯ x, xε ) ≤ λ, ε ✭✐✐✮ f (¯ x) + d(¯ x, xε ) ≤ f (xε ), λ ε ✭✐✐✐✮ f (x) + d(¯ x, x) > f (¯ x), ∀x ∈ X \ { x} ữợ õ t t ữợ ố rữợ t t ♥❤➢❝ ❧↕✐ r➡♥❣ X ∗ ❧➔ ✤è✐ ♥❣➝✉ tæ♣æ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ X ✳ ●✐→ trà ❝õ❛ ♣❤✐➳♠ ❤➔♠ x∗ ∈ X ∗ t↕✐ x ∈ X ✤÷đ❝ ❦➼ ❤✐➺✉ ❜ð✐ x∗ , x ❈→❝ ❦➼ ❤✐➺✉ ω, ω ∗ ❞ị♥❣ ∗ ✤➸ ❝❤➾ tỉ♣ỉ ✈➔ tỉ♣ỉ ②➳✉ ❝õ❛ ❝➦♣ ✤è✐ ♥❣➝✉ (X, X ∗ ) ✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ❈❤♦ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ f : X → R ∪ {+∞} ❧➔ ❤➔♠ ♥❤➟♥ ❣✐→ trà tữ rở ữợ rt f t x ¯ ∈ ❞♦♠ f ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ∂f (¯ x) = f (x) − f (¯ x) − x∗ , x − x¯ ≥0 x→¯ x, x=¯ x x − x¯ x∗ ∈ X ∗ : lim inf download by : skknchat@gmail.com ✻ ✣➸ t❤✉➟♥ ❧đ✐✱ ❝❤ó♥❣ t❛ s t tt ỳ ữợ q ữợ rt ộ tỷ ữợ ✈✐ ♣❤➙♥ ❋r➨❝❤❡t ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♣❤➛♥ tû ❝õ❛ ữợ rt q rt x ởt ✤✐➸♠ s❛♦ ❝❤♦ f (¯ x) = ∞✱ ❦❤✐ ✤â t❛ ✤➦t ∂f (¯ x) = ∅✳ ❚❛ ❝á♥ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ ♠ët ♣❤➛♥ tû x∗ ❧➔ ởt ữợ rt f t x ♥➳✉ ◦( x − x¯ ) = x→¯ x x − x¯ f (x) ≥ f (¯ x) + x∗ , x − x¯ + ◦( x − x¯ ) ❦❤✐ lim ✣à♥❤ ❧➼ ✶✳✼✳ ✭◗✉② t➢❝ tê♥❣ ♠í✮ ❈❤♦ ❳ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❆s♣❧✉♥❞ ✈➔ ϕ1 , ϕ2 : X → R∪{∞} s❛♦ ❝❤♦ x1 ❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ q✉❛♥❤ x¯ ∈ dom ϕ1 ∩dom ϕ2 ✈➔ ϕ2 ♥û❛ tử ữợ q x õ ợ > t❛ ❝â ¯ (x, r), ∂(ϕ1 + ϕ2 )(¯ x) ⊂ ∩{∂ϕ1 (x1 ) + ∂ϕ2 (x2 )| xi ∈ x¯ + γ ❇ ¯ (x, r) |ϕi (xi ) − ϕi (¯ x)| ≤ γ, i = 1, 2} + γ ❇ ❱ỵ✐ C ⊆ X ❧➔ t➟♣ ❝♦♥ ❦❤æ♥❣ ré♥❣✱ ✤â♥❣✱ ❦➼ ❤✐➺✉ ❤➔♠ ❝❤➾ ❧✐➯♥ t ợ C C ữủ ổ t❤ù❝✿ ♥➳✉ x ∈ C, δC (x) = ∞ ♥❣♦➔✐ r❛✳ ◆â♥ ♣❤→♣ t✉②➳♥ ❋r➨❝❤❡t ❝❤♦ ❈ t↕✐ x ¯ ✤÷đ❝ ❦➼ ❤✐➺✉ ❜ð✐ N (C, x) t ỗ õ tr X ữủ ①→❝ ✤à♥❤ ❜ð✐ ∂δC ✳ ▼ët ❝→❝❤ t÷ì♥❣ ✤÷ì♥❣✱ ♠ët ✈❡❝t♦r x∗ ∈ X ∗ ❧➔ ♠ët ✈❡❝t♦ ♣❤→♣ t✉②➳♥ ❋r➨❝❤❡t ❝õ❛ ❈ t↕✐ x ¯ ♥➳✉ x∗ , x − x¯ ≤ ◦( x − x¯ ), ∀x ∈ C, tr♦♥❣ ✤â ◦( x − x¯ ) = x→¯ x x − x¯ lim ✣à♥❤ ♥❣❤➽❛ ✶✳✽ ✭✣↕♦ t ữợ r ởt g : X → Y ❣✐ú❛ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✤÷đ❝ ❣å✐ r t x t ữợ u ợ s tỗ t g( x + tw) − g(¯ x) = Dg(¯ x)(u) w→u, t↓0 t lim download by : skknchat@gmail.com ✼ ❍✐➸♥ ♥❤✐➯♥✱ ♥➳✉ g ❧➔ ▲✐♣s❝❤✐t③ ❣➛♥ x ¯ t❤➻ g ❧➔ ❦❤↔ ✈✐ ❍❛❞❛♠❛r❞ t x t ữợ Dg( x)(0) = ✣à♥❤ ♥❣❤➽❛ ✶✳✾✳ ❈❤♦ F : X ⇒ Y ❧➔ →♥❤ ①↕ ✤❛ trà ✈➔ (¯x, y¯) ∈ ❣♣❤ F ✳ ❑❤✐ ✤â✱ ✤è✐ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t t↕✐ (¯ x, y¯) ❝õ❛ ❋ ❧➔ →♥❤ ①↕ ✤❛ trà D∗ F (¯ x, y¯) : Y ∗ ⇒ X ∗ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ D∗ F (¯ x, y¯)(y ∗ ) := {x∗ ∈ X ∗ |(x∗ , −y ∗ ) ∈ N (❣♣❤ F, (¯ x, y¯))} download by : skknchat@gmail.com ✷✽ ▲➜② zn ∈ ❇(xn , ηn2 ), ωn ∈ F (zn ) + g(zn ) s❛♦ ❝❤♦ yn − ωn γ ❑❤✐ ✤â✱ yn − ωn γ ηn2 ≤ ϕ(xn , yn )γ + ≤ ϕ(z, yn ) + δn z − xn ηn2 + , ¯ (xn , ηn ) ∀z ∈ ❇ ❇ð✐ ϕγ (z, yn ) ≤ [d(yn , (F + g)(z))]γ ≤ yn − ω γ + δ❣♣❤ (F +g) (z, ω), ∀ω ∈ Y ❚❛ ❝â yn − ωn γ ≤ yn − ω γ + δ❣♣❤ (F +g) (z, ω) + δn z − zn + (δn + 1) ηn2 , ¯ (xn , ηn ) × Y ✈ỵ✐ ♠å✐ (z, ω) ∈ ❇ ⑩♣ ❞ư♥❣ ◆❣✉②➯♥ ỵ (z, ) yn − ω + δ❣♣❤ (F +g) (z, ω) + δn z − zn ¯ (xn , ηn )×Y ✱ t❛ ❝â t❤➸ ❝❤å♥ (z , ω ) (zn , n )+ n XìY ợ (z , ω ) ∈ ❣♣❤ (F +g) tr➯♥ ❇ n n n n s❛♦ ❝❤♦ yn − γ ωn1 + δn zn1 − zn ≤ yn − ωn γ ηn2 ≤ ϕ (xn , yn ) + ; γ ✭✷✳✹✹✮ ✈➔ s❛♦ ❝❤♦ ❤➔♠ (z, ω) → yn − ω γ + δgph (F +g) (z, ω) + δn z − zn + (δn + 1)ηn (z, ω) − (zn1 , ωn1 ) ✭✷✳✹✺✮ ¯ (xn , ηn ) × Y t↕✐ (z , ω ) ❚❛ t❤➜② r➡♥❣ ❝→❝ ❤➔♠ ✤↕t ❝ü❝ t✐➸✉ tr➯♥ ❇ n n (z, ω) → yn − ω γ , (z, ω) → z − zn ✈➔ (z, ω) → (z, ω) − (zn1 , ωn1 ) ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ q✉❛♥❤ (zn1 , ωn1 )✱ sû ❞ư♥❣ q✉② t➢❝ tê♥❣ ♠í t❛ ❝â t❤➸ ❝❤å♥ ωn2 ∈ ❇Y ωn1 , ηn ; zn3 , ωn3 ωn2∗ ∈ ∂( yn − γ )(ωn2 ); ∈ ❇X×Y zn3∗ , −ωn3∗ zn1 , ωn1 , ηn ∩ ❣♣❤ (F + g); ∈ N ❣♣❤ (F + g), zn3 , ωn3 download by : skknchat@gmail.com ✷✾ t❤ä❛ ♠➣♥ ωn3∗ = ωn2∗ + (δn + 2)ηn ωn4∗ , ωn2∗ − ωn3∗ < (δn + 2)ηn , ✈➔ zn3∗ ≤ δn + (δn + 2)ηn ❑❤✐ yn − ωn2 ≥ yn − ωn − ωn2 − ωn ≥ ϕ(xn , yn )(1 − δn ) − 2ηn > 0✱ γ−1 ωn2∗ ∈ ∂( yn − γ )(ωn2 ) = γ yn − ωn2 ✈➔ ✈➻ t❤➳ ∂( yn − ) ωn2 , γ−1 ωn2∗ = γ yn − ωn2 t2∗ n 2 ✈ỵ✐ t2∗ = ✈➔ t2∗ n n , ωn − yn = yn − ωn ❉♦ ✤â✱ tø ✭✷✳✹✻✮ ❦➨♦ t❤❡♦ ωn3∗ ≥ ωn2∗ − (δn + 2)ηn γ−1 = γ yn − ωn2 t2∗ − (δn + 2)ηn n γ−1 = γ yn − ωn2 − (δn + 2)ηn > 0; ωn3∗ ≤ ωn2∗ + (δn + 2)ηn γ−1 = γ yn − ωn2 t2∗ + (δn + 2)ηn n γ−1 = γ yn − ωn2 + (δn + 2)ηn ❑❤✐ tn ≥ (xn , yn ) − (¯ x, y¯ + g(¯ x)) − zn3 − xn ≥ (xn , yn ) − (¯ x, y¯ + g(¯ x)) − ηn2 5ηn − > 4 download by : skknchat@gmail.com ✭✷✳✹✻✮ ✸✵ ◆❣❤➽❛ ❧➔ t❛ ♣❤↔✐ ✤➦t tn = x, y¯ + g(¯ x)) ; zn3 , yn − (¯ x) zn3 − x¯, yn − y¯ − g(¯ (un , ) = ; ζn = tn ✈➔ yn∗ = ωn3∗ ; ωn3∗ ωn3 − yn x∗n = , 1/γ tn ✭✷✳✹✼✮ zn3∗ ωn3∗ x, y¯ + g(¯ x)) ✈ỵ✐ ♥ ✤õ ❧ỵ♥✱ (tn ) ↓ ❦❤✐ n → ∞✳ x, y¯ + g(¯ x)) ✈➔ (zn3 , yn ) = (¯ ❑❤✐ (zn3 , yn ) → (¯ ❑❤✐ ϕ(xn , yn )(1 − δn ) ≤ d(yn , (F + g)(¯ x + tn un )) ≤ t1/γ ζn n ≤ yn − ωn1 + ηn ≤ ϕ(xn , yn ) + t❛ ❝â ζn ≤ ❑❤✐ ηn2 + ηn , ηn2 ϕ(xn , yn ) + + ηn 5ηn ηn2 − (xn , yn ) − (¯ x, y¯ + g(¯ x)) − 4 1/γ ηnγ ϕγ (xn , yn ) → ❝ơ♥❣ ♥❤÷ → 0✱ t❛ ✤÷đ❝ (xn , yn ) − (¯ x, y¯ + g(¯ x)) (xn , yn ) − (¯ x, y¯ + g(¯ x)) lim ζn = n→∞ ✭✷✳✹✽✮ ❚✐➳♣ t❤❡♦✱ ❜➡♥❣ ❝→❝❤ sû ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ tê♥❣ ❝❤♦ ✤è✐ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t ❝õ❛ F + g ✱ t❛ ❝â ∗ x∗n ∈ D∗ (F + g) x¯ + tn un , y¯ + g(¯ x) + tn + t1/γ n ζn (yn ) ∗ ∗ x + tn un )yn∗ = D∗ F x¯ + tn un , y¯ + g(¯ x) − g(¯ x + tn un ) + tn + t1/γ n ζn (yn ) + Dg (¯ ▼➦t ❦❤→❝✱ x + tn un , y¯ zn∗ :=x∗n − Dg ∗ (¯ x + tn un )yn∗ ∈ D∗ F (¯ ∗ + g(¯ x) − g(¯ x + tn un ) + tn + t1/γ n ζn )(yn ) download by : skknchat@gmail.com ✭✷✳✹✾✮ ✸✶ ❚ø ✭✷✳✹✼✮ ✈➔ ❝→❝ ❦➳t q✉↔ tø ✭✷✳✹✷✮✱ t❛ ✤÷đ❝ lim ζn = 0; g(¯ x + tn un ) − g(¯ x) = tn Dg(¯ x)(un ) + 0(tn ) n→∞ ❚❛ s✉② r❛ un , g(¯ x) − g(¯ x + tn un ) + tn ∈ ❝♦♥❡ ❇((u, v), εn ), ✭✷✳✺✵✮ ✈➔ tn un , g(¯ x) − g(¯ x + tn un ) + tn + t1/γ n ζn ∈ ❝♦♥❡ ❇((u, v), εn ), ✈ỵ✐ ♠ët sè ❞➣② {εn } ↓ ❱➻ t❤➳✱ t❤❡♦ ✭✷✳✹✶✮ t❛ ❝â ✭✈ỵ✐ ♥ ✤õ ❧ỵ♥✮ x + tn un ) ≥ (1 − c) zn∗ x∗n = zn∗ + Dg ∗ (¯ x + tn un )yn∗ ≥ zn∗ − Dg ∗ (¯ ❚ø ✭✷✳✹✷✮✱ t❛ s✉② r❛ r➡♥❣ tn(γ−1)/γ ζn γ−1 x∗n = t(γ−1)/γ ζ n γ−1 zn3∗ / ωn3∗ (γ−1)/γ ζn γ−1 (δn + (δn + 2)ηn ) γ yn − ωn2 γ−1 − (δn + 2)ηn yn − ωn3 γ−1 (δn + (δn + 2)ηn ) ≤ γ( yn − ωn2 − 2ηn )γ−1 − (δn + 2)ηn ≤ tn ✭✷✳✺✶✮ ❚❛ t❤➜② r➡♥❣ yn − ωn2 ≥ yn − ωn − ωn − ωn2 ≥ ϕ(xn , yn ) − δn − 2ηn > ❑❤✐ ✤â✱ lim n→∞ ηn ηn ≤ lim = n→∞ yn − ωn ϕ(xn , yn ) − δn − 2ηn ❱➻ t❤➳✱ tø ✭✷✳✺✶✮ t❛ s✉② r❛ lim t(γ−1)/γ ζn n n→∞ γ−1 ζn zn∗ ≤ lim (1 − c)−1 t(γ−1)/γ n n→∞ γ−1 x∗n = ✭✷✳✺✷✮ ❚✐➳♣ t❤❡♦✱ t❛ ❝â yn∗ , ζn y ∗ , ω − yn = n 3n ζn ω n − yn ωn3∗ , ωn3 − yn ≥ ωn3∗ ωn3 − yn ωn2∗ , ωn2 − yn − ωn2∗ − ωn3∗ ωn2 − yn − ωn2∗ ωn2 − ω ≥ ωn2∗ ωn2 − yn + ωn2∗ ωn3 − ωn2 + ωn3 − ωn2 ωn2∗ − ωn3∗ + ωn2∗ − ωn3∗ ωn2 − yn γ ωn2 − yn γ − (δn + 2)ηn (ϕ(xn , yn ) + δn + 2ηn ) − 2ηn γ yn − ωn2 γ−1 ≥ γ ωn2 − yn γ + 2ηn γ yn − ωn2 γ−1 + 2ηn2 (δn + 2) + (δn + 2)ηn (ϕ(xn , yn ) + δn + 2ηn ) download by : skknchat@gmail.com ✸✷ Ð ✤➙②✱ t❛ ❞ò♥❣ ❝→❝ ✤➥♥❣ t❤ù❝ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ωn2∗ , ωn2 − yn = γ ωn2 − yn γ ωn2∗ = γ ωn2 − yn ; (γ−1) yn − ωn2 ≥ yn − ωn − ωn − ωn2 ≥ ϕ(xn , yn )(1 − δn ) − 2ηn > ❱➻ ✈➟②✱ ❦❤✐ ηn ηn ≤ → 0, yn − ωn ϕ(xn , yn )(1 − δn ) − 2ηn t❛ ✤÷đ❝ lim n→∞ ❦❤✐ n → ∞, yn∗ , ζn = ζn ✭✷✳✺✸✮ ❈→❝ q✉❛♥ ❤➺ ✭✷✳✹✽✮✱✭✷✳✹✾✮✱✭✷✳✺✵✮✱✭✷✳✺✷✮✱✭✷✳✺✸✮ ❝❤ù♥❣ tä (0, 0) ∈ ❈rγ F ((¯ x, y¯), (u, v)) ♥➯♥ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ▼➺♥❤ ✤➲ ✷✳✶✼✳ ❈❤♦ ❳✱❨ ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ (u, v) ∈ X × Y, γ ∈ (0, 1] ✭✐✮ ◆➳✉ F : X Y tr ỗ õ ỗ t õ ởt t ỗ õ q tr r t ữợ t ( x, y) F t ữợ (u, v) t❤➻ (0, 0) ∈ / ❈rγ F ((x, y), (u, v)); ✭✐✐✮ ◆➳✉ F : X ⇒ Y ❧➔ ❤➔♠ tr õ q tr t ữợ t ( x, y) F t ữợ (u, v) t❤➻ (0, 0) ∈ / ❈r F ((x, y), (u, v)) ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❳➨t ❞➣② {tn } ↓ 0; {εn } ↓ 0; (un , ) ∈ ❝♦♥❡ ❇((u, v), εn ) ✈ỵ✐ (un , ) = 1; ζn ∈ Y ; yn∗ ∈ SY ∗ ✈➔ x∗n ∈ X ∗ s❛♦ ❝❤♦ ζn → 0; yn∗ , ζn → ❦❤✐ n → ∞, ζn ∗ x∗n ∈ D∗ F x¯ + tn un , y¯ + tn + t1/γ n ζn (yn ) ❱➻ F ỗ xn , x x tn un − yn∗ , y − y¯ − tn − t1/γ n ζn ≤ ∀(x, y) ∈ ❣♣❤ F ✭✷✳✺✹✮ ❱➻ F ❧➔ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ❍☎ ♦❧❞❡r t ữợ t ( x, y) F t ữợ (u, v) ợ số > ✈➔ d(¯ y + tn , F (¯ x + tn un )) ≤ ζn → 0, (tn (un , ) )1/γ download by : skknchat@gmail.com ✸✸ ♥➯♥ ✈ỵ✐ ♥ ✤õ ❧ỵ♥✱ t❛ ❝â d(¯ x + tn un , F −1 (¯ y + tn )) ≤ τ d(¯ y + tn , F (¯ x + tn un ))γ ≤ τ y¯ + tn − y¯ − tn − t1/γ ζn γ = τ tn ζn γ ❉♦ ✈➟②✱ t❛ ❝â t❤➸ t➻♠ xn ∈ F −1 (¯ y + tn ) s❛♦ ❝❤♦ xn − x¯ − tn un ≤ τ (1 + tn )tn ζn γ ❚ø ✭✷✳✺✹✮ t❛ ✤÷đ❝ x∗n τ (1 + tn )tn ζn γ ≥ x∗n yn∗ , ζn xn − x¯ − tn un ≥ x∗n , x¯ + tn un − xn ≥ t1/γ n ❱➻ t❤➳✱ lim inf x∗n t(γ−1)/γ ζn n γ−1 n→∞ ≥ lim τ −1 (1 + tn )−1 n→∞ yn∗ , ζn = τ −1 ζn ❇➜t ✤➥♥❣ t❤ù❝ tr➯♥✱ ❝❤ù♥❣ tä (0, 0) ∈ / ❈rγ F ((¯ x, y¯), (u, v)) sỷ q tr t ữợ t ( x, y) F t ữợ (u, v) ❈❤♦ ❝→❝ ❞➣② {tn } ↓ 0; {εn } ↓ 0; (un , ) ∈ ❝♦♥❡ ❇((u, v), εn ) ✈ỵ✐ (un , ) = 1; ζn ∈ Y ; yn∗ ∈ SY ∗ ✈➔ x∗n ∈ X ∗ s❛♦ ❝❤♦ ζn → 0; yn∗ , ζn → ❦❤✐ n → ∞, ζn x∗n ∈ D∗ F (¯ x + tn un , y¯ + tn + tn n )(yn ) õ tỗ t ♠ët ❞➣② sè t❤ü❝ ❦❤ỉ♥❣ ➙♠ {δn } ✈ỵ✐ δn ∈ (0, tn ) s❛♦ ❝❤♦ x∗n , x − x¯ − tn un − yn∗ , y − y¯ − tn − tn ζn ≤ εn (x, y) − (¯ x + tn un , y¯ + tn + tn ζn ) , ∀(x, y) ∈ ❣♣❤ F ✈ỵ✐ (x, y) − (¯ x + tn un , y¯ + tn + tn ζn ≤ δn ✭✷✳✺✺✮ ✣➦t yn = y¯ + tn un + (tn − δn )ζn , (¯ x + tn un , yn ) −→ (¯ x, y¯) ❉♦ ✤â t❤❡♦ t q (u,v) tr t ữợ F t ( x, y) t ữợ (u, v) ợ ởt số số > ợ ợ tỗ t↕✐ xn ∈ F −1 (yn ) s❛♦ ❝❤♦ xn − x¯ − tn un ≤ τ (1 + tn )d(yn , F (¯ x + tn un )) ≤ τ (1 + tn ) yn − y¯ − tn − tn ζn = τ (1 + tn )δn ζn download by : skknchat@gmail.com ✸✹ ❉♦ ✈➟②✱ (xn , yn ) − (¯ x + tn un , y¯ + tn + tn ζn ) < δn ✈ỵ✐ ♥ ✤õ ❧ỵ♥ ✈➔ ❞ü❛ ✈➔♦ ✭✷✳✺✺✮✱ t❛ s✉② r❛ x∗n τ (1 + tn )δn ζn ≥ δn yn∗ , ζn , ❜➜t ✤➥♥❣ t❤ù❝ ♥➔② ❦➨♦ t❤❡♦ lim inf x∗n ≥ τ −1 , n→∞ ✈➔ t❛ ❤♦➔♥ t❤➔♥❤ ❝❤ù♥❣ ♠✐♥❤✳ download by : skknchat@gmail.com ✸✺ ❈❤÷ì♥❣ ✸ Ù♥❣ ❞ư♥❣ ❝õ❛ t➼♥❤ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ❍☎♦❧❞❡r ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ù♥❣ ❞ö♥❣ ❝õ❛ t➼♥❤ ❝❤➼♥❤ q tr r t ữợ tr ố ữ t➼❝❤ ❜✐➳♥ ♣❤➙♥ ✭❳❡♠ ❍✉②♥❤ ❱❛♥ ◆❣❛✐✱ ◆❣✉②❡♥ ❍✉✉ ❚r♦♥✱ ▼✐❝❤❡❧ ❚❤❡r❛ ❬✹❪✮✳ ❱ỵ✐ X, Y ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ f : X × Y → R ✈➔ g : X → Y ❧➔ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝✳ ●✐↔ sỷ t ỗ õ r ❝❤÷ì♥❣ ♥➔②✱ t❛ ①➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ♣❤ư t❤✉ë❝ t❤❛♠ sè ❝â ❞↕♥❣ (Py ) f (x, y) s.t y ∈ g(x) − K, x∈X ✭✸✳✶✮ ♣❤ö t❤✉ë❝ ✈➔♦ ♠ët t❤❛♠ sè y ∈ Y ✳ ❚➟♣ ❦❤↔ t❤✐ ❝õ❛ (Py ) ✤÷đ❝ ❦➼ ❤✐➺✉ ❜ð✐ φ(y) := {x ∈ X : y ∈ g(x) − K} ✭✸✳✷✮ ợ y = tữỡ ự t (P0 ) ❦➼ ❤✐➺✉ ❜➡♥❣ (P) ✤÷đ❝ ①❡♠ ♥❤÷ ❧➔ ❜➔✐ t♦→♥ ❦❤æ♥❣ ♥❤✐➵✉✳ ❚❛ ✤➦t f (x) := f (x, 0), φ0 := φ(0) ✈➔ t❛ ❦➼ ❤✐➺✉ v(y) ❧➔ ❤➔♠ ❣✐→ trà tè✐ ÷✉ ❝õ❛ (Py ) ✈➔ S(y) ❧➔ t➟♣ ❧✐➯♥ ❦➳t ❝õ❛ ❝→❝ ♥❣❤✐➺♠ tè✐ ÷✉✿ v(y) := inf f (x, y); x∈φ(y) S(y) := ❛r❣ f (x, y) x∈φ(y) download by : skknchat@gmail.com ✭✸✳✸✮ ✭✸✳✹✮ ✸✻ ◆❤➢❝ ❧↕✐ xε ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♥❣❤✐➺♠ ε − tè✐ ÷✉ ❝õ❛ (Py ) ♥➳✉ xε ∈ φ(y) ✈➔ f (xε , y) ≤ v(y) + ε ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ t❛ →♣ ❞ư♥❣ ❦❤→✐ ♥✐➺♠ t➼♥❤ ❝❤➼♥❤ q✉② tr t ữợ t t t ữợ tr tố ữ v(y) ú t ❣✐↔ sû f ✈➔ ❣ ❧➔ ❝→❝ →♥❤ ①↕ t❤✉ë❝ ợ C t ởt ữớ y(t) t ợ ữợ d Y õ ổ tự y(t) = td + ◦(t)✱ ✈ỵ✐ t ∈ R+ ❚❛ ữợ t x0 (0) ởt ữợ h X ữủ ữợ t t x0 ố ợ ữợ d Y ợ t ữớ y(t) = td + ◦(t) ✈ỵ✐ t ≥ tr♦♥❣ tỗ t r(t) = (t) tr s x0 + th + r(t) (y(t)) ú ỵ ởt ữợ t ố ợ d ∈ Y t↕✐ x0 ✱ t❤➻ Dg(x0 )h − d ∈ TK (g(x0 )) ✭✸✳✺✮ ❱ỵ✐ TK (g(x0 )) = {d ∈ Y : d(g(x0 ) + td, K) = ◦(t), t ≥ 0} ❧➔ ♥â♥ t✐➳♣ ❧✐➯♥ ✤è✐ ✈ỵ✐ t ỗ K t g(x0 ) ữủ t õ ỗ s x0 (0) ♥➳✉ ✭✸✳✺✮ ✤ó♥❣✱ ✈➔ ♥➳✉ t❤➯♠ ✈➔♦ G(x) := g(x) K q tr t ữợ (h, d) t (x0 , 0) t h ởt ữợ t ố ợ d ự sỷ tỗ t τ, ε > s❛♦ ❝❤♦ d(x, φ(y)) ≤ τ d(y, g(x) − K), ∀(x, y) ∈ ❇((x0 , 0), ε) ∩ [(x0 , 0) + ❝♦♥❡ ❇((h, d), ε)], d(y, g(x) − K) ≤ ε (x, y) − (x0 , 0) ✭✸✳✻✮ ▲➜② y(t) = td + ◦(t) ✈➔ ✤➦t x(t) = x0 + th❀ t❛ ❝â g(x0 ) + tDg(x0 )h − td + ◦(t) ∈ K, ❦❤✐ t ↓ ❑❤✐ g(x(t)) = g(x0 ) + tDg(x0 )h + ◦(t), t❛ ❝â g(x(t)) − td + ◦(t) ∈ K tø ✤➙② t❛ ✤÷đ❝ d(y(t), g(x(t)) − K) = ◦(t) download by : skknchat@gmail.com ✸✼ ❱➻ ✈➟②✱ ✈ỵ✐ t > ✤õ ♥❤ä✱ (x(t), y(t)) ∈ ❇((x0 , 0), ε) ∩ [(x0 , 0) + ❝♦♥❡ ❇((h, d), ε)], d(y(t), g(x(t)) − K) ≤ ε (x(t), y(t)) (x0 , 0) tỗ t x ¯(t) := x0 + th + ◦(t) ∈ φ(y(t)), ❦❤✐ t > 0✳ ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❇ê ✤➲ ✸✳✹✳ ●✐↔ sû r➡♥❣ ❨ ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉✳ ▲➜② x0 ∈ φ(0) ✈➔ d ∈ Y \ {0}, h ∈ X s❛♦ ❝❤♦ ❝â ✤÷đ❝ ✭✸✳✺✮✳ ◆➳✉ ● q tr t ữợ t (x0 , 0) t ữợ (h, d) õ t õ d ✐♥t{Dg(x0 )X − TK (g(x0 ))} ❈❤ù♥❣ ♠✐♥❤✳ ▲➜② τ > 0, ε ∈ (0, 1) s❛♦ ❝❤♦ ✭✸✳✻✮ ①↔② r❛✳ ▲➜② < δ < ✤à♥❤ d˜ ∈ ❇(d, δ) ❱➻ ❝â ✤÷đ❝ ✭✸✳✺✮ ♥➯♥ t❛ ❝â ✭✸✳✼✮ ε d ✈➔ ❝è g(x0 ) + tDg(x0 )h − td + ◦(t) ∈ K, ❦❤✐ t > ❱➻ ✈➟②✱ g(x0 + th) − td + ◦(t) ∈ K, ❦❤✐ t > ❍ì♥ ♥ú❛✱ t❛ ❝â δ ≤ ε d ε(δ + d˜ ) ε d˜ < ε d˜ ❦❤✐ δ < < ❱➟② ✈ỵ✐ t ✤õ ♥❤ä✱ 2(1 − δ/2) 2 ˜ g(x0 + th) − K) ≤ tδ + ◦(t) < εt( h + d˜ ) d(td, ˜ s❛♦ ❝❤♦ ❚❤❡♦ ✭✸✳✻✮✱ t❛ ❝❤å♥ x(t) ∈ φ(td) x0 + th − x(t) ≤ τ tδ + ◦(t) ✣➦t h(t) = x(t) − x0 ✱ t❛ ❝â t h − h(t) ≤ τ δ + ◦(t) t ˜ ❑❤✐ x(t) ∈ φ(td), td˜ ∈ g(x0 + h(t)) − K, download by : skknchat@gmail.com ✸✽ ✈➔ ✈➻ t❤➳✱ ◦(t) K − g(x0 ) d˜ ∈ Dg(x0 )(h(t)) + − t t ❑❤✐ ❨ ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉✱ t❛ ❝â t❤➸ ❝❤å♥ ♠ët ❞➣② (tn )n∈N ↓ s❛♦ ❝❤♦ ❞➣② (Dg(x0 )h(tn ))n∈N ❤ë✐ tö ✤➳♥ ♠ët sè ω ∈ Dg(x0 )X ✳ ❑❤✐ ✤â✱ ♥❤í ✈➔♦ ❦➳t q✉↔ ❇ê ✤➲ ✸✳✸ t❛ ✤÷đ❝ d˜ ∈ Dg(x0 )X − TK (g(x0 )), t❛ ❤♦➔♥ t❤➔♥❤ ❝❤ù♥❣ ♠✐♥❤ ❜ê ✤➲✳ ❚❛ ❦➼ ❤✐➺✉ L(x, λ, y) ✈➔ Λ(x0 ) ❧➛♥ ❧÷đt t÷ì♥❣ ù♥❣ ❧➔ ❤➔♠ ▲❛❣r❛♥❣❡ ❝õ❛ (Py ) ✈➔ t➟♣ ❝→❝ ♥❤➙♥ tû ▲❛❣r❛♥❣❡ ❝õ❛ ❜➔✐ t♦→♥ (P0 ) ✈ỵ✐ x0 ∈ S(0)✳ ❈❤➼♥❤ ①→❝ ❤ì♥✱ ♥➳✉ NK (g(x0 )) ❧➔ ♥â♥ ♣❤→♣ t✉②➳♥ t ỗ K t g(x0 ) t t õ L(x, λ, y) = f (x, y) + λ, g(x) − y , (x, λ) ∈ X × Y ∗ ; Λ(x0 ) = {λ ∈ NK (g(x0 )) : Dx L(x0 , , 0) = 0} ởt ữợ d✱ t❛ ①➨t t✉②➳♥ t➼♥❤ ❤♦→ ❝õ❛ ❇➔✐ t♦→♥ (Py ) s❛✉✿ (PLd ) Df (x0 , 0)(h, d) s❛♦ ❝❤♦ Dg(x0 )h − d ∈ TK (g(x0 )) h∈X ◗✉❛♥ s→t r➡♥❣ ✭✸✳✼✮ ❧➔ ✤✐➲✉ ❦✐➺♥ ❝❤✉➞♥ ❤♦→ r➔♥❣ ❜✉ë❝ ❘♦❜✐♥s♦♥ ❝❤♦ ❇➔✐ t♦→♥ (PLd ) ✈➔ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ❝õ❛ (PLd ) ❧➔ (DLd ) max Dy L(x0 , λ, 0)d λ∈Λ(x0 ) ❚ø ❇ê ✤➲ ✸✳✹ t❛ ✤÷đ❝ ✤↕t ✤÷đ❝ ❦➳t q✉↔ ✤è✐ ♥❣➝✉ ❝❤♦ ❇➔✐ t♦→♥ (DLd ) s❛✉✿ ❇ê ✤➲ ✸✳✺✳ ❈❤♦ x0 ∈ S(0) ✈➔ d ∈ Dg(x0 )X − Tk (g(x0 ))✳ ●✐↔ sû ❨ ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✈➔ G = g K q tr t ữợ t (x0 , 0) t ữợ (h, d) ởt số ữợ h X ợ Dg(x0 )hd TK (g(x0 )) ❑❤✐ ✤â ❦❤æ♥❣ ❝â ❦❤♦↔♥❣ ❝→❝❤ ✭♥♦ ❣❛♣✮ ❣✐ú❛ ❤❛✐ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ (PLd ) ✈➔ (DLd )✳ ❍ì♥ ♥ú❛✱ ❣✐→ trà tè✐ ÷✉ ❝❤✉♥❣ ❧➔ ❤ú✉ ❤↕♥✱ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ t➟♣ Λ(x0 ) ❧➔ ❦❤æ♥❣ ré♥❣❀ ✈➔ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ t➟♣ ❝→❝ ❣✐→ trà ♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛ (DLd ) ❧➔ t➟♣ ❝♦♠♣❛❝t✳ ✣à♥❤ ỵ s ữ r ởt t q q t r t ữợ tr tè✐ ÷✉ v(y)✳ download by : skknchat@gmail.com ✸✾ ✣à♥❤ ❧➼ ỳ ữợ d ∈ Y ✈ỵ✐ {d, −d} ⊆ Dg(x)X − TK (g(x)) ✈ỵ✐ ♠å✐ x ∈ S(x0 ) ●✐↔ sû ✭✐✮ ❤➔♠ ✤❛ trà G = g − K ❧➔ ❝❤➼♥❤ q✉② tr t ữợ ợ ữợ (0, d) (0, −d) t↕✐ ♠å✐ ✤✐➸♠ (x, 0) ✈ỵ✐ x ∈ S(0)❀ ✭✐✐✮ ✈ỵ✐ ❜➜t ❦ý ❞➣② yn = tn d + (t) ợ tn tỗ t ởt ◦(tn ) − ❝→❝ ♥❣❤✐➺♠ tè✐ ÷✉ (xn ) ❝õ❛ (Pyn ), ❤ë✐ tö ✤➳♥ ♠ët sè x0 ∈ S(0) ❑❤✐ ✤â✱ ❦➼ ❤✐➺✉ v− (0, d) ✈➔ v+ (0, d) ữợ tr t ữợ r v t t ữợ t❛ ❝â v− (0, d) ≥ inf inf Dy L(x, λ, y)d; v+ (0, d) ≤ inf sup Dy L(x, λ, y)d x∈S(0) λ∈Λ(x) x∈S(0) λ∈Λ(x) ✭✸✳✽✮ ❚ø ✤â s✉② r❛✱ ♥➳✉ Λ(x) ❧➔ ♠ët ♥❤➙♥ tû {λ(x)} ✈ỵ✐ ♠å✐ x S(0) t t ữợ r ợ ữợ v(y) t tỗ t v (0, d) = inf Dy L(x, λ(x), 0)d x∈S(0) ❈❤ù♥❣ x S(0) ữợ h X s❛♦ ❝❤♦ Dg(x)h − d ∈ TK (g(x)) ◆➳✉ G q tr t ữợ t t (x, 0) ợ ữợ (0, d) t G ụ q tr t ữợ t (x, 0) ợ ữợ (h, d) h ữợ t ố ✈ỵ✐ ❞✱ ✐✳❡✳✱ x + th + ◦(t) ∈ φ(td), t ↓ ❱➻ t❤➳✱ v(td) ≤ f (x + th + ◦(t), td) = f (x, 0) + tDf (x, 0)(h, d) + ◦(t) ✈➔ ❞♦ ✤â✱ lim sup t↓0 v(td) − v(0) ≤ Df (x, 0)(h, d) t x S(0) tũ ỵ h ởt ✤✐➸♠ ❦❤↔ t❤✐ ❜➜t ❦ý ❝õ❛ (PLd )✱ ❜➜t ✤➥♥❣ t❤ù❝ t❤ù ❤❛✐ tr♦♥❣ ✭✸✳✽✮ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ download by : skknchat@gmail.com ✹✵ ❱ỵ✐ ❜➜t ✤➥♥❣ t❤ù❝ ♥❤➜t tr♦♥❣ ✭✸✳✽✮✱ ❝❤♦ tn ↓ 0; yn = tn d + ◦(tn ) ✈➔ (xn ) ❧➔ ♠ët ❞➣② ◦(tn ) − ❝→❝ ♥❣❤✐➺♠ ❝õ❛ (Pyn ) ♥❤÷ tr♦♥❣ ✭✐✐✮ ♠➔ ♥â ❤ë✐ tö ✤➳♥ x0 ∈ S(0)✳ ▲➜② h ∈ X s❛♦ ❝❤♦ Dg(x0 )h + d ∈ TK (g(x0 )); t÷ì♥❣ ✤÷ì♥❣✱ g(x0 ) + tDg(x0 )h + td + ◦(t) ∈ K, t ↓ ❑❤✐ g(xn ) − yn K ụ ữ K ỗ ợ ❜➜t ❦➻ t > 0✱ ❦❤✐ n ✤õ ❧ỵ♥ ✈➔ s❛♦ ❝❤♦ tn < 1✳ ❚❛ ❝â t (1 − tn /t)[g(xn ) − yn ] + tn /t[g(x0 ) + tDg(x0 )h + td + ◦(t)] = g(xn ) − yn + tn /t[g(x0 ) − g(xn )] + tn Dg(x0 )h + tn d + tn ◦ (t)/t ∈ K ❱➻ t❤➳✱ ✈ỵ✐ ε > 0✱ ❦❤✐ ♥ ✤õ ❧ỵ♥✱ t❛ ❝â d(g(xn ) + tn Dg(x0 )h, K) ≤ tn ε ❑❤✐ g(xn + tn h) = g(xn ) + tn Dg(x0 )h + ◦(tn ), t❛ ✤÷đ❝✱ d(g(xn + tn h), K) ≤ tn ε + ◦(tn ) ❱➻ G ❧➔ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ♥➯♥ t❛ t➻♠ ✤÷đ❝ ♠ët sè zn ∈ φ0 = G−1 (0) s❛♦ ❝❤♦ xn + tn h − zn = ◦(tn ε) ❉♦ ✤â✱ f (xn , yn ) − f (zn , 0) + ◦(tn ) v(yn ) − v(0) ≥ tn tn f (xn , yn ) − f (xn + tn h, 0) − ◦(tn ε) = tn ◦(tn ε) = −Df (x0 , 0)(h − d) − tn ❈✉è✐ ❝ị♥❣✱ t❤❡♦ ❇ê ✤➲ ✸✳✺✱ ❦❤ỉ♥❣ ❝â ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ ❤❛✐ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ (PL−d ) (DLd ) > tũ ỵ t s✉② r❛ lim inf n→∞ v(yn ) − v(0) ≥ inf Dy L(x, λ, 0)d λ∈Λ(x0 ) tn ❚ø ✤➙② t❛ s✉② r❛ ✤÷đ❝ ❜➜t ✤➥♥❣ t❤ù❝ t❤ù ♥❤➜t ❝õ❛ ✭✸✳✽✮✳ download by : skknchat@gmail.com ✹✶ ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ♠ët sè ✈➜♥ ✤➲ s❛✉✿ ✲ ❚r➻♥❤ ❜➔② ❧↕✐ ❤➺ t❤è♥❣ ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì sð ❧✐➯♥ q✉❛♥ ✤➳♥ t➼♥❤ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ✤➸ tø ✤â tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ t➼♥❤ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ❍☎ ♦❧❞❡r t❤❡♦ ữợ r trữ t q tr r t ữợ trữ q ✤ë ❞è❝ ♠↕♥❤✱ t➼♥❤ ê♥ ✤à♥❤ ♥❤✐➵✉ ✈➔ t➼♥❤ ê♥ t q tr r t ữợ ữợ ❝→❝ ✤✐➲✉ ❦✐➺♥ t✐➳♣ t✉②➳♥ − ✤è✐ ✤↕♦ ❤➔♠✳ ✲ ❚r➻♥❤ ❜➔② ù♥❣ ❞ö♥❣ ❝õ❛ t➼♥❤ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ❍☎♦❧❞❡r tr♦♥❣ ❝→❝ ❜➔✐ t♦→♥ ❚è✐ ÷✉ ✈➔ ●✐↔✐ t➼❝❤ ❜✐➳♥ ♣❤➙♥✳ ❇➯♥ ❝↕♥❤ ♥❤ú♥❣ ❦➳t q✉↔ ✤↕t ✤÷đ❝✱ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳ ✈➔ t❤✐➳✉ sât✳ t ữủ sỹ ỗ qỵ t ❝ỉ ✈➔ ❝→❝ ❜↕♥ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ỡ ữủ t ợ sỹ q t ❣✐ó♣ ✤ï ✈➔ ❝❤➾ ❜↔♦ t➟♥ t➻♥❤ ❝õ❛ t❤➛② ❚❙✳ ◆❣✉②➵♥ ❍ú✉ ❚rå♥✳ ❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ ♥❤➜t ✤➳♥ ❚❤➛②✳ ❚æ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❇❛♥ ●✐→♠ ❍✐➺✉ tr÷í♥❣ ỡ ỏ ũ qỵ t❤➛② ❝ỉ ❣✐→♦ tr♦♥❣ ❑❤♦❛ ❚♦→♥ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï ✈➔ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ✤➸ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✳ download by : skknchat@gmail.com ✹✷ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ❋r❛♥❦♦✇s❦❛✱ ❍✳✱◗✉✐♥❝❛♠♣♦✐①✱ ▼✳✿ ❍☎ ♦❧❞❡r ♠❡tr✐❝ r❡❣✉❧❛r✐t② ♦❢ s❡t✲✈❛❧✉❡❞ ♠❛♣s✳ ▼❛t❤✳ Pr♦❣r❛♠✳ ✶✸✷✭✶✲✷✱ ❙❡r✳❆✮✱ ✸✸✸✲✸✺✹ ✭✷✵✶✷✮✳ ❬✷❪ ■♦❢❢❡✱ ❆✳❉✳✿ ◆♦♥❧✐♥❡❛r r❡❣✉❧❛r✐t② ♠♦❞❡❧s✳ ▼❛t❤✳ Pr♦❣r❛♠✳ ✶✸✾✭✶✲✷✮✱ ✷✷✸✲✷✹✷ ✭✷✵✶✸✮ ❬✸❪ ■♦❢❢❡✱ ❆✳✿ ▼❡tr✐❝ r❡❣✉❧❛r✐t②✿ ❚❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✿ ❛ s✉r✈❡②✳ ■♥ ♣r❡♣❛r❛t✐♦♥ ✭✷✵✶✺✮✳ ❬✹❪ ❍✉②♥❤ ❱❛♥ ◆❣❛✐✱ ◆❣✉②❡♥ ❍✉✉ ❚r♦♥✱ ▼✐❝❤❡❧ ❚❤❡r❛✱ ❉✐r❡❝t✐♦♥❛❧ ❍☎♦❧❞❡r ♠❡tr✐❝ r❡❣✲ ✉❧❛r✐t②✱ ❏✳ ❖♣t✐♠ ❚❤❡♦r② ❆♣♣❧✱ ❉❡❝❡♠❜❡r ✷✵✶✻✱ ❱♦❧✉♠❡ ✶✼✶✱ ■ss✉❡ ✸✱ ✼✽✺✕✽✶✾✳ download by : skknchat@gmail.com ... , (¯ x + tn un , yn ) −→ (¯ x, y¯) ❉♦ ✤â t❤❡♦ t q (u,v) tr t ữợ F t ( x, y) t ữợ (u, v) ợ ởt số số > ợ ợ tỗ t↕✐ xn ∈ F −1 (yn ) s❛♦ ❝❤♦ xn − x¯ − tn un ≤ τ (1 + tn )d(yn , F (¯ x + tn un ))... ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ♥❤✐➲✉ ♥❤→♥❤ ❝õ❛ t♦→♥ ❤å❝ ♥❤÷ tè✐ ÷✉✱ ❝→❝ ❦➳t ❧✉➟♥ ❦❤↔ ✈✐✱ ỵ tt ữỡ số tr ♥❤✐➲✉ ❜➔✐ t♦→♥ ●✐↔✐ t➼❝❤✳ ❍✐➺♥ ♥❛②✱ ❦❤→✐ ♥✐➺♠ ❝❤➼♥❤ q✉② ♠❡tr✐❝ ✤❛♥❣ ✤÷đ❝ sü q✉❛♥ t➙♠ ♥❣❤✐➯♥... tr➻♥❤ ❜➔② ♣❤✐➯♥ ❜↔♥ t q tr r t ữợ t ❤✐➸✉ ❝→❝ ✤➦❝ tr÷♥❣ ❝õ❛ t➼♥❤ ❝❤➼♥❤ q✉② ♠❡tr✐❝ t❤❡♦ ữợ q ởt số ổ tr ✈ü❝ ♥➔②✳ ▲✉➟♥ ✈➠♥ s➩ ❤➺ t❤è♥❣ ❤â❛ ✈➔ ❝❤✐ t✐➳t ❤â❛ ♥❤ú♥❣ ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ ♥❤ú♥❣ ❦➳t q✉↔
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