i BỘ GIÁO DỤC VÀ ĐÀO TẠO TRƯỜNG ĐẠI HỌC KINH TẾ TP.HCM ĐỀ TÀI NGHIÊN CỨU KHOA HỌC CẤP TRƯỜNG CÁC ĐIỀU KIỆN TỐI ƯU CẤP HAI VỚI HIỆN TƯỢNG ENVELOPE-LIKE TRONG CÁC BÀI TỐN QUY HOẠCH ĐA MỤC TIÊU KHƠNG TRƠN Mã số: CS – 2013 - 37 Chủ nhiệm: TS Nguyễn Đình tuấn Tp Hồ Chí Minh 12 – 2013 LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ▼Ö❈ ệ ệ ệ ữỡ ỵ ✤➲ t➔✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✸ ✷✳ ▼ö❝ t✐➯✉ ✈➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✹ ✸✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✹ ✹✳ ❑➳t ❝➜✉ ❝õ❛ ✤➲ t➔✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✹ ❈❤÷ì♥❣ ✶✿ ▼ët sè ❝ỉ♥❣ ❝ư tr♦♥❣ ❣✐↔✐ t➼❝❤ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ ❦❤ỉ♥❣ trì♥ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✺ ❈❤÷ì♥❣ lờ ổ ữợ ụ ữ ✈❡❝tì ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✼ ❈❤÷ì♥❣ ✸✿ t ữợ tr ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳✳✳✳✶✶ ❈❤÷ì♥❣ ✹✿ ❈→❝ ✤✐➲✉ tố ữ ợ tữủ ❈❤÷ì♥❣ ✺✿ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ ❝➜♣ ❤❛✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✷✺ t ữợ ự rở t ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✷✾ ❈→❝ ❜➔✐ ❜→♦ ❦❤♦❛ ❤å❝ ❧✐➯♥ q✉❛♥ trü❝ t✐➳♣ ✤➳♥ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✸✶ ✶ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❈❤÷ì♥❣ ♠ð ✤➛✉ ỵ t r q t ❤å❝✱ ✈➔ tê♥❣ q✉→t ❤ì♥ tr♦♥❣ tè✐ ÷✉ ❤â❛✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐ ❝❤✐➳♠ ♠ët ✈à tr➼ q✉❛♥ trå♥❣✱ ✈➻ ♥â ❝✉♥❣ ❝➜♣ t❤æ♥❣ t✐♥ t❤➯♠ q✉❛♥ tr♦♥❣ ❝❤♦ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ♠ët✳ ✣➸ ✤→♣ ù♥❣ ❝❤♦ sü ♣❤➙♥ ❧♦↕✐ ù♥❣ ❞ö♥❣ t❤ü❝ t➳✱ ❝→❝ ❜➔✐ t♦→♥ tè✐ ÷✉ ✤÷đ❝ ①❡♠ ①➨t✱ ✈➔ ❞♦ ✤â ❝→❝ ❝ỉ♥❣ ❝ư ✈➔ ❦ÿ t❤✉➟t ♥❣❤✐➯♥ ❝ù✉✱ ♥❣➔② ❝➔♥❣ trð ♥➯♥ ♣❤ù❝ t↕♣ ❤ì♥✳ ❚✉② ♥❤✐➯♥ ❝❤ó♥❣ t❛ ❝â t❤➸ ♥❤➟♥ t❤➜② r➡♥❣✱ tr♦♥❣ ✤❛ sè ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❧✐➯♥ q✉❛♥ ✤➣ ❝â✱ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❝→❝ ❦➳t q✉↔ ✈➲ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐ ❝â t❤➸ ✤÷đ❝ ❦❤➥♥❣ ✤à♥❤ ♠ët ❝→❝❤ t÷ì♥❣ tü ♥❤÷ tr♦♥❣ ❦➳t q✉↔ ❝ê ✤✐➸♥ ❧➔ ✤↕♦ ❤➔♠ ❝➜♣ ❤❛✐ ❝õ❛ ❝→❝ ❤➔♠ ♠ö❝ t✐➯✉ ✭❤♦➦❝ ❝→❝ ❤➔♠ ▲❛❣r❛♥❣❡ tr♦♥❣ ❝→❝ ❜➔✐ t♦→♥ ❝â r➔♥❣ ❜✉ë❝✮ t↕✐ ❝→❝ ✤✐➸♠ ❝ü❝ t✐➸✉ ❧➔ ❦❤æ♥❣ ➙♠✳ ❑❛✇❛s❛❦✐ ❬✶✼❪ ❧➔ ♥❤➔ ♥❣❤✐➯♥ ❝ù✉ ✤➛✉ t✐➯♥ ❝❤♦ t❤➜② r➡♥❣ ❝→❝ ✤↕♦ ❤➔♠ t❤❡♦ ữợ r õ t t ỹ t t ữợ ❤➔♠ ❦➳t ❤đ♣ ❜ð✐ ❤➔♠ ♠ư❝ t✐➯✉ ✈➔ ❝→❝ r➔♥❣ ❜✉ë❝ ♥➡♠ tr➯♥ ♣❤➛♥ ✤➦❝ ❜✐➺t ❝õ❛ ❜✐➯♥ ❝õ❛ ♥â♥ ❤đ♣ ➙♠ tr♦♥❣ t➼❝❤ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ↔♥❤✳ ➷♥❣ ➜② ❣å✐ ❤✐➺♥ t÷đ♥❣ ♥➔② ❧➔ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡✳ ❈→❝ ❦➳t q✉↔ ❝õ❛ ❑❛✇❛s❛❦✐ ✤➣ ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ ♥❣❤✐➯♥ ❝ù✉ ♣❤→t tr✐➸♥ tr♦♥❣ ❬✻✱ ✽✱ ✷✺✱ ✷✻❪✱ ❧✉æ♥ ❧✉æ♥ ①❡♠ ①➨t t q ổ ữợ tở ợ C ✱ ❣✐è♥❣ ♥❤÷ tr♦♥❣ ❬✶✼❪✳ ❚r♦♥❣ q✉② ❤♦↕❝❤ ✤❛ ♠✉❝ t✐➯✉✱ ❝→❝ ❦➳t q✉↔ ✤➛✉ t✐➯♥ t❤✉ë❝ ❦✐➸✉ ♥➔② ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❬✶✹✱ ✶✺❪ ❝ơ♥❣ ①➨t ❝❤♦ ❝→❝ trữớ ủ trỡ ố ợ q t ❦❤ỉ♥❣ trì♥✱ ●✉t✐➨rr❡③✲❏✐♠➨♥❡③✲◆♦✈♦ ❬✶✸❪ ✤➣ ❞ị♥❣ ❝→❝ ✤↕♦ ❤➔♠ t❤❡♦ ữợ r tr tt tố ữ ợ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡✳ ❍å ①❡♠ ①➨t ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ❋r➨❝❤❡t ♠➔ ✤↕♦ ❤➔♠ ❝õ❛ ♥â ❧➔ ❧✐➯♥ tö❝ ❤♦➦❝ ê♥ ✤à♥❤ t↕✐ ✤✐➸♠ ♥❣❤✐➯♥ ❝ù✉✳ ❚✉② ♥❤✐➯♥✱ ✈➝♥ ❝á♥ ♥❤✐➲✉ t→❝ ❣✐↔ ❝❤÷❛ ♥❤➟♥ r❛ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐✳ ✣✐➲✉ ♥➔② ❝â t❤➸ ❞➝♥ ✤➳♥ ♠ët sè s❛✐ ❧➛♠ ❦❤æ♥❣ ❜✐➳t✳ ❍ì♥ ♥ú❛✱ tr♦♥❣ ❝→❝ ❜➔✐ ❜→♦ ♥â✐ tr➯♥✱ ✤ỉ✐ ❦❤✐ ❦❤ỉ♥❣ ①→❝ ✤à♥❤ ✤÷đ❝ ❦❤✐ ♥➔♦ t❤➻ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣✲❧✐❦❡ ①↔② r❛ ✈➔ ❦❤✐ ♥➔♦ t❤➻ ❦❤æ♥❣✳ ❈→❝ q✉❛♥ s→t tr ỗ ự ❝ù✉ ✤➛✉ t✐➯♥ ❝õ❛ ❝❤ó♥❣ tỉ✐ tr♦♥❣ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ♥➔② ❧➔ ❧➔♠ rã ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐✳ ▼➦t ❦❤→❝✱ ♠ët ❝→❝❤ t✐➳♣ ❝➟♥ ❝❤➼♥❤ ❝❤♦ tè✐ ÷✉ ❦❤ỉ♥❣ trì♥ ❧➔ ✤➲ ①✉➜t ✈➔ →♣ ❞ö♥❣ ❝→❝ ✤↕♦ ❤➔♠ s✉② rë♥❣ t❤➼❝❤ ❤ñ♣ ✤➸ t❤❛② t❤➳ ✤↕♦ ❤➔♠ ●➙t❡❛✉① ✈➔ ❋r➨❝❤❡t ❝ê ổ tỗ t tt tè✐ ÷✉✳ ◆❤✐➲✉ ❧♦↕✐ ✤↕♦ ❤➔♠ ✤➣ ✤÷đ❝ ❞ị♥❣✱ ♠é✐ ❧♦↕✐ ✤➲✉ ❝â t❤✉➟♥ ❧ñ✐ r✐➯♥❣ tr♦♥❣ ♠ët sè t➼♥❤ ❤✉è♥❣ ❝ư t❤➸ ♥❤÷♥❣ ❦❤ỉ♥❣ t❤✉➟♥ ❧đ✐ ❝❤♦ t➜t ❝↔ ❝→❝ tr÷í♥❣ ❤đ♣✳ ●➛♥ ✤➙②✱ ❝→❝ ✤↕♦ ❤➔♠ ✤❛ trà ❝❤♦ ❤➔♠ ✈❡❝tì ✤ì♥ trà ✤➣ ✤÷đ❝ sû ❞ư♥❣ ❤✐➺✉ q✉↔ ✤➸ ❝✉♥❣ ❝➜♣ ❝→❝ q✉② t➢❝ ♥❤➙♥ tû tr♦♥❣ ❝→❝ q✉② ❤♦↕❝❤ ❦❤ỉ♥❣ trì♥✱ ①❡♠ ❬✺✱ ✶✵✱ ✶✶✱ ✶✸✱ ✶✾✱ ✷✸❪ ✭♥❤÷♥❣ tr♦♥❣ ❬✺✱ ✶✵✱ ✶✶✱ ✶✾✱ ✷✸❪✱ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ❦❤ỉ♥❣ ①↔② r❛✮✳ ❈→❝ q✉❛♥ s→t ♥➔② ❧➔ ỗ ự t t ự t❤ù ❤❛✐ ❝õ❛ ❝❤ó♥❣ tỉ✐ tr♦♥❣ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ t ữợ r ữủ t tr ũ ợ t ❝❤➜t l✲ê♥ ✤à♥❤ ✭✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❬✷✱ ✸✱ ✹✱ ✶✷❪✮ ✤➸ ✤↕t ✤÷đ❝ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐ ♠ỵ✐ ❝↔✐ t❤✐➺♥ ✈➔ ♠ð rë♥❣ ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❣➛♥ ✤➙②✳ ❱➻ ❣✐→ trà ❝õ❛ ✤↕♦ t ữợ r t ởt t ợ ỡ tr t ữợ ❝➜♣ ❤❛✐ ❉✐♥✐ ✈➔ ♣❛r❛❜♦❧✐❝✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❝õ❛ ❝❤ó♥❣ tỉ✐ ♠↕♥❤ ❤ì♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tr♦♥❣ ❬✶✸❪✳ ❍ì♥ ♥ú❛✱ ❝❤ó♥❣ tỉ✐ ♥ỵ✐ ❧ä♥❣ ✸ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❝→❝ ❣✐↔ t❤✐➳t ❝❤➼♥❤ ✤➦t r❛ tr♦♥❣ ❬✶✸❪✿ t❤❛② t❤➳ ❧➛♥ ❧÷đt t➼♥❤ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ✈➔ ê♥ ✤à♥❤ ❜ð✐ t➼♥❤ ❦❤↔ ✈✐ ❝❤➦t ✈➔ l✲ê♥ ✤à♥❤✳ ✷✳ ▼ö❝ t✐➯✉ ✈➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉✳ ❈❤ó♥❣ tỉ✐ ①❡♠ ①➨t ❜➔✐ t♦→♥ q✉② ❤♦↕❝❤ ✤❛ ♠ö❝ t✐➯✉ s❛✉ ✤➙②✳ ❈❤♦ ❝→❝ ❤➔♠ f : Rn → Rm ✱ g : Rn → Rp ✈➔ h : Rn Rr C õ ỗ õ tr Rm K t ỗ tr Rp t ữợ sỹ t ú tổ ❧➔ ✭P✮ ♠✐♥ f (x)✱ s❛♦ ❝❤♦ g(x) ∈ −K ✱ h(x) = 0✳ ◆➳✉ K = Rn+ ✱ t❤➻ r➔♥❣ ❜✉ë❝ g(x) ∈ −K trð t❤➔♥❤ r➔♥❣ ❜✉ë❝ ❜➜t ✤➥♥❣ t❤ù❝ t❤ỉ♥❣ t❤÷í♥❣✳ ❈❤ó♥❣ tỉ✐ ❞ị♥❣ ❝→❝ ✤↕♦ ❤➔♠ t ữợ tr r ữợ tt ❦❤↔ ✈✐ ❝❤➦t ✭tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥✮ ❤❛② l✲ê♥ ✤à♥❤ ✭tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ✮ ✤➸ t❤✐➳t ❧➟♣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐ ợ ợ t t ữủ ró ỡ ❜➔✐ t♦→♥ q✉② ❤♦↕❝❤ ✤❛ ♠ư❝ t✐➯✉ ❦❤ỉ♥❣ trì♥ ✭P✮✳ ❈ö t❤➸✱ ✤➲ t➔✐ t❤ü❝ ❤✐➺♥ ❝→❝ ♠ö❝ t✐➯✉ ♥❣❤✐➯♥ ❝ù✉ s❛✉ ✤➙②✳ ✰ ❑❤→✐ ♥✐➺♠ ❤➔♠ l✲ê♥ ✤à♥❤ ✈æ ữợ ụ ữ tỡ ởt số t t ú t ữợ tr ❝➜♣ ❤❛✐ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ ✰ ❈→❝ tố ữ ợ tữủ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ❝❤♦ ❝→❝ ♥❣❤✐➺♠ ②➳✉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ✭P✮✳ ✰ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ ❝➜♣ ❤❛✐ ❝❤♦ ❝→❝ ♥❣❤✐➺♠ ❝❤➢❝ ❝❤➢♥ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ✭P✮✳ ❈→❝ ❦➳t q✉↔ ❝õ❛ ✤➲ t➔✐ ❤♦➔♥ t❤✐➺♥ ❝→❝ ❦➳t q✉↔ ✤➣ ❝â tr♦♥❣ ❧➽♥❤ ✈ü❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ tr♦♥❣ ❝→❝ ❜➔✐ t♦→♥ q✉② ❤♦↕❝❤ ✤❛ ♠ư❝ t✐➯✉ ❦❤ỉ♥❣ trì♥✳ ❈→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ♥➔② ✤➣ ✤÷đ❝ t→❝ ❣✐↔ ✈➔ ●❙✳❚❙❑❍✳ P❤❛♥ ◗✉è❝ ❑❤→♥❤✱ tr÷í♥❣ ✣↕✐ ❤å❝ ◗✉è❝ t➳✱ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❚♣✳ ❍❈▼ ❝æ♥❣ ❜è tr➯♥ ♠ët t↕♣ ❝❤➼ ❦❤♦❛ ❤å❝ q✉è❝ t➳ tr♦♥❣ ❤➺ t❤è♥❣ ■❙■✳ ✸✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉✳ ✣➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ❞ị♥❣ ❝→❝ ❝ỉ♥❣ ❝ư ✈➔ ❦ÿ t❤✉➟t tr♦♥❣ ❣✐↔✐ t➼❝❤ ❦❤ỉ♥❣ trì♥✱ ❣✐↔✐ t➼❝❤ ✤❛ trà ✈➔ ❣✐↔✐ t➼❝❤ ❤➔♠✳ t t t ỗ ữỡ ã ữỡ ỵ tỹ ✤➲ t➔✐✱ ♠ö❝ t✐➯✉ ✈➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ t ã ữỡ ởt số ổ tr ❣✐↔✐ t➼❝❤ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ ❦❤ỉ♥❣ trì♥ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ❝➜♣ ã ữỡ lờ ổ ữợ ụ ữ tỡ ởt số t t ú ã ữỡ t ữợ ✤❛ trà ❝➜♣ ❤❛✐ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ • ❈❤÷ì♥❣ ✹✿ ❈→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❝➜♣ ợ tữủ ã ữỡ ❦✐➺♥ tè✐ ÷✉ ✤õ ❝➜♣ ❤❛✐✳ ✹ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❈❤÷ì♥❣ ✶✿ ▼ët sè ❝ỉ♥❣ ❝ư tr♦♥❣ ❣✐↔✐ t➼❝❤ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ ❦❤ỉ♥❣ trì♥ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ N ✈➔ R ❧➛♥ ❧÷đt ❧➔ ❝→❝ t➟♣ ❤đ♣ ❝→❝ sè tü ♥❤✐➯♥ ✈➔ sè t❤ü❝✳ ❱ỵ✐ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ X ✱ X ∗ ❧➔ ✤è✐ ♥❣➝✉ t♦♣♦ ❝õ❛ ♦❢ X ❀ , ❧➔ t➼❝❤ ✤è✐ ♥❣➝✉✳ ❧➔ ❝❤✉➞♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ❜➜t ❦ý ✈➔ d(y, S) ❧➔ ❦❤♦↔♥❣ ❝→❝❤ tø ✤✐➸♠ y ✤➳♥ t➟♣ S ✳ Bn (x, r) = {y ∈ Rn : x − y < r}❀ Sn = {y ∈ Rn : y = 1}❀ Sn∗ = {y ∈ (Rn )∗ : y = 1}❀ L(X, Y ) ỵ ổ t t ❜à ❝❤➦♥ tø X ✈➔♦ Y ✱ tr♦♥❣ ✤â X ✈➔ Y ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✳ ❱ỵ✐ ♥â♥ C Rn ỵ C = {c ∈ (Rn )∗ : c∗ , c ≥ 0, ∀c ∈ C} ❧➔ ♥â♥ ✤è✐ ❝ü❝ ❝õ❛ C ✳ ❱ỵ✐ A Rn ỵ rA tA A ❜❞A✱ ❝♦♥✈A✱ ❝♦♥❡A ✈➔ ▲✐♥A ❧➛♥ ❧÷đt ❧➔ ♣❤➛♥ tr♦♥❣ tữỡ ố tr õ ỗ ♥â♥ ❝õ❛ A ✈➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ s✐♥❤ ❜ð✐ A✳ ❱ỵ✐ t > ✈➔ r ∈ N✱ o(tr ) ỵ ởt tở t s❛♦ ❝❤♦ o(tr )/tr → ❦❤✐ t → 0+ ✳ ❈❤ó♥❣ t❛ ❤➣② ♥❤ỵ ❧↕✐ ♠ët sè ✤à♥❤ ♥❣❤➽❛ s❛✉ ✤➙②✳ ⑩♥❤ ①↕ f : Rn → X ✱ tr♦♥❣ ✤â X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ❝❤➦t t↕✐ x ∈ Rn ♥➳✉ ♥â ❝â ✤↕♦ ❤➔♠ ❋r➨❝❤❡t f (x) t↕✐ x ✈➔ limy→x,t→0+ suph∈Sn (f (y + th) − f (y)) − f (x)h = 0✳ t ❱ỵ✐ ❤➔♠ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ f : Rn → Rm ✱ ❏❛❝♦❜✐❛♥ s✉② rë♥❣ ❈❧❛r❦❡ ❝õ❛ f t↕✐ x ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ∂f (x) = conv{limf (xk ) : xk ∈ Ω, xk → x}✱ tr♦♥❣ ✤â f ❦❤↔ ✈✐ tr♦♥❣ Ω✱ ợ t trũ t ỵ r ▼ët ✈➔✐ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❏❛❝♦❜✐❛♥ s✉② rë♥❣ ❈❧❛r❦❡ ✤÷đ❝ ❧✐➺t ❦➯ tr♦♥❣ ♠➺♥❤ ✤➲ s❛✉ ✤➙②✳ ▼➺♥❤ ✤➲ ✶✳✶ ✭❬✼❪✮✳ ❈❤♦ f : R n → Rm ❧➔ ❤➔♠ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ t↕✐ x✳ ❑❤✐ ✤â✱ ✭✐✮ f (x) t ỗ rộ tr L(Rn , Rm ); ✭✐✐✮ ∂f (x) ❧➔ t➟♣ ♠ët ✤✐➸♠ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ f ❧➔ ❦❤↔ ✈✐ ❝❤➦t t↕✐ x✿ ∂f (x) = {f (x)}; ✭✐✐✐✮ ∂f (x) = {limk→∞ vk : vk ∈ ∂f (xk ), xk → x}✱ ♥â✐ ❝→❝❤ ❦❤→❝ ✭✈➻ ∂f (x) ❧➔ ❝♦♠♣➠❝✮✱ →♥❤ ①↕ ∂f (.) ❧➔ ♥ú❛ ❧✐➯♥ tö❝ tr➯♥ t↕✐ x; ỵ tr tr r f st ữỡ tr ởt ỗ U ❝õ❛ x ✈➔ a, b ∈ U ✱ t❤➻ f (b) − f (a) ∈ conv(∂f ([a, b])(b − a)) m = tỗ t ởt c ∈ (a, b) s❛♦ ❝❤♦ f (b) − f (a) ∈ ∂f (c)(b − a) ●✐í ✤➙②✱ t❛ ♥❤➢❝ ❧↕✐ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ ♥â♥ t✐➳♣ ①ó❝ ✈➔ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ❤❛✐ s➩ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ♣❤➛♥ s❛✉✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ❈❤♦ x , u ∈ R n ✈➔ M ⊂ Rn ✳ ✭❛✮ ◆â♥ ❝♦♥t✐♥❣❡♥t ❝õ❛ M t↕✐ x0 ❧➔ ✺ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com T (M, x0 ) = {v ∈ Rn : ∃tk → 0+ , ∃vk → v, ∀k ∈ N, x0 + tk vk ∈ M }✳ ✭❜✮ ◆â♥ t✐➳♣ ①ó❝ tr♦♥❣ ❝õ❛ M t↕✐ x0 ❧➔ IT (M, x0 ) = {v ∈ Rn : ∀tk → 0+ , ∀vk → v, ∀k ✤õ ❧ỵ♥, x0 + tk vk ∈ M }✳ ✭❝✮ ❚➟♣ ❝♦♥t✐♥❣❡♥t ❝➜♣ ❤❛✐ ❝õ❛ M t↕✐ (x0 , u) ❧➔ T (M, x0 , u) = {w ∈ Rn : ∃tk → 0+ , ∃wk → w, ∀k ∈ N, x0 + tk u + 12 t2k wk ∈ M }✳ ✭❞✮ ◆â♥ t✐➳♣ ①ó❝ ❝➜♣ ❤❛✐ t✐➺♠ ❝➟♥ ❝õ❛ M t↕✐ (x0 , u) ❧➔ T (M, x0 , u) = {w ∈ Rn : ∃(tk , rk ) → (0+ , 0+ ) : tk /rk → 0, ∃wk → w✱ ∀k ∈ N, x0 + tk u + 12 tk rk wk ∈ M }✳ ✭❡✮ ❚➟♣ ❦➲ ❝➜♣ ❤❛✐ ❝õ❛ M t↕✐ (x0 , u) ❧➔ A2 (M, x0 , u) = {w ∈ Rn : ∀tk → 0+ , ∃wk → w, ∀k ∈ N, x0 + tk u + 21 t2k wk ∈ M }✳ ✭❢✮ ❚➟♣ t✐➳♣ ①ó❝ tr♦♥❣ ❝➜♣ ❤❛✐ ❝õ❛ M t↕✐ (x0 , u) ❧➔ IT (M, x0 , u) = {w ∈ Rn : ∀tk → 0+ , ∀wk → w, ∀k ✤õ ❧ỵ♥, x0 + tk u + 12 t2k wk ∈ M }✳ ▼➺♥❤ ✤➲ s❛✉ ✤➙② tâ♠ t➢t ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ❤❛✐ tr➯♥✳ ▼➺♥❤ ✤➲ ✶✳✸✳ ❈❤♦ M ⊂ R ✱ x n ∈ Rn ✈➔ u ∈ Rn ✳ ❑❤✐ ✤â✱ ✭✐✮ IT (M, x0 , u) ⊂ A2 (M, x0 , u) ⊂ T (M, x0 , u) ⊂ clcone[cone(M − x0 ) − u]; ✭✐✐✮ ♥➳✉ u ∈ T (M, x0 )✱ t❤➻ T (M, x0 , u) = ∅✳ t ỳ M ỗ tM = u ∈ T (M, x0 )✱ t❤➻ ✭①❡♠ ❬✶✺✱ ✷✹✱ ✷✽❪✮ ✭✐✐✐✮ ✐♥t❝♦♥❡(M − x0 ) = IT (intM, x0 ); ✭✐✈✮ ♥➳✉ A2 (M, x0 , u) = ∅✱ t❤➻ IT (M, x0 , u) = intA2 (M, x0 , u), clIT (M, x0 , u) = A2 (M, x0 , u); ✭✈✮ ♥➳✉ u ∈ cone(M − x0 )✱ t❤➻ ✭❛✮ IT (M, x0 , u) = intcone[cone(M − x0 ) − u]❀ ✭❜✮ A2 (M, x0 , u) = clcone[cone(M − x0 ) − u]✳ ✻ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ữỡ lờ ổ ữợ ụ ♥❤÷ ✈❡❝tì ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣ ❍➔♠ h : Rn → Rm ✤÷đ❝ ❣å✐ ❧➔ ê♥ ✤à♥❤ t x Rn tỗ t ởt U ❝õ❛ x ✈➔ ϑ > s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ y ∈ U ✱ h(y) − h(x) ≤ ϑ y − x ✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✶ ✭❬✷✱ ✶✷❪✮ ✭✐✮ t ữợ ữợ tữỡ ự tr : Rn R t x t ữợ u ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ϕl (x, u) = lim inf t→0+ (ϕ(x + tu) − ϕ(x)) t ✭t÷ì♥❣ ù♥❣✱ ϕu (x, u) = lim supt→0+ (ϕ(x + tu) − ϕ(x))✮✳ t ✭✐✐✮ ❍➔♠ ϕ ✤÷đ❝ ❣å✐ ❧➔ l✲ê♥ tữỡ ự uờ t x tỗ t ♠ët ❧➙♥ ❝➟♥ U ❝õ❛ x ✈➔ ϑ > s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ y ∈ U ✈➔ u ∈ Sn ✱ |ϕl (y, u)−ϕl (x, u)| ≤ ϑ y −x ✭✷✳✶✮ ✭t÷ì♥❣ ù♥❣✱ |ϕu (y, u) − ϕu (x, u)| ≤ ϑ y − x ✮✳ ✭✷✳✷✮ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ l✲ê♥ ✤à♥❤ ϕ : R → R ✤÷đ❝ tâ♠ t➢t tr♦♥❣ ♠➺♥❤ ✤➲ s❛✉✳ n ▼➺♥❤ ✤➲ ✷✳✷ ✭✐✮ ✭❬✷✱ ✹❪✮ ❍➔♠ l✲ê♥ ✤à♥❤ ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ ✈➔ ❦❤↔ ✈✐ ❝❤➦t✳ ✭✐✐✮ ✭❬✶✷❪✮ ϕ ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ϕ ❧➔ rt t x tỗ t ởt ❝➟♥ U ❝õ❛ x s❛♦ ❝❤♦ ϕ ❧➔ ▲✐♣s❝❤✐t③ tr➯♥ U tỗ t > s ϕ (y) − ϕ (x) ≤ ϑ y − x ❤➛✉ ❤➳t tr♦♥❣ U ✭t❤❡♦ ♥❣❤➽❛ ✤ë ✤♦ ▲❡❜❡s❣✉❡✮✳ ✭✐✐✐✮ ✭❬✶✷❪✮ ❈→❝ ❦❤→✐ ♥✐➺♠ l✲ê♥ ✤à♥❤ ✈➔ u✲ê♥ ✤à♥❤ ❧➔ t÷ì♥❣ ✤÷ì♥❣❀ ❤ì♥ ♥ú❛ ❧➙♥ ❝➟♥ U ✈➔ ❤➡♥❣ sè ϑ ✤÷đ❝ →♣ ❞ư♥❣ ❣✐è♥❣ ♥❤❛✉ tr♦♥❣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✮ ❤♦➦❝ ✭✷✳✷✮✳ ❇ð✐ ▼➺♥❤ ✤➲ ✷✳✷ ✭✐✐✐✮✱ tr♦♥❣ s t sỷ t ữợ ữợ lờ ỏ t ữợ tr ✈➔ u✲ê♥ ✤à♥❤ ✤÷đ❝ ♥❤➢❝ ✤➳♥ ❦❤✐ ❝➛♥ t❤✐➳t✳ ❑❤→✐ ♥✐➺♠ l✲ê♥ ✤à♥❤ ✤÷đ❝ ♠ð rë♥❣ ❝❤♦ ❝→❝ ❤➔♠ ✈❡❝tì ữ s t ữợ ữợ tữỡ ự tr : R m n Rm t x t ữợ u ố ợ (R ) ữủ ∗ ξ , Φ(x + tu) − Φ(x) t ✭t÷ì♥❣ ù♥❣✱ Φuξ∗ (x, u) = lim supt→0+ ξ ∗ , Φ(x + tu) − Φ(x) ✮✳ t ❚➼♥❤ ❝❤➜t ❣✐→ trà tr✉♥❣ ❜➻♥❤ s❛✉ ✤➙② ❝❤♦ ❝→❝ ❤➔♠ ✈❡❝tì ❧✐➯♥ tư❝ s➩ ❝➛♥ ✤➳♥✳ Φlξ∗ (x, u) = lim inf t→0+ ▼➺♥❤ ✤➲ ✷✳✹ ✭❬✸❪✮✳ ❈❤♦ ❤➔♠ Φ : R → Rm ❧✐➯♥ tö❝ tr➯♥ ♠ët t➟♣ ♠ð U ⊂ Rn ❝❤ù❛ ✤♦↕♥ [a, b] ✈➔ ξ ∗ (Rm ) õ tỗ t γ1 , γ2 ∈ (a, b) s❛♦ ❝❤♦ n ✼ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com Φlξ∗ (γ1 , b − a) ≤ ξ ∗ , Φ(b) − Φ(a) ≤ Φlξ∗ (γ2 , b − a)✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✺✳ ❈❤♦ ❤➔♠ Φ : R n ∗ → Rm ✈➔ Γ = C ∗ ∩ Sm ✳ ✭✐✮ ✭❬✸❪✮ sỷ C Rm õ ỗ õ ợ tC = ữủ lờ t x t rPstr tỗ t ♠ët ❧➙♥ ❝➟♥ U ❝õ❛ x ✈➔ ϑ > s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ y ∈ U ✱ u ∈ Sn ✱ ✈➔ ξ ∗ ∈ Γ✱ |Φlξ∗ (y, u) − Φlξ∗ (x, u)| ≤ ϑ y − x ✳ ✭✐✐✮ ✭❬✶✷❪✮ Φ ✤÷đ❝ ❣å✐ ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x t❤❡♦ ♥❣❤➽❛ ❝õ❛ ●✐♥❝❤❡✈ ♥➳✉✱ ✈ỵ✐ ♠å✐ ξ ∗ ∈ (R ) ổ ữợ (.) := ∗ , Φ(.) ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x✳ m ∗ ổ ữợ tỡ f õ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t f ❧➔ ê♥ ✤à♥❤✱ t❤➻ f ❧➔ l✲ê♥ ✤à♥❤✳ ▼ët ✈➔✐ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ❤➔♠ ✈❡❝tì lờ ữủ t ủ ữợ ▼➺♥❤ ✤➲ ✷✳✻ ✭✐✮ ✭❬✶✷❪✮ Φ : Rn → Rm ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x t❤❡♦ ♥❣❤➽❛ ❝õ❛ ●✐♥❝❤❡✈ ♥➳✉ tỗ t ởt U x ✈➔ ϑ > s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ y ∈ U ✱ u ∈ Sn ✱ ✈➔ ξ ∗ ∈ (Rm )∗ ✱ |Φlξ∗ (y, u) − Φlξ∗ (x, u)| ≤ ϑ ξ ∗ y−x ✳ ✭✐✐✮ ✭❬✶✷❪✮ Φ : Rn → Rm ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x t❤❡♦ ♥❣❤➽❛ ❝õ❛ ●✐♥❝❤❡✈ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♥â ❦❤↔ ✈✐ rt t x tỗ t ởt U ❝õ❛ x ✈➔ ϑ > s❛♦ ❝❤♦ Φ ❧➔ ▲✐♣s❝❤✐t③ tr➯♥ U ✱ ✈➔✱ ✈ỵ✐ ❤➛✉ ❤➳t y ∈ U ✱ Φ (y) − Φ (x) ≤ ϑ y − x ✳ ✭✐✐✐✮ ✭❬✸✱ ✶✷❪✮ ◆➳✉ ❤➔♠ Φ : R → Rm ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x t❤❡♦ ♥❣❤➽❛ ❝õ❛ ❇❡❞♥❛✞r➼❦✲P❛st♦r ❤♦➦❝ ●✐♥❝❤❡✈✱ t❤➻ Φ ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ t↕✐ x ✈➔ ❦❤↔ ✈✐ ❝❤➦t t↕✐ x✳ n ❙❛✉ ✤➙②✱ t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ❤❛✐ ✤à♥❤ ♥❣❤➽❛ tr➯♥ ✈➲ l✲ê♥ ✤à♥❤ ❝❤♦ ❝→❝ ❤➔♠ ✈❡❝tì ❧➔ t÷ì♥❣ ✤÷ì♥❣ ♥❤❛✉✳ ▼➺♥❤ ✤➲ ✷✳✼✳ ◆➳✉ Φ : R → Rm ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x t❤❡♦ ♥❣❤➽❛ ❝õ❛ ●✐♥❝❤❡✈✱ t❤➻ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ ✈➲ l✲ê♥ ✤à♥❤ ❝õ❛ ❇❡❞♥❛✞r➼❦✲P❛st♦r t❤ä❛ ♠➣♥✳ ◆➳✉ C ❧➔ ♥❤å♥ ✈➔ ✐♥tC = ∅✱ t❤➻ ❤❛✐ ✤à♥❤ ♥❣❤➽❛ tr➯♥ ❧➔ t÷ì♥❣ ✤÷ì♥❣ ♥❤❛✉✳ n ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ▼➺♥❤ ✤➲ ✷✳✻ ✭✐✮✱ t❛ s✉② r❛ r➡♥❣ Φ ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x t❤❡♦ ♥❣❤➽❛ tỗ t ởt ❝➟♥ U ❝õ❛ x ✈➔ ϑ > s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ ∗ y ∈ U, u ∈ Sn ✱ ✈➔ ξ ∗ ∈ Sm ✱ |Φlξ∗ (y, u) − Φlξ∗ (x, u)| ≤ ϑ y − x ∗ ❱➻ t❤➳✱ ❜➜t ✤➥♥❣ t❤ù❝ ♥➔② t❤ä❛ ♠➣♥ ❝❤♦ ξ ∗ ∈ Γ ⊂ Sm ♥❤÷ ②➯✉ ❝➛✉ tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❇❡❞♥❛✞r➼❦✲P❛st♦r✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû C ❧➔ ♥❤å♥✱ ✐♥tC = ∅ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ t❤ä❛ ♠➣♥ ❝❤♦ ξ ∗ ∈ Γ✳ ∗ ∗ ❱➻ ▲✐♥Γ = (Rm )∗ ✱ ✈ỵ✐ ♠å✐ i ∈ {1, 2, , m} tỗ t i,1 i,r i,1 , , αi,ri ∈ R i ri ∗ ∗ ∗ m ∗ ✇✐t❤ ri = 1, , m s❛♦ ❝❤♦ ei = j=1 αi,j ξi,j ✱ tr♦♥❣ ✤â ei ∈ (R ) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ e∗i , x = xi ✱ ✈ỵ✐ x = (x1 , x2 , , xm )✳ ❈❤♦ M = maxi,j |αi,j | ✈➔ Φi ❧➔ t❤➔♥❤ ♣❤➛♥ t❤ù i ❝õ❛ Φ✳ ❱➻ Φ ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x t❤❡♦ ♥❣❤➽❛ ❝õ❛ ❇❡❞♥❛✞r➼❦✲P❛st♦r✱ t❛ t❤➜② r➡♥❣ Φle∗i (x, u) = Φue∗i (x, u) = e∗i , Φ (x)u ✳ ❱ỵ✐ ♠å✐ ✽ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com y ∈ U, u ∈ Sn ✱ ✈➔ i ∈ {1, 2, , m}✱ t❛ ❝â Φli (y, u) − Φli (x, u) = Φle∗i (y, u) − Φle∗i (x, u) = lim inf t→0+ e∗i , (Φ(y + tu) − Φ(y)) − Φ (x)u t i ∗ , (Φ(y + tu) − Φ(y)) − Φ (x)u = lim inf t→0+ rj=1 αi,j ξi,j t ri l l ≥ − j=1 |αi,j ||Φξi,j ∗ (y, u) − Φξ ∗ (x, u)| ≥ −ϑ1 y − x ✱ i,j tr♦♥❣ ✤â ϑ1 = mM ϑ✳ ❍ì♥ ♥ú❛✱ ❜ð✐ sü t÷ì♥❣ ✤÷ì♥❣ ❝õ❛ ❝→❝ ❦❤→✐ ♥✐➺♠ l✲ê♥ ✤à♥❤ ✈➔ u✲ê♥ ✤à♥❤✱ t❛ ❝â Φli (y, u) − Φli (x, u) ≤ Φue∗i (y, u) − Φue∗i (x, u) = lim supt→0+ e∗i , (Φ(y + tu) − Φ(y)) − Φ (x)u t i ∗ = lim supt→0+ rj=1 αi,j ξi,j , (Φ(y + tu) − Φ(y)) − Φ (x)u t ri u u ≤ j=1 |αi,j ||Φξi,j ∗ (y, u) − Φξ ∗ (x, u)| ≤ ϑ1 y − x ✳ i,j ❍❛✐ ❜➜t ✤➥♥❣ t❤ù❝ ✤↕t ✤÷đ❝ tr➯♥ ❝❤ù♥❣ tä r➡♥❣ Φi ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x t❤❡♦ ♥❣❤➽❛ ❝õ❛ ●✐♥❝❤❡✈ ✈ỵ✐ ♠å✐ i = 1, , m✳ ❉♦ ✤â✱ ❤➔♠ ✈❡❝tì Φ ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x✳ ❇ð✐ ♠➺♥❤ ✤➲ tr➯♥✱ tø ✤➙② ✈➲ s❛✉ t❛ ❝❤➾ q✉❛♥ t➙♠ ✤➳♥ ❦❤→✐ ♥✐➺♠ l✲ê♥ ✤à♥❤ t❤❡♦ ♥❣❤➽❛ ❝õ❛ ●✐♥❝❤❡✈✳ ✾ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✶✵ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❇➡♥❣ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tü✱ t❛ ✤↕t ✤÷đ❝ g(x0 + tk uk + 21 t2k w) − g(x0 ) − tk g (x0 )u → z0 + g (x0 )w✱ t2k /2 h(x0 + tk uk + 21 t2k w) − h(x0 ) − tk h (x0 )u → w0 + h (x0 )w = 0✳ t2k /2 ❱➻ h(x0 ) = ✈➔ h (x0 )u = 0✱ tø ❣✐ỵ✐ ❤↕♥ s❛✉ ❝ị♥❣✱ t❛ s✉② r❛ r➡♥❣ h(x0 + tk uk + 12 t2k w) t2k /2 tt ữợ q tr ợ k ợ tỗ t yk H s ❝❤♦ x0 + tk uk + 21 t2k w − yk ≤ µ h(x0 + tk uk + 21 t2k w) + o(t2k )✱ ✈➔ ✈➻ t❤➳ (x0 + tk uk + 21 t2k w − yk )/ 12 t2k → 0✳ ✣➦t wk := (yk − x0 − tk uk )/ 12 t2k ✱ t❛ ✤÷đ❝ wk → w ✈➔ x0 +tk uk + 12 t2k wk ∈ H ✳ ✭✹✳✷✮ ▼➦t ❦❤→❝✱ ❜ð✐ ✭✹✳✶✮ ✈➔ t➼♥❤ ❝❤➜t ▲✐♣s❝❤✐t③ ❝õ❛ f ✈➔ g ✱ f (x0 + tk uk + 12 t2k wk ) − f (x0 ) − tk f (x0 )u → y0 + f (x0 )w ∈ −✐♥t❝♦♥❡[C + f (x0 )u]✱ t2k /2 ✭✹✳✸✮ g(x0 + tk uk + 21 t2k wk ) − g(x0 ) − tk g (x0 )u → z0 +g (x0 )w ∈ IT (−K, g(x0 ), g (x0 )u)✳✭✹✳✹✮ t2k /2 ❱➻ IT (−✐♥tC, f (x0 )u) = −✐♥t❝♦♥❡(C + f (x0 )u) ✭❜ð✐ ▼➺♥❤ ✤➲ ✶✳✸ ✭✐✐✐✮✮✱ ✭✹✳✸✮ ❞➝♥ ✤➳♥✱ ✈ỵ✐ k ❧ỵ♥✱ f (x0 + tk uk + 12 t2k wk ) − f (x0 ) − tk f (x0 )u f (x0 )u + tk ∈ −✐♥tC ✱ t2k /2 ✈➔ ❞♦ ✤â f (x0 +tk uk + 12 t2k wk )−f (x0 ) ∈ −intC ✳ ✭✹✳✺✮ ▼ët ❝→❝❤ t÷ì♥❣ tü✱ ❜ð✐ ✭✹✳✹✮ ✈➔ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ IT ✱ ✈ỵ✐ k ✤õ ❧ỵ♥ t❛ ❝â g(x0 + tk uk + 12 t2k wk ) − g(x0 ) − tk g (x0 )u g(x0 ) + tk g (x0 )u + t2k ∈ −K ✳ t2k /2 ❉♦ ✤â✱ g(x0 + tk uk + 12 t2k wk ) ∈ −K ✳ ✭✹✳✻✮ ❈→❝ ❝æ♥❣ t❤ù❝ ✭✹✳✷✮✱ ✭✹✳✺✮ ✈➔ ✭✹✳✻✮ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t ✈➲ x0 ✳ ✭✐✐✐✮ ❱ỵ✐ w ∈ T (M, x0 , u) tỗ t (tk , rk ) → (0+ , 0+ ) : tk /rk → 0✱ ✈➔ xk ∈ M s❛♦ ❝❤♦ wk := (xk − x0 − tk u)/ 12 tk rk → w✳ ❇ð✐ ▼➺♥❤ ✤➲ ✸✳✹ ✭✐✐✐✮✱ t❛ ❝â f (xk ) − f (x0 ) − tk f (x0 )u → f (x0 )w✳ tk rk /2 ✶✼ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ◆➳✉ f (x0 )w ∈ −✐♥t❝♦♥❡[C + f (x0 )u]✱ t❤➻✱ ✈ỵ✐ k ✤õ ❧ỵ♥✱ f (xk ) − f (x0 ) − tk f (x0 )u ∈ −✐♥tC ✱ f (x0 )u + rk tk rk /2 ✤✐➲✉ ♥➔② s✉② r❛ r➡♥❣ f (xk ) − f (x0 ) ∈ −✐♥tC ✿ t ữ ỵ r t ỵ ✹✳✷ ♥â✐ ✈➲ t➜t ❝↔ ❝→❝ ✤✐➸♠ t❤✉ë❝ t➟♣ D2 (f, g, h)(x0 , u) ❧ỵ♥ ❤ì♥ t➟♣ d2 (f, g, h)(x0 , u) ữủ ũ tr ỵ ❝õ❛ ❬✶✸❪✱ ✈➻ t❤➳ ❦➳t q✉↔ ❝õ❛ ❝❤ó♥❣ tỉ✐ ♠↕♥❤ ỡ ỡ ỳ ỵ rở ✤➲ ✸✳✺ ❝õ❛ ❬✺❪ ✈➔ ✹✳✶ ❝õ❛ ❬✶✾❪✱ ♠➔ ❝❤➾ ①➨t tr÷í♥❣ ❤đ♣ h = ✈➔ g (x0 )u K(g(x0 )) t ữủ tứ ỵ ♠ët ❞↕♥❣ ✤è✐ ♥❣➝✉ t❤❡♦ ❝→❝ ♥❤➙♥ tû ▲❛❣r❛♥❣❡✱ t❛ ỵ t ủ tỷ rt Λ(x0 ) := {(c∗ , k ∗ , h∗ ) ∈ (Rm )∗ × (Rp )∗ × (Rr )∗ : (c∗ , k ∗ , h∗ ) = (0, 0, 0), c∗ ◦ f (x0 )+ k ∗ ◦ g (x0 ) + h∗ ◦ h (x0 ) = 0, c∗ ∈ C ∗ , k ∗ ∈ N (−K, g(x0 ))} ỵ t s ỵ C , C Rn t ỗ s C1 õ tỗ t ởt s t r✐➯♥❣ C1 ✈➔ C2 ✈➔ ❦❤æ♥❣ ❝❤ù❛ C2 ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ C1 ∩ riC2 = ∅✳ ✣à♥❤ ỵ tC tK rộ tt ợ ỵ ✹✳✷ t❤ä❛ ♠➣♥✱ ✈➔ x0 ❧➔ ♥❣❤✐➺♠ ②➳✉ ✤à❛ ♣❤÷ì♥❣ P õ tỗ t (c , k ∗ , h∗ ) ∈ Λ(x0 ) s❛♦ ❝❤♦ (c∗ , k ∗ ) = (0, 0); ✭✐✐✮ ✈ỵ✐ ♠å✐ u ∈ Rn ✈ỵ✐ (f, g, h) (x0 )u ∈ −[C × clK(g(x0 )) \ int(C × K(g(x0 )))] × {0}✱ ✈➔ (y0 , z0 , w0 ) ∈ D2 (f, g, h)(x0 , u) tỗ t (c , k ∗ , h∗ ) ∈ Λ(x0 ) s❛♦ ❝❤♦ c∗ , y0 + k ∗ , z0 + h∗ , w0 ≥ supk∈A2 (−K,g(x0 ),g (x0 )u) k ∗ , k ✱ ✈➔ ♥➳✉✱ t❤➯♠ ♥ú❛✱ • h = 0✱ t❤➻ (c∗ , k ∗ ) = (0, 0); • ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ❝➜♣ ❤❛✐ ✭❚❘u ✮ (g, h) (x0 )Rn − T (T (−K, g(x0 )), g (x0 )u) × {0} = Rp × Rr ✱ t❤ä❛✱ t❤➻ c∗ = 0; ✭✐✐✐✮ ♥➳✉ w ∈ T (M, x0 , u) t tỗ t c C \ {0} ✈ỵ✐ c∗ , f (x0 )u = s❛♦ ❝❤♦ c∗ , f (x0 )w ≥ 0✳ ❈❤ù♥❣ ♠✐♥❤✳ ỵ ✹✳✸ ✈ỵ✐ C1 = (f, g, h) (x0 )Rn ✈➔ C2 = −int[C ×K(g(x0 ))]×{0}✱ t❛ ❝â (c∗ , k ∗ , h∗ ) ∈ (Rm )∗ ×(Rp )∗ ×(Rr )∗ ✈ỵ✐ (c∗ , k ∗ , h∗ ) = (0, 0, 0) ✈➔ α ∈ R s❛♦ ❝❤♦✱ ∀(y, z, t) ∈ (f, g, h) (x0 )Rn ✱ ∀(c, k) ∈ −(C × K(g(x0 )))✱ c∗ , y + k ∗ , z + h∗ , t ≥ α✱ ✭✹✳✼✮ c∗ , c + k ∗ , k ≤ α ✭✹✳✽✮ ✈➔ s✐➯✉ ♣❤➥♥❣ H := {(y, z, t) ∈ Rm × Rp × Rr ) : c∗ , y + k ∗ , z + h∗ , t = α} ❦❤æ♥❣ ❝❤ù❛ C2 ✳ ❱➻ (f, g, h) (x0 )Rn ✈➔ C × K(g(x0 )) ❧➔ ❝→❝ ♥â♥✱ α = 0✳ ❑❤✐ ✤â ✭✹✳✼✮ s✉② r❛ r➡♥❣ c∗ ◦ f (x0 ) + k ∗ ◦ g (x0 ) + h∗ ◦ h (x0 ) = 0✳ ❈❤♦ k = tr♦♥❣ ✭✹✳✽✮ t❛ ✤↕t ✤÷đ❝ ✶✽ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com c∗ ∈ C ∗ ✳ ✣➦t c = tr♦♥❣ ✭✹✳✽✮ t❛ ❝â k ∗ ∈ K(g(x0 ))∗ = N (−K, g(x0 ))✳ ❱➻ s✐➯✉ ♣❤➥♥❣ H ❦❤æ♥❣ ❝❤ù❛ C2 ✱ (c∗ , k ∗ ) = (0, 0)✳ ✭✐✐✮ ●✐↔ sû A2 (−K, g(x0 ), g (x0 )u) = ∅ ✭♥➳✉ ❦❤æ♥❣✱ t q t tữớ ỵ →♣ ❞ư♥❣ ❇ê ✤➲ ✹✳✸ ✈ỵ✐ C1 = (f, g, h) (x0 )Rn +(y0 , z0 , w0 ) ✈➔ C2 = −intcone[C + f (x0 )u] × IT (−K, g(x0 ), g (x0 )u) × {0}✱ t❛ ✤↕t ✤÷đ❝ (c∗ , k ∗ , h∗ ) ∈ (Rm ) ì (Rp ) ì (Rr ) ợ (c , k ∗ , h∗ ) = (0, 0, 0) ✈➔ α ∈ R s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ (y, z, t) ∈ (f, g, h) (x0 )Rn ✱ w ∈ −✐♥t❝♦♥❡[C + f (x0 )u] ✈➔ k ∈ IT (−K, g(x0 ), g (x0 )u)✱ c∗ , y + k ∗ , z + h∗ , t + c∗ , y0 + k ∗ , z0 + h∗ , w0 ≥ α✱ ✭✹✳✾✮ c∗ , w + k ∗ , k ≤ α ✭✹✳✶✵✮ ✈➔ s✐➯✉ ♣❤➥♥❣ H ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ tr➯♥ ❦❤ỉ♥❣ ❝❤ù❛ C2 ✳ ❱➻ (f, g, h) (x0 )Rn ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥✱ tø ✭✹✳✾✮ t❛ ❝â✱ ✈ỵ✐ ♠å✐ (y, z, t) ∈ (f, g, h) (x0 )Rn ✱ c∗ , y + k ∗ , z + h∗ , t = 0✱ ✈➔ ✈➻ t❤➳ c∗ ◦ f (x0 ) + k ∗ ◦ g (x0 ) + h∗ ◦ h (x0 ) = ✈➔ c∗ , y0 + k ∗ , z0 + h∗ , w0 ≥ α✳ ✭✹✳✶✶✮ ❱➻ −✐♥t❝♦♥❡[C +f (x0 )u] ❧➔ ♥â♥✱ ✭✹✳✶✵✮ s✉② r❛ r➡♥❣ c∗ , w ≤ 0✱ ✈ỵ✐ ♠å✐ w ∈ −✐♥t❝♦♥❡[C + f (x0 )u]✱ ♠➔ s✉② r❛ r➡♥❣ c∗ ∈ C ∗ ✈➔ c∗ , f (x0 )u = 0✳ ❈❤♦ w ❤ë✐ tö ✤➳♥ ❣è❝ tr♦♥❣ ✭✹✳✶✵✮✱ t❛ ❝â k ∗ , k ≤ α✱ ✈ỵ✐ ♠å✐ k ∈ IT (−K, g(x0 ), g (x0 )u) ữ t ợ k ∈ A2 (−K, g(x0 ), g (x0 )u) ✭①❡♠ ▼➺♥❤ ✤➲ ✶✳✸ ✭✐✈✮✮✳ ✣✐➲✉ ♥➔② ❝ị♥❣ ✈ỵ✐ ✭✹✳✶✶✮ s✉② r❛ r➡♥❣ c∗ , y0 + k ∗ , z0 + h∗ , w0 ≥ supk∈A2 (−K,g(x0 ),g (x0 )u) k ∗ , k ✳ ✣➸ t❤➜② r➡♥❣ (c∗ , k ∗ , h∗ ) ∈ Λ(x0 ) ♥❤÷ ②➯✉ ❝➛✉✱ t❛ q✉❛♥ s→t tø ▼➺♥❤ ✤➲ ✸✳✶ ❬✽❪✱ A2 (−K, g(x0 ), g (x0 )u) + T (T (−K, g(x0 )), g (x0 )u) ⊂ A2 (−K, g(x0 ), g (x0 )u)✳ ❉♦ ✤â✱ k ∗ , k+k1 ≤ α✱ ✈ỵ✐ k ∈ A2 (−K, g(x0 ), g (x0 )u) ✈➔ k1 ∈ T (T (−K, g(x0 )), g (x0 )u)✳ ❱➻ T (T (−K, g(x0 )), g (x0 )u) ❧➔ ♥â♥✱ tø ❝æ♥❣ t❤ù❝ ✭✷✳✶✶✵✮ tr♦♥❣ ❬✻❪ s✉② r❛ r➡♥❣ k ∗ ∈ −[T (T (−K, g(x0 )), g (x0 )u)]∗ = {k ∗ ∈ N (−K, g(x0 )) : k ∗ , g (x0 )u = 0}✳ ✭✹✳✶✷✮ ❇➙② ❣✐í✱ ❣✐↔ sû h = 0✳ ◆➳✉ (c∗ , k ∗ ) = (0, 0)✱ t❤➻ ❜ð✐ ✭✹✳✾✮ ✈➔ ✭✹✳✶✵✮✱ α = ✈➔ ❞♦ ✤â s✐➯✉ ♣❤➥♥❣ H ❝❤ù❛ C2 ✿ ♠➙✉ t❤✉➝♥✳ ●✐↔ sû ✤✐➲✉ ❦✐➺♥ ✭❚❘u ✮ t❤ä❛ ♠➣♥✱ tù❝ ❧➔✱ ✈ỵ✐ ♠å✐ (y, z) Rp ì Rr tỗ t x Rn ✈➔ k ∈ T (T (−K, g(x0 )), g (x0 )u) s❛♦ ❝❤♦ (g, h) (x0 )x − (k, 0) = (y, z)✳ ❉♦ ✤â✱ (k ∗ , h∗ ), (y, z) = k ∗ , g (x0 )x + h∗ , h (x0 )x − k ∗ , k ✳ ◆➳✉ c∗ = 0✱ t❤➻ ✈➳ ♣❤↔✐ ❝õ❛ ✤➥♥❣ t❤ù❝ tr➯♥ ❜➡♥❣ − k ∗ , k ✱ ✈➻ (c∗ , k ∗ , h∗ ) ∈ Λ(x0 )✳ ❇ð✐ ✭✹✳✶✷✮ k ∗ , k ≤ 0✳ ❱➻ (y, z) tũ ỵ (k , h ) = (0, 0)✿ ♠➙✉ t❤✉➝♥✳ ✭✐✐✐✮ ✣✐➲✉ ♥➔② s✉② r❛ tø ỵ ỵ t tổ tữớ q trỹ t ỵ ✹✳✹ ❦❤✐ h = 0✳ ❍➺ q✉↔ ✹✳✺✳ ❈❤♦ ❝→❝ tt ỵ tọ h = 0✱ ✈➔ x ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ✭P✮✳ ❑❤✐ ✤â✱ ❧➔ ♥❣❤✐➺♠ ②➳✉ ✭✐✮ t➟♣ s❛✉ ✤➙② ❦❤→❝ ré♥❣✿ ✶✾ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com Λ1 (x0 ) := {(c∗ , k ∗ ) ∈ (Rm )∗ × (Rp )∗ : (c∗ , k ∗ ) = (0, 0), c∗ ◦ f (x0 ) + k ∗ ◦ g (x0 ) = 0, c∗ ∈ C ∗ , k ∗ ∈ N (−K, g(x0 ))}; ✭✐✐✮ ✈ỵ✐ ♠å✐ u ∈ Rn ✈ỵ✐ (f, g) (x0 )u ∈ −[C×❝❧K(g(x0 ))\ ✐♥t(C × K(g(x0 )))] ✈➔ (y0 , z0 ) D2 (f, g)(x0 , u) tỗ t (c , k ∗ ) ∈ Λ1 (x0 ) s❛♦ ❝❤♦ c∗ , y0 + k ∗ , z0 ≥ supk∈A2 (−K,g(x0 ),g (x0 )u) k ∗ , k ✱ ✈➔ c∗ = ♥➳✉✱ t❤➯♠ ♥ú❛✱ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② s❛✉ ✤➙② t❤ä❛ ✭❚❘✶u ✮ g (x0 )Rn − T (T (−K, g(x0 )), g (x0 )u) = Rp ; ✭✐✐✐✮ ♥➳✉ w ∈ T (M, x0 , u)✱ t❤➻ tỗ t c C \ {0} ợ c , f (x0 )u = s❛♦ ❝❤♦ c , f (x0 )w ữ ỵ tr ỵ ✭✈➔ ❍➺ q✉↔ ✹✳✺✮ ✤ó♥❣ ❝❤♦ ❝→❝ ❤➔♠ g ✈➔ h tũ ỵ ỡ ỳ t t t t❤✐➳t tr➯♥ (g, h) ♥❤÷ s❛✉✿ (g, h) ❧➔ l✲ê♥ t x0 ữợ q tr t ữợ t (x0 , u) ố ợ K ì {0} t ❦❤➥♥❣ ✤à♥❤ ✭✐✐✐✮ trð ♥➯♥ ♠↕♥❤ ❤ì♥ ✈➔ ✤÷đ❝ ❞✐➵♥ t↔ t❤❡♦ g ✈➔ h ♥❤÷ s❛✉✿ ♥➳✉ g (x0 )w ∈ T (−K, g(x0 ), g (x0 )u) ✈➔ h (x0 )w = t tỗ t c C ∗ \ {0} ✈ỵ✐ c∗ , f (x0 )u = s❛♦ ❝❤♦ c∗ , f (x0 )w ≥ t ữợ ợ (g, h) t g K ì {0} t❤❛② ✈➻ −K ✱ ✈ỵ✐ M = (g, h)−1 (−K × {0}) = G ∩ H ✈➔ ❜ð✐ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ T ✱ t❛ ❝â T (M, x0 , u) = {w ∈ Rn : (g, h) (x0 )w ∈ T (−K × {0}, (g, h)(x0 ), (g, h) (x0 )u)} = {w ∈ Rn : g (x0 )w ∈ T (−K, g(x0 ), g (x0 )u), h (x0 )w = 0}✳ ✭✐✐✮ ▼➦❝ ❞ò t❛ ✤➣ ❜✐➳t ❜✐➸✉ t❤ù❝ s t tr ỵ ổ ❞÷ì♥❣✿ supk∈A2 (−K,g(x0 ),g (x0 )u) k ∗ , k ≤ 0✱ ✭✹✳✶✸✮ t❛ ♠✉è♥ ✤÷❛ r❛ ❧í✐ ❣✐↔✐ t❤➼❝❤ ✤ì♥ ❣✐↔♥ ✈➻ t➛♠ q✉❛♥ trå♥❣ ❝õ❛ ♥â✳ ❇ð✐ ▼➺♥❤ ✤➲ ✶✳✸ ✭✐✮✱ t❛ ❝â A2 (−K, g(x0 ), g (x0 )u) ⊂ ❝❧❝♦♥❡❬❝♦♥❡(−K − g(x0 )) − g (x0 )u❪✳ ▼➦❝ ❦❤→❝ ✭①❡♠ ♣❤➛♥ ❝✉è✐ ❝õ❛ ♣❤➨♣ ❝❤ù♥❣ ♠✐♥❤ ỵ k ∗ ∈ −[T (T (−K, g(x0 )), g (x0 )u)]∗ = −[clcone(cone(−K − g(x0 )) − g (x0 )u)]∗ ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ ✭✹✳✶✸✮✳ ❈æ♥❣ t❤ù❝ ♥➔② ♣❤↔♥ →♥❤ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣✲❧✐❦❡✱ ❜ð✐ ✈➻ ♥â ❝â t❤➸ t❤ä❛ ♠➣♥ ♥❤÷ ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ❝❤➦t✳ ❙✉♣r❡♠✉♠ tr♦♥❣ ✭✹✳✶✸✮ tr✐➺t t✐➯✉ ✭✈➔ ❣✐è♥❣ tr÷í♥❣ ❤đ♣ ❝ê ✤✐➸♥✮ ♥➳✉ ∈ A2 (−K, g(x0 ), g (x0 )u)✳ ✣➦❝ ❜✐➺t✱ ✤✐➲✉ ♥➔② ①↔② r❛ ♥➳✉ t❛ ①➨t ❝→❝ ✤✐➲✉ ❦✐➺♥ ữợ u Rn ợ g (x0 )u ∈ ❝♦♥❡(−K − g(x0 )) = −K(g(x0 )) ✭①❡♠ ▼➺♥❤ ✤➲ ✶✳✸ ✭✈✮ ✭❜✮✮✱ ♥❤÷ ♥❤✐➲✉ t→❝ tỹ ỵ t ữợ u ✈ỵ✐ g (x0 )u ∈ ❝❧❝♦♥❡(−K − g(x0 )) = −clK(g(x0 ))✳ ◆❣❤➽❛ ❧➔✱ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣✲❧✐❦❡ ①↔② r❛ ❝❤♦ ❝→❝ ✤✐➸♠ u tr♦♥❣ ❧ê ❤ê♥❣ ❜❛♦ ✤â♥❣ ❞÷í♥❣ ♥❤÷ ♥❤ä ♥➔②✳ ❈❤ó♥❣ t❛ ♥❤➜♥ ♠↕♥❤ r➡♥❣ ❦❤ỉ♥❣ ❝â ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣✲❧✐❦❡ ①↔② r❛ ❝❤♦ g (x0 )u ∈ −K(g(x0 ))✱ ♥❣❛② ❝↔ ❦❤✐ ✤✐➸♠ ♥➔② ♥➡♠ tr➯♥ ❜✐➯♥✳ ❍ì♥ ♥ú❛✱ ♥➳✉ K ❧➔ t➟♣ ✤❛ ❞✐➺♥✱ t❤➻ −K(g(x0 )) ❧➔ ✤â♥❣✱ ✈➔ ✈➻ t❤➳ ❝ơ♥❣ ❦❤ỉ♥❣ ❝â ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣✲❧✐❦❡ ①↔② r❛✳ ❍➺ q✉↔ ✹✳✺ ♠ð rë♥❣ ✣à♥❤ ỵ ỵ ❞♦ ①➨t t❤➯♠ ❝→❝ ✤✐➸♠ u tr♦♥❣ ❧ê ❤ê♥❣ ♥â✐ tr➯♥ ✈➔ ①↔② r❛ ❤✐➺♥ t÷đ♥❣✳ ❍ì♥ ♥ú❛✱ ♥➳✉ h = (h1 , , hr ) ❦❤↔ ✈✐ ❝➜♣ ❤❛✐ ✷✵ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com t↕✐ x0 ✈ỵ✐ h1 (x0 ), , hr (x0 ) ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✱ t❤➻ ✤✐➲✉ ❦✐➺♥ ✭▼❋✮ t❤ä❛ ♠➣♥ ✈ỵ✐ S = {0} ✈➔✱ ❞♦ ✤â✱ ✤✐➲✉ ❦✐➺♥ ✭❉▼❙❘u ✮ ụ tọ t ỵ tữỡ ự ỵ t r tữỡ ự ỵ tr õ tt ợ ❤↕♥ ❤ì♥ ♥❤÷ s❛✉✿ f ✈➔ g ❧✐➯♥ tư❝ t↕✐ x0 ✱ d2 (f, g)(x0 , u) ✤÷đ❝ ❞ị♥❣✱ ✈➔ h = (h1 , , hr ) ❦❤↔ ✈✐ ❋r➨❝❤❡t ❝➜♣ ❤❛✐ t↕✐ x0 ✈ỵ✐ h1 (x0 ), , hr (x0 ) ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✳ ❍ì♥ ♥ú❛✱ D2 (f, g, h)(x0 , u) ữủ ũ tr ỵ ✹✳✷ ✈➔ ✹✳✹ ❧ỵ♥ ❤ì♥ d2 (f, g, h)(x0 , u) tr♦♥❣ ❬✶✸❪✳ ❑➳t q✉↔ s❛✉ ✤➙② ❝❤♦ t❛ ♠ët ✤➦❝ tr÷♥❣ ❝õ❛ ♥â♥ t✐➳♣ ①ó❝ ❝➜♣ ❤❛✐ t✐➺♠ ❝➟♥ ❝õ❛ t➟♣ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ M = g −1 (−K)✳ ▼➺♥❤ ✤➲ ✹✳✻✳ ❈❤♦ ❤➔♠ g ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x ữợ q tr t ữợ t (x0 , u) ✤è✐ ✈ỵ✐ −K ✳ ❑❤✐ ✤â✱ ✈ỵ✐ M = g −1 (−K) t❛ ❝â T (M, x0 , u) = {w ∈ Rn : g (x0 )w ∈ T (−K, g(x0 ), g (x0 )u)}✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ w T (M, x0 , u) tỗ t (tk , rk ) → (0+ , 0+ ) : tk /rk → ✈➔ wk → w s❛♦ ❝❤♦ xk := x0 + tk u + 12 tk rk wk ∈ M ✳ ❇ð✐ ▼➺♥❤ ✤➲ ✸✳✹ ✭✐✐✐✮✱ t❛ ❝â g(xk ) − g(x0 ) − tk g (x0 )u → g (x0 )w✳ tk rk /2 ❱➻ g(xk ) ∈ −K ✱ t❛ ❝â g (x0 )w ∈ T (−K, g(x0 ), g (x0 )u)✳ ❇➙② ❣✐í ❣✐↔ sû g (x0 )w ∈ T (−K, g(x0 ), g (x0 )u) õ tỗ t (tk , rk ) (0+ , 0+ ) : tk /rk → ✈➔ zk → g (x0 )w s❛♦ ❝❤♦ g(x0 ) + tk g (x0 )u + 21 tk rk zk ∈ K ợ k tt ữợ ❝❤➼♥❤ q✉② ♠❡tr✐❝✱ ✈ỵ✐ k ✤õ ❧ỵ♥✱ t❛ ❝â tk ∈ (0, ρ) ✈➔ uk := u+ 12 rk w ∈ Bn (u, ρ) s❛♦ ❝❤♦ d(x0 + tk uk , M ) ≤ µd(g(x0 + tk uk ), −K) ≤ µ g(x0 + tk uk ) − g(x0 ) − tk g (x0 )u − 21 tk rk zk ≤ µ( g(x0 + tk uk ) − g(x0 ) − g (x0 )(tk uk ) + ≤ µϑ tk uk t r g k k (x0 )w − 12 tk rk zk ) + 12 µtk rk g (x0 )w − zk = 12 µtk rk (2ϑ(tk /rk ) uk + g (x0 )w − zk ) ✭❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ❝ị♥❣ ✤÷đ❝ s✉② r❛ tø ❇ê ✤➲ ✸✳✷✮✳ ❱➻ zk → g (x0 )w✱ t tự tỗ t xk M s❛♦ ❝❤♦ x0 + tk uk − xk / 21 tk rk → 0✳ ❉♦ ✤â✱ w ∈ T (M, x0 , u) ✈➻ xk − x0 − tk u xk − x0 − tk uk wk := = + w → w✳ tk rk /2 tk rk /2 ❱➼ ❞ư s❛✉ ✤➙② ❝❤♦ t❤➜② ♠ët tr÷í♥❣ ❤đ♣ ♠➔ ỵ õ t ữủ tr ♠ët sè ❦➳t q✉↔ ❣➛♥ ✤➙② ❦❤ỉ♥❣ →♣ ❞ư♥❣ ✤÷đ❝✳ ❱➼ ❞ö ✹✳✶✳ ❈❤♦ ϕ : [0, +∞) → R ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ♥➳✉ s > 1, ϕ(s) = 1/(q + 1) ♥➳✉ 1/(q + 1) < s ≤ 1/q, q ∈ N, ♥➳✉ s = ❱➻ ϕ ❦❤æ♥❣ ❣✐↔♠ tr➯♥ [0, +∞)✱ ❤➔♠ θ(x) = |x| ϕ(s)ds ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❝❤♦ x ∈ R ✭ϕ ✈➔ θ ✤÷đ❝ ❧➜② tø ❬✺❪✮✳ ữ ỵ r ỗ ✷✶ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✈➔ ❝â ❝→❝ ✤↕♦ ❤➔♠ ❜➯♥ ♣❤↔✐ ✈➔ ❜➯♥ tr→✐✱ ✈ỵ✐ q ∈ N✱ θ+ (1/q) = 1/q ✱ θ− (1/q) = 1/(q + 1)✳ ❈❤♦ C = K = R+ ✱ x0 = (0, 0) ✈➔ f ✱g, h : R2 → R ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ f (x1 , x2 ) = x21 + x2 ♥➳✉ x1 ≥ 0, θ(x1 ) + x2 ♥➳✉ x1 < 0, g(x1 , x2 ) = 12 x21 + x1 + x32 ✱ h(x1 , x2 ) = −x21 − x2 ✳ ❑❤✐ ✤â✱ g ✈➔ h ❧➔ ❝→❝ ❤➔♠ t❤✉ë❝ ❧ỵ♣ C ✈➔ t❛ ❝â f (x0 ) = (0, 1)✱ g (x0 ) = (1, 0)✱ h (x0 ) = (0, −1)✳ ❱➻ ✤✐➲✉ ❦✐➺♥ ✭▼❋✮ ✤è✐ ✈ỵ✐ S = {0} tọ h ữợ q tr t ữợ t (x0 , u) ố ợ t ợ ♠å✐ u ∈ R2 ✳ ❚➟♣ ❤ñ♣ ❝→❝ ♥❤➙♥ tû ❋r✐t③ ❏♦❤♥ ❝õ❛ ✭P✮ ❧➔ Λ(x0 ) = {(c∗ , k ∗ , h∗ ) ∈ R3 : c∗ = h∗ > 0, k ∗ = 0}✳ ❍➔♠ f ❦❤æ♥❣ ❦❤↔ ✈✐ tr♦♥❣ U \ {x0 }✱ ✈ỵ✐ ❜➜t ❝ù ❧➙♥ ❝➟♥ U ❝õ❛ x0 ✳ ❉♦ ✤â✱ t❛ ❦❤æ♥❣ t ũ ỵ r➡♥❣ f ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x0 ✳ ❚❤➟t ✈➟②✱ ❝❤♦ U ❧➔ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ x0 ✱ x = (x1 , x2 ) ∈ U ✱ ✈➔ u = (u1 , u2 ) ∈ S2 ✳ ◆➳✉ x1 > 0✱ t❤➻ |f l (x, u) − f l (x0 , u)| = |2x1 u1 | ≤ x − x0 ◆➳✉ x1 < 0✱ t❤➻ |f l (x, u) − f l (x0 , u)| = |θ± (x1 )u1 | ≤ lims→x±1 ϕ(s) ≤ −x1 ≤ x − x0 ◆➳✉ x1 = 0✱ t❤➻ |f l (x, u) − f l (x0 , u)| = ◆❤÷ t❤➳✱ f ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x0 ✱ ✈➻ |f l (x, u) − f l (x0 , u)| ≤ x − x0 ✈ỵ✐ ♠å✐ x ∈ U ✈➔ u ∈ S2 ✳ ❈❤å♥ u = (−1, 0)✳ ❑❤✐ ✤â (f, g, h) (x0 )u = (0, −1, 0) ∈ −[C×❝❧K(g(x0 ))\ ✐♥t(C × K(g(x0 )))] × {0}✱ A2 (−K, g(x0 ), g (x0 )u) = R✱ D2 (f, g, h)(x0 , u) = {(1 + α, β, −2 − α) : α, β ∈ R}✳ ✭✣➸ t➼♥❤ t❤➔♥❤ ♣❤➛♥ ✤➛✉ t✐➯♥ ❝õ❛ D2 (f, g, h)(x0 , u)✱ q✉❛♥ s→t r➡♥❣ |x| − ln(1 + |x|) ≤ |x| |x| ϕ(s)ds ≤ sds = x2 /2✳✮ ❱ỵ✐ (c∗ , k ∗ , h∗ ) ∈ Λ(x0 )✱ tù❝ ❧➔✱ c∗ = h∗ > 0, k ∗ = 0✱ ✈➔ (y0 , z0 , w0 ) ∈ D2 (f, g, h)(x0 , u)✱ tù❝ ❧➔✱ y0 = + α✱ z0 = β ✱ w0 = −2 − α ✈ỵ✐ α, β ∈ R✱ t❛ ❝â c∗ , y0 + k ∗ , z0 + h∗ , w0 = c∗ (1 + α) + c∗ (−2 − ) = c < ỵ ✭✈➻ ❦❤ỉ♥❣ ❝â ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ①↔② r❛✮✱ x0 ❦❤ỉ♥❣ ❧➔ ♥❣❤✐➺♠ ②➳✉ ✤à❛ ♣❤÷ì♥❣✳ ❚r♦♥❣ ✈➼ ❞ư s❛✉ ✤➙②✱ ❍➺ q✉↔ ✹✳✺ t❤✉➟♥ t✐➺♥ ❤ì♥ ♥❤✐➲✉ ❦➳t q✉↔ ❣➛♥ ✤➙②✳ ❱➼ ❞ö ✹✳✷✳ ❈❤♦ C = R+ ✱ K = {(k1 , k2 , k3 ) ∈ R3 : k2 k3 ≥ 2k12 , k2 ≤ 0, k3 ≤ 0}✱ x0 = (0, 0)✱ ✈➔ f : R2 → R ✈➔ g : R2 → R3 ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ f (x1 , x2 ) = −3x21 + x2 ♥➳✉ x1 ≥ 0, θ(x1 ) + x2 ♥➳✉ x1 < 0, g(x1 , x2 ) = (x1 , + x21 , x2 )✱ tr♦♥❣ ✤â θ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ tr♦♥❣ ❱➼ ❞ư ✹✳✶ ✭✈➔ K ✈➔ g ✤÷đ❝ ❧➜② tø ❱➼ ❞ö ✶ ❝õ❛ ❬✶✸❪✮✳ ❑❤✐ ✤â✱ g ❧➔ ❤➔♠ t❤✉ë❝ ❧ỵ♣ C t↕✐ x0 ✱ ✈➔ t❛ ❝â ✷✷ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com f (x0 ) = (0, 1)✱ g (x0 ) ❂ 0 0✱ g(x0 ) = (0, 1, 0) ∈ −K ✱ ❘ã r➔♥❣ K õ ỗ õ ợ tK = ❣✐í t❛ t➼♥❤ K(g(x0 )) = cone(K + g(x0 ))✳ ❱ỵ✐ (t1 , t2 , t3 ) ∈ K(g(x0 ))✱ (t1 , t2 , t3 ) = α[(k1 , k2 , k3 ) + (0, 1, 0)] ⇔ t1 = αk1 , t2 = α(k2 + 1), t3 = αk3 ✈ỵ✐ α ≥ ✈➔ (k1 , k2 , k3 ) ∈ K ✳ ◆➳✉ α = 0✱ t❤➻ (t1 , t2 , t3 ) = (0, 0, 0)✳ ✭✹✳✶✹✮ ◆➳✉ α > 0✱ t❤➻ k1 = t1 /α, k2 = (t2 /α) − 1, k3 = t3 /α✳ ❚❛ ❝â k2 ≤ ⇔ t2 ≤ α✱ k3 ≤ ⇔ t3 ≤ 0✱ k2 k3 ≥ 2k12 ⇔ (t2 − α)t3 ≥ 2t21 ✳ ❚❛ ❝â ❤❛✐ tr÷í♥❣ ❤ñ♣✳ ◆➳✉ t3 = 0✱ t❤➻ ❞➵ ❞➔♥❣ t❤➜② r➡♥❣ t1 = 0✳ ❱➻ t2 ≤ α✱ t❛ ✤÷đ❝ (t1 , t2 , t3 ) = (0, β2 , 0) ✈ỵ✐ β2 ∈ R✳ ✭✹✳✶✺✮ ◆➳✉ t3 < 0✱ t❤➻ t2 ≤ α + (2t21 /t3 )✳ ❱➻ t❤➳ (t1 , t2 , t3 ) = (β1 , β2 , β3 ) ✈ỵ✐ β1 , β2 ∈ R, β3 < 0✳ ✭✹✳✶✻✮ ❚ø ✭✹✳✶✹✮ ✲ ✭✹✳✶✻✮✱ t❛ ✤↕t ✤÷đ❝ K(g(x0 )) = {(k1 , k2 , k3 ) ∈ R3 : k3 < 0} ∪ {(0, k2 , 0) : k2 ∈ R} ❍ì♥ ♥ú❛✱ T (−K, g(x0 )) = {(k1 , k2 , k3 ) ∈ R3 : k3 ≥ 0}, N (−K, g(x0 )) = {λ(0, 0, −1) : λ ∈ R}, Λ1 (x0 ) = {(c∗ , k ∗ ) ∈ R × R3 : c∗ = α > 0, k ∗ = α(0, 0, −1)} ❚÷ì♥❣ tü ♥❤ö tr♦♥❣ ❱➼ ❞ö ✹✳✶✱ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ f ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x0 ✈➔ ❦❤æ♥❣ ❦❤↔ ✈✐ tr♦♥❣ U \ {x0 }✱ ✈ỵ✐ ❜➜t ❝ù ❧➙♥ ❝➟♥ U ❝õ❛ x0 ✳ ❈❤♦ u = (u1 , u2 ) ∈ R2 s❛♦ ❝❤♦ (f, g) (x0 )u ∈ −[C×❝❧K(g(x0 ))\ ✐♥t(C × K(g(x0 )))]✳ ❑❤✐ ✤â✱ u = (u1 , 0) ✈ỵ✐ u1 ∈ R ✈➔ g (x0 )u = (u1 , 0, 0)✳ ◆➳✉ u = 0✱ tù❝ ❧➔✱ u1 = t❤➻ g (x0 )u ∈ −K(g(x0 ))✳ ◆➳✉ u = 0✱ t❤➻ g (x0 )u ∈ −K(g(x0 )) ✈➔ D2 (f, g)(x0 , u) {(β, γ, 0, β) : β, γ ∈ R}✳ ❱➻ t❤➳✱ ✈ỵ✐ ♠å✐ (y0 , z0 ) ∈ D2 (f, g)(x0 , u) tỗ t (c , k ) (1, 0, 0, −1) ∈ C ∗ × K(g(x0 ))∗ \ {(0, 0)} s❛♦ ❝❤♦ c∗ ◦ f (x0 ) + k ∗ ◦ g (x0 ) = ✈➔ c∗ , y0 k ∗ , z0 = 0✳ ❉♦ õ ỵ ỵ ❝õ❛ ❬✺❪ ❦❤æ♥❣ t❤➸ ❜→❝ ❜ä x0 ✳ 0✱ = = + ▼➦❝ ❦❤→❝✱ ❝❤å♥ u = (1, 0)✱ t❛ ❝â A2 (−K, g(x0 ), g (x0 )u) = {(k1 , k2 , k3 ) ∈ R3 : k3 ≥ 4}✱ D2 (f, g)(x0 , u) = {(−6 + β, γ, 2, β) : β, γ ∈ R}✳ ❱ỵ✐ α > 0✱ ❝❤♦ (c∗ , k ∗ ) = (α, 0, 0, −α) ∈ Λ1 (x0 )✱ t❛ ❝â supk∈A2 (−K,g(x0 ),g (x0 )u) k ∗ , k = −4α✱ ✈➔✱ ✈ỵ✐ (y0 , z0 ) ∈ D2 (f, g)(x0 , u)✱ c∗ , y0 + k ∗ , z0 = α(−6 + β) − αβ = −6α < −4α✳ ❇ð✐ ❍➺ q✉↔ ✹✳✺ ✭✐✐✮✱ x0 ❦❤æ♥❣ ❧➔ ♥❣❤✐➺♠ ②➳✉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✭P✮✳ ✷✸ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❱➼ ❞ö t✐➳♣ t❤❡♦ ♠✐♥❤ ❤å❛ ✈✐➺❝ →♣ ❞ö♥❣ ❦❤➥♥❣ ✤à♥❤ ✭✐✐✐✮ ❝õ❛ ❍➺ q✉↔ ✹✳✺ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♠➔ ❦❤➥♥❣ ✤à♥❤ ✭✐✐✮ ❦❤ỉ♥❣ →♣ ❞ư♥❣ ✤÷đ❝✳ ❱➼ ❞ư ✹✳✸✳ ❈❤♦ C = R = {0}✱ x0 = (0, 0)✱ f ♥❤÷ tr♦♥❣ ❱➼ ❞ư ✹✳✶ ✈➔ g : R2 → R ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ g(x1 , x2 ) = x31 + x22 ✳ ❑❤✐ ✤â✱ f ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x0 ✈➔ f (x0 ) = (0, 1)✳ ❱➻ ✐♥tK = ∅✱ ❍➺ q✉↔ ✹✳✺ ✭✐✐✮ ❦❤ỉ♥❣ t❤➸ →♣ ❞ư♥❣ ✤÷đ❝✳ ❚❤û ❦✐➸♠ tr❛ ❦❤➥♥❣ ✤à♥❤ ✭✐✐✐✮✱ t❛ ❝â +, K M = {(x1 , x2 ) ∈ R2 : x31 + x22 = 0}✱ T (M, x0 ) = {(u1 , u2 ) ∈ R2 : u1 ≤ 0, u2 = 0}✳ ❱ỵ✐ u = (−1, 0) ∈ T (M, x0 )✱ T (M, x0 , u) = R2 ✳ ▲➜② w = (w1 , w2 ) ✈ỵ✐ w2 < 0✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ c∗ ∈ C ∗ \ {0} ✈ỵ✐ c∗ , f (x0 )u = 0✱ tù❝ ❧➔✱ c∗ > 0✱ t❛ ❝â c∗ , f (x0 )w = c∗ w < 0✳ ❇ð✐ ✭✐✐✐✮✱ x0 ❦❤æ♥❣ ❧➔ ♥❣❤✐➺♠ ữỡ ữ ỵ r ữủ ✤à♥❤ ♥❣❤➽❛ tr♦♥❣ ❬✶✱ ✶✻❪✮ ❧➔ ❝→❝ ❧♦↕✐ ✤↕♦ ❤➔♠ r➜t tê♥❣ q✉→t tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✭①❡♠ ❬✶✽✱ ✷✵✱ ✷✶❪✮✳ ❱➼ ❞ư s❛✉ ✤➙② ❝❤ù♥❣ tä r➡♥❣ ❍➺ q✉↔ ✹✳✺ ❝â t❤➸ →♣ ❞ö♥❣ ✤÷đ❝ tr♦♥❣ ❦❤✐ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ t❤❡♦ ❝→❝ ①➜♣ tr ỵ ổ ✤÷đ❝✳ ❱➼ ❞ư ✹✳✹✳ ❈❤♦ C = R +, K f (x) = = R✱ x0 = 0✱ ✈➔ f, g : R → R ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ g = ✈➔ −x2 /2 ♥➳✉ x ≥ 0, √ x2 /2 − (2/3)x −x ♥➳✉ x < 0, ✭f ✤÷đ❝ ❧➜② tø ❱➼ ❞ư ✹✳✶ ❝õ❛ ❬✶✾❪✮✳ ❑❤✐ ✤â✱ f ❦❤↔ ✈✐ ❧✐➯♥ tö❝ t↕✐ x0 ✈➔ f (x0 ) = 0✳ ❚➟♣ ❤ñ♣ ❝→❝ ♥❤➙♥ tû ❋r✐t③ ❏♦❤♥ ❝õ❛ ✭P✮ ❧➔ Λ1 (x0 ) = {(c∗ , k ∗ ) ∈ R2 : c∗ > 0, k ∗ = 0}✳ ❱ỵ✐ u = 1✱ A2 (−K, g(x0 ), g (x0 )u) = R✱ D2 (f, g)(x0 , u) = {(−1, 0)}✱ ✈➔ ✈ỵ✐ ♠å✐ (c∗ , k ∗ ) ∈ Λ1 (x0 )✱ t❛ ❝â c∗ , −1 + k ∗ , = −c∗ < supk∈A2 (−K,g(x0 ),g (x0 )u) k ∗ , k = 0✳ ❇ð✐ ❍➺ q✉↔ ✹✳✺ ✭✐✐✮✱ x0 ❦❤æ♥❣ ❧➔ ♥❣❤✐➺♠ ②➳✉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✭P✮✳ ❇➙② ❣✐í t❛ t❤û ❞ị♥❣ ỵ r ợ u ∈ Rn , T (M, x0 , u) = T (M, x0 , u) = R✳ ❚❛ t➼♥❤ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ f t↕✐ x0 ✈➔ ❝→❝ t➟♣ ❧✐➯♥ q✉❛♥ ♥❤÷ s❛✉✿ Bf (x0 ) = {−1/2} ∪ (α, +∞) ✈ỵ✐ α > 0✱ ❝❧Bf (x0 ) = {−1/2} ∪ [α, +∞) ✈➔ Bf (x0 )∞ = [0, +∞)✳ ❱➻ t❤➳✱ x0 t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ tr ỵ ổ ❜→❝ ❜ä✳ ✷✹ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❈❤÷ì♥❣ ✺✿ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ ❝➜♣ ❤❛✐ ❚r♦♥❣ ♣❤➛♥ ♥➔②✱ t❛ ①➨t ❜➔✐ t♦→♥ ✭P✮ ✈ỵ✐ h = 0✱ ✈➔ C ✱ K ❝â t❤➸ ❦❤æ♥❣ ỗ ợ tr rộ s ữủ sû ❞ö♥❣ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤õ ❝➜♣ ❤❛✐✳ ❇ê ✤➲ ✺✳✶✳ ●✐↔ sû x ∈ M ⊆ Rn ✳ ◆➳✉ xk ∈ M \ {x0 } ❤ë✐ tử x0 t tỗ t u T (M, x0 ) \ {0} ✈➔ ♠ët ❞➣② ❝♦♥✱ ❝ò♥❣ ỵ xk s (xk x0 )/tk → u✱ tr♦♥❣ ✤â tk = xk − x0 ; ✭✐✐✮ ✭❬✶✺❪✱ ❇ê ✤➲ ✸✳✹✮ ❤♦➦❝ z ∈ T (M, x0 , u)u tỗ t s (xk −x0 −tk u)/ 21 t2k → z ❤♦➦❝ z ∈ T (M, x0 , u) ∩ u⊥ \ {0} rk 0+ tỗ t s tk /rk → 0+ ✈➔ (xk − x0 − tk u)/ 12 tk rk → z ✱ tr♦♥❣ ✤â u⊥ ❧➔ ♣❤➛♥ ũ trỹ u Rn ỵ ✺✳✷✳ ❳➨t ❜➔✐ t♦→♥ ✭P✮ ✈ỵ✐ h = 0✱ ❝❤♦ f ✈➔ g ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x ∈ M ✳ ❑❤✐ ✤â✱ ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ ✤õ ✤➸ x0 ❧➔ ♥❣❤✐➺♠ ❝❤➢❝ ❝❤➢♥ ✤à❛ ♣❤÷ì♥❣ ❝➜♣ ❤❛✐✳ ✭✐✮ ∀u ∈ Sn ✱ ∃(c∗ , k ∗ ) ∈ C ∗ × K(g(x0 ))∗ s❛♦ ❝❤♦ c∗ , f (x0 )u + k ∗ , g (x0 )u > 0✳ ✭✐✐✮ ∀u ∈ Sn ✈ỵ✐ u ∈ T (M, x0 ) ✈➔ f (x0 )u ∈ −C ✱ t❛ ❝â ✭❛✮ ∀w ∈ T (M, x0 , u) ∩ u⊥ ✱ ∀(y0 , z0 ) ∈ d2 (f, g)(x0 , u) ✈ỵ✐ g (x0 )w + z0 ∈ T (−K, g(x0 ), g (x0 )u)✱ ∃(c∗ , k ∗ ) ∈ Λ1 (x0 ) ✭✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ tr♦♥❣ ❍➺ q✉↔ ✹✳✺✮ t❤ä❛ ♠➣♥ c∗ , y0 + k ∗ , z0 > k ∗ , g (x0 )w + z0 ; ✭✺✳✶✮ ✭❜✮ ∀w ∈ T (M, x0 , u) ∩ u⊥ \ {0}✱ ∃c∗ ∈ C ∗ \ {0} ✈ỵ✐ c∗ , f (x0 )u = t❤ä❛ ♠➣♥ c∗ , f (x0 )w > ự ỵ tr sỷ ự r tỗ t↕✐ xk ∈ M ∩ Bn (x0 , k1 ) \ {x0 } ✈➔ ck ∈ C s❛♦ ❝❤♦ f (xk ) − f (x0 ) + ck ∈ Bm (0, k1 t2k )✱ ✭✺✳✷✮ tr♦♥❣ ✤â tk = xk − x0 ✳ ❚❛ ❝â t❤➸ ❣✐↔ sû (xk − x0 )/tk → u ∈ T (M, x0 ) ∩ Sn ✳ ❈❤✐❛ ✭✺✳✷✮ ❝❤♦ tk ✈➔ q✉❛ ❣✐ỵ✐ ❤↕♥✱ t❛ ✤÷đ❝ f (x0 )u ∈ −C ✳ ❇ð✐ ❇ê ✤➲ ✺✳✶✱ ❝❤➾ ❝➛♥ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣ s❛✉ ❧➔ rữớ ủ ởt ỗ t w T (M, x0 , u)∩u⊥ s❛♦ ❝❤♦ wk := (xk −x0 −tk u)/ 21 t2k → w✳ ❇ð✐ ▼➺♥❤ ✤➲ ✸✳✹ ✭✐✮✱ t❛ ❝â (yk , zk ) := (f, g)(xk ) − (f, g)(x0 ) − tk (f, g) (x0 )u → (y, z)✱ t2k /2 ✈ỵ✐ (y, z) ∈ D2p (f, g)(x0 , u, w)✳ ❇ð✐ ▼➺♥❤ ✤➲ ✸✳✺✱ (y, z) = (f, g) (x0 )w + (y0 , z0 )✳ ✈ỵ✐ (y0 , z0 ) ∈ d2 (f, g)(x0 , u)✳ ❱➻ g(xk ) ∈ −K ✈➔ zk = (g(xk ) − g(x0 ) − tk g (x0 )u)/ 21 t2k → z = g (x0 )w + z0 ✱ s✉② r❛ r➡♥❣ g (x0 )w + z0 ∈ T (−K, g(x0 ), g (x0 )u)✱ ✈➔ ❞♦ ✤â g (x0 )u ∈ T (−K, g(x0 ))✳ tt tỗ t (c , k ∗ ) ∈ Λ1 (x0 ) t❤ä❛ ♠➣♥ ✭✺✳✶✮✳ ❈ë♥❣ t❤➯♠ c∗ , f (x0 )w ✷✺ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✈➔♦ ❤❛✐ ✈➳ ❝õ❛ ✭✺✳✶✮ ✈➔ ❞ò♥❣ ✤➥♥❣ t❤ù❝ c∗ ◦ f (x0 ) + k ∗ ◦ g (x0 ) = 0✱ ❜➜t ✤➥♥❣ tự tữỡ ữỡ ợ c , f (x0 )w + y0 > 0✳ ✭✺✳✸✮ ▼➦t ❦❤→❝✱ tø ✭✺✳✷✮ t❛ s✉② r❛ (f (xk ) − f (x0 ) − tk f (x0 )u)/ 21 t2k + (ck + tk f (x0 )u)/ 12 t2k → 0✳ ❱➻ sè ❤↕♥❣ ✤➛✉ t✐➯♥ ð ✤➙② ❧➔ yk ✱ sü ❤ë✐ tö tr➯♥ ❞➝♥ ✤➳♥ y = f (x0 )w + y0 ∈ −❝❧❝♦♥❡(C + f (x0 )u)✳ ❱➻ f (x0 )u ∈ −C ✱ g (x0 )u ∈ T (−K, g(x0 )) ✈➔ c∗ ◦ f (x0 ) + k ∗ ◦ g (x0 ) = 0✱ t❛ s✉② r❛ c∗ , f (x0 )u = 0✱ ✈➔ ❞♦ ✤â c∗ ∈ ❬❝❧❝♦♥❡✭C + f (x0 )u)]∗ ✳ ❱➻ t❤➳✱ c∗ , f (x0 )w + y0 ≤ 0✱ ♠➙✉ t❤✉➝♥ ợ rữớ ủ ỗ t rk 0+ s❛♦ ❝❤♦ tk /rk → ✈➔ wk := (xk − x0 − tk u)/ 12 tk rk → w ∈ T (M, x0 , u) ∩ u⊥ \ {0}✳ tt tỗ t c C ∗ \ {0} ✈ỵ✐ c∗ , f (x0 )u = ✈➔ c∗ , f (x0 )w > 0✳ ❇ð✐ ▼➺♥❤ ✤➲ ✸✳✹ ✭✐✐✐✮✱ t❛ ❝â (f (xk ) − f (x0 ) − tk f (x0 )u)/ 21 tk rk → f (x0 )w✳ ▼➦t ❦❤→❝✱ tø ✭✺✳✷✮ t❛ s✉② r❛ (f (xk ) − f (x0 ) − tk f (x0 )u)/ 12 tk rk + (ck + tk f (x0 )u)/ 12 tk rk → 0✳ ❱➻ t❤➳✱ f (x0 )w ∈ −❝❧❝♦♥❡(C + f (x0 )u)✳ ❉♦ ✤â✱ c∗ , f (x0 )w ≤ 0✿ ♠➙✉ t ữ ỵ tr ỵ ữủ s r ởt tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ✭❛ ✮ ∀w ∈ Rn ✱ ∀(y0 , z0 ) ∈ d2 (f, g)(x0 , u) ✈ỵ✐ g (x0 )w + z0 ∈ T (−K, g(x0 ), g (x0 )u)✱ ∃(c∗ , k ∗ ) ∈ Λ1 (x0 ) t❤ä❛ ♠➣♥ ✭✺✳✶✮✳ ✭❛ ✮ ∀(y0 , z0 ) ∈ d2 (f, g)(x0 , u)✱ ∃(c∗ , k ∗ ) ∈ Λ1 (x0 )✱ c∗ , y0 + k ∗ , z0 > supk∈T (−K,g(x0 ),g (x0 )u) k ∗ , k ✳ tr ỵ ✤÷đ❝ s✉② r❛ ❜ð✐ ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ✭❜ ✮ ∀w ∈ u⊥ \ {0} ✈ỵ✐ g (x0 )w ∈ T (−K, g(x0 ), g (x0 )u)✱ ∃(c∗ , k ∗ ) ∈ Λ1 (x0 )✱ k ∗ , g (x0 )w < 0✳ ✭❜ ✮ ∀w ∈ u⊥ \ {0} ✈ỵ✐ g (x0 )w ∈ ❝❧❝♦♥❡❬❝♦♥❡(−K − g(x0 )) − g (x0 )u]✱ ∃(c∗ , k ∗ ) ∈ Λ1 (x0 )✱ k ∗ , g (x0 )w < 0✳ ❚❤ü❝ ✈➟②✱ ❜ð✐ ▼➺♥❤ ✤➲ ✸✳✹ ✭✐✐✐✮✱ ♥➳✉ w ∈ T (M, x0 , u)✱ t❤➻ g (x0 )w ∈ T (−K, g(x0 ), g (x0 )u) ⊂ ❝❧❝♦♥❡❬❝♦♥❡(−K − g(x0 )) − g (x0 )u]✳ õ ỵ t ỵ ❝õ❛ ❬✶✸❪✱ tr♦♥❣ ✤â f ✈➔ g ✤÷đ❝ ❣✐↔ sû ❧➔ ê♥ ✤à♥❤ t↕✐ x0 ✈➔ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❛✬✮ ữủ ũ ữ ỵ r t D2 (f, g)(x0 , u) ♥❤÷ tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥✱ tr ỵ t ũ t ọ ỡ d2 (f, g)(x0 , u) ✤➸ ✤÷đ❝ ❦➳t ❧✉➟♥ ♠↕♥❤ ❤ì♥✳ ✭✐✐✐✮ ❇➙② ❣✐í t❛ ♥â✐ ✤➳♥ ❧ê ❤ê♥❣ ❣✐ú❛ ✤✐➲✉ tr ỵ tr ỵ t ữ ỵ r u ∈ T (M, x0 )✱ t❤➻ g (x0 )u ∈ T (−K, g(x0 )) = ✷✻ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com −❝❧K(g(x0 ))✳ ❱➻ t❤➳✱ ✣à♥❤ ỵ ụ ú ữợ u Sn s❛♦ ❝❤♦ (f, g) (x0 )u ∈ −[C×❝❧K(g(x0 ))] ữủ t ữ tr ỵ ỡ ỳ ỵ tữỡ ự ỵ ụ ú ✭t❤ü❝ sü ②➳✉ ❤ì♥✮ ♥➳✉ t❛ t❤❛② D2 ✭t÷ì♥❣ ù♥❣✱ d2 ✮ ❜ð✐ d2 ✭t÷ì♥❣ ù♥❣✱ D2 ✮✳ ◆❣❛② ❝↔ ợ ữ tr K tọ ♠➣♥ t➼♥❤ ❝❤➜t s❛✉ t↕✐ g(x0 )✿ T (−K, g(x0 ), u) = A2 (−K, g(x0 ), u) ✈ỵ✐ ♠å✐ u ∈ Rp ✱ ❧ê ❤ê♥❣ ♥â✐ tr➯♥ ❧➔ ♥❤ä✿ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✤÷đ❝ t❤❛② ❜ð✐ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❝❤➦t tr♦♥❣ ✤✐➲✉ ❦✐➺♥ r s ỵ ✤à♥❤ ♥❣❤✐➺♠ ❝❤➢❝ ❝❤➢♥✱ tr♦♥❣ ❦❤✐ ❝→❝ ❦➳t q✉↔ ❣➛♥ ✤➙② t❤➻ ❦❤ỉ♥❣✳ ❱➼ ❞ư ✺✳✶✳ ❳➨t ❱➼ ❞ư ✹✳✷ ữ ợ f ữủ ữ tr ❱➼ ❞ö ✹✳✶✮ f (x1 , x2 ) = −x21 + x2 ♥➳✉ x1 ≥ 0, θ(x1 ) + x2 ♥➳✉ x1 < ❚÷ì♥❣ tü ♥❤÷ tr♦♥❣ ❱➼ ❞ư ✹✳✷✱ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ f ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x0 ✈➔ f (x0 ) = (0, 1)✳ ❑❤✐ ✤â✱ ✈ỵ✐ u = (1, 0) ∈ S2 ✈➔ (c∗ , k ∗ ) ∈ C ∗ × K(g(x0 ))∗ ✱ t❛ ❝â c∗ , f (x0 )u + k ∗ , g (x0 )u = 0✳ ❉♦ ✤â✱ tr ỵ ổ tọ ❈❤♦ u = (u1 , u2 ) ∈ S2 s❛♦ ❝❤♦ (f, g) (x0 )u ∈ −[C×❝❧K(g(x0 ))]✳ ❑❤✐ ✤â✱ u = (u1 , 0) ✈ỵ✐ u1 = ±1✳ ❚❛ ❝â T (−K, g(x0 ), g (x0 )u) = A2 (−K, g(x0 ), g (x0 )u)✱ ✈➔ ❞♦ ✤â✱ ✈ỵ✐ k ∗ = (0, 0, −1) ∈ N (−K, g(x0 ))✱ supk∈T (−K,g(x0 ),g (x0 )u) k ∗ , k = −4✳ ◆➳✉ u1 = 1✱ t❤➻ ✈ỵ✐ (y0 , z0 ) ∈ d2 (f, g)(x0 , u) = {(2, 0, 2, 0)} tỗ t (c , k ∗ ) = (1, 0, 0, −1) ∈ Λ1 (x0 ) t❤ä❛ ♠➣♥ c∗ , y0 + k ∗ , z0 = −2 > supk∈T (−K,g(x0 ),g (x0 )u) k ∗ , k ✳ ◆➳✉ u1 = −1✱ t❤➻ ✈ỵ✐ (y0 , z0 ) ∈ d2 (f, g)(x0 , u) tỗ t tk 0+ s y0 = limk→+∞ ✈➔ z0 = (0, 2, 0)✳ ❱➻ t❤➳✱ ✈ỵ✐ (c∗ , k ∗ ) = (1, 0, 0, −1) ∈ Λ1 (x0 )✱ t❛ ❝â θ(−tk ) c∗ , y0 + k ∗ , z0 = limk→+∞ ≥ > supk∈T (−K,g(x0 ),g (x0 )u) k ∗ , k ✳ tk /2 θ(−tk ) t2k /2 ❉♦ ✤â✱ ✤✐➲✉ ❦✐➺♥ ✭❛ ✮ t❤ä❛ ♠➣♥✱ ✈➔ ❞♦ ✤â ỵ ụ tọ ❈❤♦ w = (w1 , w2 ) ∈ v ⊥ \ {(0, 0)}✱ tù❝ ❧➔✱ w1 = ✈➔ w2 = 0✱ ♥➳✉ g (x0 )w = (0, 0, w2 ) ∈ ❝❧❝♦♥❡❬❝♦♥❡(−K − g(x0 )) − g (x0 )u] = {(k1 , k2 , k3 ) ∈ R3 : k3 ≥ 0}✱ t❤➻ w2 > 0✳ ❈❤å♥ (c∗ , k ∗ ) = (1, 0, 0, −1) ∈ Λ1 (x0 )✱ t❛ ❝â k ∗ , g (x0 )w = −w2 < 0✳ ❱➻ t❤➳✱ ✤✐➲✉ ❦✐➺♥ ✭❜ ✮ t❤ä❛ ♠➣♥✱ ✈➔ ❞♦ ✤â ✤✐➲✉ ❦✐➺♥ ✭❜✮ ❝õ❛ ✣à♥❤ ỵ ụ tọ t q ỵ x0 ữỡ ❤❛✐ ❝õ❛ ❜➔✐ t♦→♥ ✭P✮✳ ❱➻ f ❦❤æ♥❣ ê♥ ✤à♥❤ t x0 ỵ ỵ ổ ữủ ỡ ỳ tr õ t ỵ ✹✳✶ ❝õ❛ ❬✺❪ ❦❤ỉ♥❣ →♣ ❞ư♥❣ ✤÷đ❝✳ ✷✼ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com t ữợ ♥❣❤✐➯♥ ❝ù✉ ♠ð rë♥❣ ✤➲ t➔✐ ❚r♦♥❣ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ♥➔②✱ ✤➛✉ t✐➯♥✱ ❝❤ó♥❣ tỉ✐ ❣✐ỵ✐ t❤✐➺✉ ❦❤→✐ ♥✐➺♠ lờ ổ ữợ ụ ữ tỡ s→t ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ ❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ tổ t t ữợ trà ❝➜♣ ❤❛✐ ✈➔ ✤÷❛ r❛ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ú ố ũ ũ t ữợ tr r ữợ tt t ✭tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥✮ ❤❛② l✲ê♥ ✤à♥❤ ✭tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ✮✱ ❝❤ó♥❣ tỉ✐ t❤✐➳t tố ữ ợ ❝→❝ ♥❣❤✐➺♠ ②➳✉ ✤à❛ ♣❤÷ì♥❣ ❤❛② ❝→❝ ♥❣❤✐➺♠ ❝❤➢❝ ❝❤➢♥ ữỡ ợ t t ữủ ró ỡ tr♦♥❣ ❜➔✐ t♦→♥ q✉② ❤♦↕❝❤ ✤❛ ♠ư❝ t✐➯✉ ❦❤ỉ♥❣ trì♥ ✭P✮✳ ❚r♦♥❣ ❦➳ ❤♦↕❝❤ ♥❣❤✐➯♥ ❝ù✉ t÷ì♥❣ ❧❛✐✱ ❝❤ó♥❣ tỉ✐ s rở ữợ ự t ❝→❝❤ ①➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ❦❤ỉ♥❣ trì♥ ❦❤→ tê♥❣ q✉→t s❛✉ ✤➙②✿ ✭P✶✮ ♠✐♥ f (x)✱ s✳t✳ x ∈ S ✱ g(x) ∈ −K, h(x) = 0✱ tr♦♥❣ ✤â f : X → Y ✱ g : X → Z ✱ ✈➔ h : X → W ❧➔ ❝→❝ →♥❤ ①↕✱ X ✈➔ W ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ Y ✈➔ Z ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ S X C Y õ ỗ õ K Z t ỗ ú tổ s➩ t❤✐➳t ❧➟♣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥ ✈➔ ✤õ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ ❝❤♦ ❝→❝ ♥❣❤✐➺♠ ②➳✉ ✈➔ ♥❣❤✐➺♠ ❝❤➢❝ ❝❤➢♥ ❝õ❛ ❜➔✐ t♦→♥ ✭P✶✮ ❜➡♥❣ ❝→❝ q✉② t➢❝ ♥❤➙♥ tû ❋r✐t③✲ ❏♦❤♥✲▲❛❣r❛♥❣❡ ❞ò♥❣ ❝→❝ ✤↕♦ ❤➔♠ t ữợ tr õ t ú t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ❤❛✐✳ ❈❤ó♥❣ tỉ✐ ✤➦t r❛ ❝→❝ ❣✐↔ t❤✐➳t ❣✐↔♠ ♥❤➭✿ st❡❛❞✐♥❡ss ✈➔ ❦❤↔ ✈✐ ❝❤➦t ❝❤♦ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐✱ t÷ì♥❣ ù♥❣❀ ê♥ ✤à♥❤ ✈➔ l✲ê♥ ✤à♥❤ ❝❤♦ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤õ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐✱ t÷ì♥❣ ù♥❣✳ ✷✽ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❑✳ ❆❧❧❛❧✐ ❛♥❞ ❚✳ ❆♠❛❤r♦q✱ ❙❡❝♦♥❞✲♦r❞❡r ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ ♣r✐♠❛❧ ❛♥❞ ❞✉❛❧ ♥❡❝❡s✲ s❛r② ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s✱ ❖♣t✐♠✐③❛t✐♦♥ ✭✶✾✾✼✮ ✷✷✾✲✷✹✻✳ ✹✵ ❬✷❪ ❉✳ ❇❡❞♥❛✞r➼❦ ❛♥❞ ❑✳ P❛st♦r✱ ❖♥ s❡❝♦♥❞✲♦r❞❡r ❝♦♥❞✐t✐♦♥s ✐♥ ✉♥❝♦♥str❛✐♥❡❞ ♦♣t✐♠✐③❛✲ t✐♦♥✱ ▼❛t❤✳ Pr♦❣r❛♠✳ ✭❙❡r✳ ❆✮ ✭✷✵✵✽✮ ✷✽✸✲✷✾✽✳ ✶✶✸ ❬✸❪ ❉✳ ❇❡❞♥❛✞r➼❦ ❛♥❞ ❑✳ P❛st♦r✱ ❉❡❝r❡❛s❡ ♦❢ C 1,1 ♣r♦♣❡rt② ✐♥ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥✱ ❘❆■❘❖✲ ❖♣❡r✳ ❘❡s✳ ✭✷✵✵✾✮ ✸✺✾✲✸✼✷✳ ✹✸ ❬✹❪ ❉✳ ❇❡❞♥❛✞r➼❦ ❛♥❞ ❑✳ P❛st♦r✱ l✲st❛❜❧❡ ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♥t✐♥✉♦✉s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✭✷✵✵✾✮ ✷✸✶✼✲✷✸✷✹✳ ✼✵ ❬✺❪ ❉✳ ❇❡❞♥❛✞r➼❦ ❛♥❞ ❑✳ P❛st♦r✱ ❖♥ s❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✐♥ ❝♦♥str❛✐♥❡❞ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♦♣t✐♠✐③❛t✐♦♥✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✭✷✵✶✶✮ ✶✸✼✷✲✶✸✽✷✳ ✼✹ ❬✻❪ ❏✳❋✳ ❇♦♥♥❛♥s ❛♥❞ ❆✳ ❙❤❛♣✐r♦✱ P❡rt✉r❜❛t✐♦♥ ❆♥❛❧②s✐s ♦❢ ❖♣t✐♠✐③❛t✐♦♥ Pr♦❜❧❡♠s✱ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦ ✭✷✵✵✵✮✳ ❬✼❪ ❋✳❍✳ ❈❧❛r❦❡✱ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ◆♦♥s♠♦♦t❤ ❆♥❛❧②s✐s✱ ❲✐❧❡② ■♥t❡rs❝✐❡♥❝❡✱ ◆❡✇ ❨♦r❦ ✭✶✾✽✸✮✳ ❬✽❪ ❘✳ ❈♦♠✐♥❡tt✐✱ ▼❡tr✐❝ r❡❣✉❧❛r✐t②✱ t❛♥❣❡♥t s❡ts ❛♥❞ s❡❝♦♥❞ ♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s✱ ❆♣♣❧✳ ▼❛t❤✳ ❖♣t✐♠✳ ✭✶✾✾✵✮ ✷✻✺✲✷✽✼✳ ✷✶ ❬✾❪ ❆✳▲✳ ❉♦♥❝❤❡✈✱ ❘✳❚✳ ❘♦❝❦❛❢❡❧❧❛r✱ ■♠♣❧✐❝✐t ❋✉♥❝t✐♦♥s ❛♥❞ ❙♦❧✉t✐♦♥ ▼❛♣♣✐♥❣s✱ ❙♣r✐♥❣❡r✱ ❉♦r❞r❡❝❤t ✭✷✵✵✾✮✳ ❬✶✵❪ ■✳ ●✐♥❝❤❡✈✱ ❆✳ ●✉❡rr❛❣❣✐♦ ❛♥❞ ▼✳ ❘♦❝❝❛✱ ❙❡❝♦♥❞✲♦r❞❡r ❝♦♥❞✐t✐♦♥s ❢♦r C 1,1 ❝♦♥✲ str❛✐♥❡❞ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥✱ ▼❛t❤✳ Pr♦❣r❛♠✳ ✭❙❡r✳ ❇✮ ✭✷✵✵✺✮ ✸✽✾✲✹✵✺✳ ✶✵✹ ❬✶✶❪ ■✳ ●✐♥❝❤❡✈✱ ❆✳ ●✉❡rr❛❣❣✐♦ ❛♥❞ ▼✳ ❘♦❝❝❛✱ ❋r♦♠ s❝❛❧❛r t♦ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥✱ ❆♣♣❧✳ ▼❛t❤✳ ✭✷✵✵✻✮ ✺✲✸✻✳ ✺✶ ❬✶✷❪ ■✳ ●✐♥❝❤❡✈✱ ❖♥ s❝❛❧❛r ❛♥❞ ✈❡❝t♦r l✲st❛❜❧❡ ❢✉♥❝t✐♦♥s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✶✾✹✳ ✼✹ ✭✷✵✶✶✮ ✶✽✷✲ ❬✶✸❪ ❈✳ ●✉t✐➨rr❡③✱ ❇✳ ❏✐♠➨♥❡③ ❛♥❞ ❱✳ ◆♦✈♦✱ ❖♥ s❡❝♦♥❞ ♦r❞❡r ❋r✐t③ ❏♦❤♥ t②♣❡ ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✐♥ ♥♦♥s♠♦♦t❤ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♣r♦❣r❛♠♠✐♥❣✱ ▼❛t❤✳ Pr♦❣r❛♠✳ ✭❙❡r✳ ❇✮ ✭✷✵✶✵✮ ✶✾✾✲✷✷✸✳ ✶✷✸ ❬✶✹❪ ❇✳ ❏✐♠➨♥❡③ ❛♥❞ ❱✳ ◆♦✈♦✱ ❙❡❝♦♥❞ ♦r❞❡r ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ✐♥ s❡t ❝♦♥str❛✐♥❡❞ ❞✐❢❢❡r❡♥t✐❛❜❧❡ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥✱ ▼❛t❤✳ ▼❡t❤♦❞s ❖♣❡r✳ ❘❡s✳ ✭✷✵✵✸✮ ✷✾✾✲✸✶✼✳ ✺✽ ❬✶✺❪ ❇✳ ❏✐♠➨♥❡③ ❛♥❞ ❱✳ ◆♦✈♦✱ ❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✐♥ ❞✐❢❢❡r❡♥t✐❛❜❧❡ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ✈✐❛ s❡❝♦♥❞✲♦r❞❡r t❛♥❣❡♥t s❡ts✱ ❆♣♣❧✳ ▼❛t❤✳ ❖♣t✐♠✳ ✭✷✵✵✹✮ ✶✷✸✲✶✹✹✳ ✹✾ ❬✶✻❪ ❆✳ ❏♦✉r❛♥✐ ❛♥❞ ▲✳ ❚❤✐❜❛✉❧t✱ ❆♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ ♠❡tr✐❝ r❡❣✉❧❛r✐t② ✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠♠✐♥❣ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✱ ▼❛t❤✳ ❖♣❡r✳ ❘❡s✳ ✭✶✾✾✷✮ ✸✾✵✲✹✵✵✳ ✶✽ ❬✶✼❪ ❍✳ ❑❛✇❛s❛❦✐✱ ❆♥ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ❡❢❢❡❝t ♦❢ ✐♥❢✐♥✐t❡❧② ♠❛♥② ✐♥❡q✉❛❧✐t② ❝♦♥str❛✐♥ts ♦♥ s❡❝♦♥❞✲♦r❞❡r ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ❢♦r ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s✱ ▼❛t❤✳ Pr♦❣r❛♠✳ ✭✶✾✽✽✮ ✼✸✲✾✻✳ ✹✶ ❬✶✽❪ P✳◗✳ ❑❤❛♥❤ ❛♥❞ ◆✳❉✳ ❚✉❛♥✱ ❋✐rst ❛♥❞ s❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✉s✐♥❣ ✷✾ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r ♥♦♥s♠♦♦t❤ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✭✷✵✵✻✮ ✷✽✾✲✸✵✽✳ ✶✸✵ ❬✶✾❪ P✳◗✳ ❑❤❛♥❤ ❛♥❞ ◆✳❉✳ ❚✉❛♥✱ ❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ♥♦♥s♠♦♦t❤ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♦♣t✐♠✐③❛t✐♦♥ ✉s✐♥❣ ❍❛❞❛♠❛r❞ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡s✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✭✷✵✵✼✮ ✸✹✶✲✸✺✼✳ ✶✸✸ ❬✷✵❪ P✳◗✳ ❑❤❛♥❤ ❛♥❞ ◆✳❉✳ ❚✉❛♥✱ ❋✐rst ❛♥❞ s❡❝♦♥❞✲♦r❞❡r ❛♣♣r♦①✐♠❛t✐♦♥s ❛s ❞❡r✐✈❛t✐✈❡s ♦❢ ♠❛♣♣✐♥❣s ✐♥ ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ♥♦♥s♠♦♦t❤ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥✱ ❆♣♣❧✳ ▼❛t❤✳ ❖♣t✐♠✳ ✭✷✵✵✽✮ ✶✹✼✲✶✻✻✳ ✺✽ ❬✷✶❪ P✳◗✳ ❑❤❛♥❤ ❛♥❞ ◆✳❉✳ ❚✉❛♥✱ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✉s✐♥❣ ❛♣♣r♦①✐♠❛✲ t✐♦♥s ❢♦r ♥♦♥s♠♦♦t❤ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ✉♥❞❡r ✐♥❝❧✉s✐♦♥ ❝♦♥str❛✐♥ts✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✭✷✵✶✶✮ ✹✸✸✽✲✹✸✺✶✳ ✼✹ ❬✷✷❪ P✳◗✳ ❑❤❛♥❤ ❛♥❞ ◆✳❉✳ ❚✉❛♥✱ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✇✐t❤ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ❡❢❢❡❝t ❢♦r ♥♦♥s♠♦♦t❤ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ✐♥ ✐♥❢✐♥✐t❡ ❞✐♠❡♥s✐♦♥s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✭✷✵✶✸✮ ✶✸✵✲✶✹✽✳ ✼✼ ❬✷✸❪ ▲✳ ▲✐✉✱ P✳ ◆❡✐tt❛❛♥♠☎❛❦✐ ❛♥❞ ▼✳ ❑✞r➼✞③❡❦✱ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ♥♦♥❞♦♠✐♥❛t❡❞ s♦❧✉t✐♦♥s ♦❢ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♣r♦❣r❛♠♠✐♥❣ ✇✐t❤ C 1,1 ❞❛t❛✱ ❆♣♣❧✳ ▼❛t❤✳ ✭✷✵✵✵✮ ✸✽✶✲✸✾✼✳ ✹✺ ❬✷✹❪ ❨✳ ▼❛r✉②❛♠❛✱ ❙❡❝♦♥❞✲♦r❞❡r ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ❢♦r ♥♦♥❧✐♥❡❛r ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜✲ ❧❡♠s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s t♦ ❛♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠✱ ▼❛t❤✳ ❖♣❡r✳ ❘❡s✳ ✭✶✾✾✵✮ ✹✻✼✲✹✽✷✳ ✶✺ ❬✷✺❪ ❏✳P✳ P❡♥♦t✱ ❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠♠✐♥❣ ❛♥❞ ❝♦♠♣♦s✐t❡ ♦♣t✐♠✐③❛t✐♦♥✱ ▼❛t❤✳ Pr♦❣r❛♠✳ ✭✶✾✾✹✮ ✷✷✺✲✷✹✺✳ ✻✼ ❬✷✻❪ ❏✳P✳ P❡♥♦t✱ ❙❡❝♦♥❞ ♦r❞❡r ❝♦♥❞✐t✐♦♥s ❢♦r ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ✇✐t❤ ❝♦♥str❛✐♥ts✱ ❙■❆▼ ❏✳ ❈♦♥tr♦❧ ❖♣t✐♠✳ ✭✶✾✾✾✮ ✸✵✸✲✸✶✽✳ ✸✼ ❬✷✼❪ ❘✳❚✳ ❘♦❝❦❛❢❡❧❧❛r✱ ❈♦♥✈❡① ❆♥❛❧②s✐s✱ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✱ Pr✐♥❝❡t♦♥✱ ◆❡✇ ❏❡r✲ s❡② ✭✶✾✼✵✮✳ ❬✷✽❪ ❉✳ ❊✳ ❲❛r❞✱ ❈❛❧❝✉❧✉s ❢♦r ♣❛r❛❜♦❧✐❝ s❡❝♦♥❞✲♦r❞❡r ❞❡r✐✈❛t✐✈❡s✱ ❙❡t ❱❛❧✉❡❞ ❆♥❛❧✳ ✭✶✾✾✸✮ ✷✶✸✲✷✹✻✳ ✶ ✸✵ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❈→❝ ❜➔✐ ❜→♦ ❦❤♦❛ ❤å❝ ❧✐➯♥ q✉❛♥ trü❝ t✐➳♣ ✤➳♥ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ❬✶❪ P✳ ◗✳ ❑❤❛♥❤ ❛♥❞ ◆✳ ❉✳ ❚✉❛♥✱ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✇✐t❤ t❤❡ ❡♥✈❡❧♦♣❡✲ ❧✐❦❡ ❡❢❢❡❝t ✐♥ ♥♦♥s♠♦♦t❤ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠♠✐♥❣✱ ■✿ l✲st❛❜✐❧✐t② ❛♥❞ s❡t✲✈❛❧✉❡❞ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡s✱ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ✭✷✵✶✸✮ ✻✾✺✲✼✵✷✳ ✹✵✸ ❬✷❪ P✳ ◗✳ ❑❤❛♥❤ ❛♥❞ ◆✳ ❉✳ ❚✉❛♥✱ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✇✐t❤ t❤❡ ❡♥✈❡❧♦♣❡✲ ❧✐❦❡ ❡❢❢❡❝t ✐♥ ♥♦♥s♠♦♦t❤ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠♠✐♥❣✱ ■■✿ ❖♣t✐♠❛❧✐t② ❝♦♥✲ ❞✐t✐♦♥s✱ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ✭✷✵✶✸✮ ✼✵✸✲✼✶✹✳ ✹✵✸ ✸✶ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com