Các điều kiện tối ưu cấp hai với hiện tượng envelope like cho các bài toán tối ưu vectơ không trơn trong các không gian vô hạn chiều

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 CHO CÁC BÀI TỐN TỐI ƯU VECTƠ KHƠNG TRƠN TRONG CÁC KHÔNG GIAN VÔ HẠN CHIỀU Mã số: CS – 2014 - 43 Chủ nhiệm: TS Nguyễn Đình Tuấn Tp Hồ Chí Minh - 2014 ▼Ư❈ ▲Ư❈ ❈❤÷ì♥❣ ♠ð ✤➛✉✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✸ ữỡ ợ t t ự ởt sè ❦✐➳♥ t❤ù❝ ❣✐↔✐ t➼❝❤ ❤➔♠ ❝ơ♥❣ ♥❤÷ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✺ ❈❤÷ì♥❣ ✷✿ ✣↕♦ ❤➔♠ s✉② rë♥❣ ❦✐➸✉ ①➜♣ ①➾ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✾ ❈❤÷ì♥❣ ✸✿ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥ ❝➜♣ ❤❛✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✶✸ ❈❤÷ì♥❣ ✹✿ ❈→❝ ✤✐➲✉ tố ữ t ữợ ♥❣❤✐➯♥ ❝ù✉ ♠ð rë♥❣ ✤➲ t➔✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✸✷ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✸✸ ữỡ ỵ t➔✐✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐ ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣ ✈➻ ♥â ❧➔♠ ❝❤♦ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ♠ët ❤♦➔♥ t❤✐➺♥ ❤ì♥ ❜➡♥❣ ♥❤ú♥❣ t❤ỉ♥❣ t✐♥ ❝➜♣ ❤❛✐ ❣✐ó♣ ➼❝❤ r➜t ♥❤✐➲✉ tr♦♥❣ ✈✐➺❝ ♥❤➟♥ r❛ ❝→❝ ♥❣❤✐➺♠ tè✐ ÷✉ ❝ơ♥❣ ♥❤÷ ✤÷❛ r❛ ❝→❝ t❤✉➟t t♦→♥ sè ✤➸ t➼♥❤ ❝→❝ ♥❣❤✐➺♠ ♥➔②✳ ❇↔♥ ❝❤➜t ❝õ❛ t❤ỉ♥❣ t✐♥ ❝➜♣ ❤❛✐ ♥➔② ❧➔ ♥❤÷ s❛✉✳ ◆â✐ ♠ët ❝→❝❤ ✤ì♥ ❣✐↔♥✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ♠ët ❦❤➥♥❣ ✤à♥❤ r➡♥❣ t↕✐ ✤✐➸♠ ❝ü❝ trà✱ ✤↕♦ ❤➔♠ t ữợ ủ t ✈➔ ❝→❝ r➔♥❣ ❜✉ë❝✱ ❦❤æ♥❣ t❤✉ë❝ ✈➲ ♣❤➛♥ tr♦♥❣ ❝õ❛ ♥â♥ ✭❤đ♣✮ ➙♠ tr♦♥❣ t➼❝❤ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ↔♥❤✳ ✣↕♦ t ữợ õ t tr ♥â♥ ♥â✐ tr➯♥✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐ ❝✉♥❣ ❝➜♣ t❤ỉ♥❣ t✐♥ t❤➯♠✿ ♥â✐ t ữợ r ❧➔ ❦❤æ♥❣ ➙♠✳ ❚✉② ♥❤✐➯♥✱ ✈➔♦ ♥➠♠ ✶✾✽✽✱ ❑❛✇❛s❛❦✐ ❬✶✹❪ ❧➔ ♥❣÷í✐ ✤➛✉ t✐➯♥ ✤➣ ♣❤→t ❤✐➺♥ r❛ r➡♥❣ ❦❤✐ t❛ ①➨t ❜❛♦ ✤â♥❣ ❝õ❛ ♥â♥ ➙♠ ♥â✐ tr➯♥✱ ✤↕♦ ❤➔♠ ❝➜♣ ❤❛✐ ❝õ❛ ❤➔♠ ▲❛❣r❛♥❣❡ ❝â t❤➸ ➙♠ ♥➳✉ t ữợ ởt ủ ♥â✐ tr➯♥ ♥➡♠ tr➯♥ ♣❤➛♥ ✤➦❝ ❜✐➺t ❝õ❛ ❜✐➯♥ ❝õ❛ ♥â♥ ➙♠✳ ➷♥❣ ❣å✐ ❤✐➺♥ t÷đ♥❣ ♥➔② ❧➔ ❤✐➺♥ t÷đ♥❣ ự ổ ú ỵ ❤✐➺♥ t÷đ♥❣ ♥➔② ✈➔ ♠➢❝ ♣❤↔✐ s❛✐ ❧➛♠ ✤→♥❣ t✐➳❝✳ ◆❤✐➲✉ t→❝ ❣✐↔ ❦❤→❝ ❝❤➾ ①➨t ♥â♥ ➙♠ ♥â✐ tr➯♥✱ ❦❤æ♥❣ ①➨t ❜❛♦ ✤â♥❣ ❝õ❛ ♥â♥ ♥➔②✱ ✈➔ ✈➻ t❤➳ ❦❤ỉ♥❣ ❝â ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ①↔② r❛✳ ✣➣ ❝â ♥❤✐➲✉ ✤â♥❣ ❣â♣ q✉❛♥ trå♥❣ ❝❤♦ ❤✐➺♥ t÷đ♥❣ t❤ó ✈à ♥➔②✳ ❈→❝ ❦➳t q✉↔ ❝õ❛ ❑❛✇❛s❛❦✐ ✤÷đ❝ ♠ð rë♥❣ ✈➔ ♣❤→t tr t q ổ ữợ ✈✐ ❝➜♣ ❤❛✐ tr♦♥❣ ❬✸✱ ✺✱ ✷✹✱ ✷✺❪✱ q✉② ❤♦↕❝❤ ✤❛ ♠ö❝ t✐➯✉ ❦❤↔ ✈✐ ❝➜♣ ❤❛✐ tr♦♥❣ ❬✶✵✱ ✶✶❪✱ q✉② ❤♦↕❝❤ ✤❛ ♠ö❝ t✐➯✉ ✭❤ú✉ ❤↕♥ ❝❤✐➲✉✮ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ❝➜♣ ♠ët tr♦♥❣ ❬✼❪ ✈➔ ❝❤♦ q✉② ❤♦↕❝❤ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ❝❤➦t tr♦♥❣ ❬✷✵✱ ✷✶❪✳ ❈❤ó♥❣ tỉ✐ ♥❤➟♥ t❤➜② r➡♥❣ ❝➛♥ ♣❤↔✐ ❣✐↔✐ t❤➼❝❤ rã r➔♥❣ ❤ì♥ ❦❤✐ ♥➔♦ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ①↔② r❛ ✈➔ ❦❤✐ ♥➔♦ ❤✐➺♥ t÷đ♥❣ ♥➔② ❦❤ỉ♥❣ ①↔② r❛✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ ❝❤ó♥❣ tỉ✐ s➩ ❧➔♠ rã ỡ ố ợ ỳ ữợ r tữủ ❍ì♥ ♥ú❛✱ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐ t♦→♥ ✈ỵ✐ ♠ù❝ ✤ë ❦❤ỉ♥❣ trì♥ ❝➜♣ ❝❛♦ ❤ì♥ ❧✉ỉ♥ ❧✉ỉ♥ ❧➔ ♠ët ♥❤✉ ❝➛✉ t❤ü❝ t➳✳ ❉♦ ✤â✱ tr♦♥❣ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ❝❤å♥ ❝→❝ ①➜♣ ①➾ ✤➣ ✤÷đ❝ ✤➲ ①✉➜t tr♦♥❣ ❬✶✱ ✶✸❪ ❞ò♥❣ ❧➔♠ ❝→❝ ✤↕♦ ❤➔♠ s✉② rë♥❣✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐ ❞ò♥❣ ❝→❝ ữủ ự tr ợ t❤✐➳t r➡♥❣ t➜t ❝↔ ❝→❝ ①➜♣ ①➾ ✤÷đ❝ sû ❞ư♥❣ ❧➔ ❝♦♠♣❛❝t✳ ❈→❝ ①➜♣ ①➾ ❝â t❤➸ ❦❤æ♥❣ ❜à ❝❤➦♥ ✤➣ ✤÷đ❝ ❞ò♥❣ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ tr♦♥❣ ❬✶✺✱ ✶✼✲✶✾❪ ❝❤♦ ♥❤✐➲✉ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ❦❤→❝ ♥❤❛✉✳ ✣↕♦ ❤➔♠ s✉② rë♥❣ t❤✉ë❝ ❧♦↕✐ ♥➔② t✐➺♥ ❧ñ✐ ð ❝❤ê ❧➔ ♥❣❛② ❝↔ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❧✐➯♥ tư❝ t↕✐ ♠ët ✤✐➸♠ ❝â t❤➸ ❝â ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❦❤æ♥❣ t➛♠ t❤÷í♥❣ t↕✐ ✤✐➸♠ ♥➔②✳ ❚✉② ♥❤✐➯♥✱ ✤➸ t➟♣ tr✉♥❣ tr➯♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➜♣ ❤❛✐ ✈➔ ①→❝ ✤à♥❤ rã ❝→❝ ữợ ữủ r tữủ ❡♥✈❡❧♦♣❡✲❧✐❦❡✱ ❝❤ó♥❣ tỉ✐ ❝❤õ ②➳✉ ①➨t ❝→❝ →♥❤ ①↕ ❦❤↔ ởt q st tr ỗ ❤ù♥❣ ❝❤♦ ♠ư❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❝❤ó♥❣ tỉ✐ tr♦♥❣ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ♥➔② ❧➔ →♣ ❞ö♥❣ ✤↕♦ ❤➔♠ s✉② rë♥❣ ❦✐➸✉ ①➜♣ ①➾ ✤➸ t❤✐➳t ❧➟♣ ❝→❝ ✤✐➲✉ tố ữ ợ ợ tữủ ❝❤♦ ❝→❝ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ❦❤ỉ♥❣ trì♥ tr♦♥❣ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✈æ ❤↕♥ ❝❤✐➲✉✳ ❈→❝ →♥❤ ①↕ tr♦♥❣ ❜➔✐ t♦→♥ ♥❣❤✐➯♥ ❝ù✉ ❧➔ ❦❤↔ ✈✐ ❝❤➦t ✭tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥✮ ❤❛② ❦❤↔ ✈✐ ✭tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ✮ ✈➔ ❦❤ỉ♥❣ ❝➛♥ ❦❤↔ ✈✐ ❧✐➯♥ tö❝✳ ❈→❝ ❦➳t q✉↔ ♥➔② ❝↔✐ t❤✐➺♥ ✈➔ ♠ð rë♥❣ ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ✸ ❣➛♥ ✤➙②✳ ✷✳ ▼ư❝ t✐➯✉ ✈➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉✳ ❈❤ó♥❣ tỉ✐ ①❡♠ ①➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì tr♦♥❣ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✈æ ❤↕♥ ❝❤✐➲✉ s❛✉ ✤➙②✳ ❈❤♦ X, Z, W ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ Y ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ C Y õ ỗ õ K Z t ỗ f : X Y ✱ g : X → Z ✱ ✈➔ h : X W t ữợ sü ①❡♠ ①➨t ❝õ❛ ❝❤ó♥❣ tỉ✐ ❧➔ ✭P✮ ♠✐♥C f (x)✱ s❛♦ ❝❤♦ g(x) ∈ −K, h(x) = 0✳ ❈❤ó♥❣ tỉ✐ ❞ò♥❣ ❝→❝ ✤↕♦ ❤➔♠ s✉② rë♥❣ t❤❡♦ ♥❣❤➽❛ ①➜♣ ợ ự ổ trỡ ữợ t❤✐➳t ❦❤↔ ✈✐ ❝❤➦t ✭tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥✮ ❤❛② ❦❤↔ ✈✐ ✭tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ✮✱ tr→♥❤ ❣✐↔ t❤✐➳t ❦❤↔ ✈✐ ❧✐➯♥ tö❝✱ ✤➸ t❤✐➳t tố ữ ợ t ❝❤➜t ❡♥✈❡❧♦♣❡✲❧✐❦❡ ❝❤♦ ❜➔✐ t♦→♥ ✭P✮✳ ❈→❝ ❦➳t q✉↔ ❝õ❛ ❝❤ó♥❣ tỉ✐ ❧➔♠ rã ❤ì♥ ✈➜♥ ✤➲ ❦❤✐ ♥➔♦ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ①↔② r❛ ✈➔ ❤♦➔♥ t❤✐➺♥ ❝→❝ ❦➳t q✉↔ ✤➣ ❝â tr♦♥❣ ❧➽♥❤ ✈ü❝ ♥❣❤✐➯♥ ❝ù✉ ♥➔②✳ ❈ö t❤➸✱ ✤➲ t➔✐ t❤ü❝ ❤✐➺♥ ❝→❝ ♠ö❝ t✐➯✉ ♥❣❤✐➯♥ ❝ù✉ s❛✉ ✤➙②✳ ✰ ❑❤→✐ ♥✐➺♠ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐✳ ✰ ❑❤→✐ ♥✐➺♠ ✤↕♦ ❤➔♠ s✉② rë♥❣ ❦✐➸✉ ①➜♣ ①➾ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ ✰ tố ữ ợ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ❝❤♦ ❝→❝ ♥❣❤✐➺♠ ②➳✉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ✭P✮✳ ✰ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ ❝➜♣ ❤❛✐ ❝❤♦ ❝→❝ ♥❣❤✐➺♠ ❝❤➢❝ ❝❤➢♥ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ✭P✮✳ ❈→❝ ❦➳t q✉↔ ❝õ❛ ✤➲ t➔✐ ❤♦➔♥ t❤✐➺♥ ❝→❝ ❦➳t q✉↔ ✤➣ ❝â tr♦♥❣ ❧➽♥❤ ✈ü❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝❤♦ ❝→❝ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ❦❤ỉ♥❣ trì♥ tr♦♥❣ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✈æ ❤↕♥ ❝❤✐➲✉✳ ❈→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ♥➔② ✤➣ ✤÷đ❝ t→❝ ❣✐↔ ✈➔ ●❙✳❚❙❑❍✳ P❤❛♥ ◗✉è❝ ❑❤→♥❤✱ tr÷í♥❣ ✣↕✐ ❤å❝ ◗✉è❝ t➳✱ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❚♣✳ ❍❈▼ ❝æ♥❣ ❜è tr♦♥❣ ♠ët ❜➔✐ ❜→♦ tr➯♥ t↕♣ ❝❤➼ ❦❤♦❛ ❤å❝ q✉è❝ t➳ tr♦♥❣ ❤➺ t❤è♥❣ ■❙■ ❬✷✷❪✿ P✳◗✳ ❑❤❛♥❤ ❛♥❞ ◆✳❉✳ ❚✉❛♥✱ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✇✐t❤ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ❡❢❢❡❝t ❢♦r ♥♦♥s♠♦♦t❤ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ✐♥ ✐♥❢✐♥✐t❡ ❞✐♠❡♥s✐♦♥s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✭✷✵✶✸✮ ✶✸✵✲✶✹✽✳ ✼✼ ✸✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉✳ ✣➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ❞ò♥❣ ❝→❝ ❝ỉ♥❣ ❝ư ✈➔ ❦ÿ t❤✉➟t tr♦♥❣ ❣✐↔✐ t➼❝❤ ❦❤ỉ♥❣ trì♥✱ ❣✐↔✐ t➼❝❤ ✤❛ trà ✈➔ ❣✐↔✐ t➼❝❤ ❤➔♠✳ ✹✳ ❑➳t ❝➜✉ ❝õ❛ ✤➲ t➔✐✳ ✣➲ t ỗ ữỡ ữỡ ỵ ❞♦ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✱ ♠ö❝ t✐➯✉ ✈➔ ❦➳t q✉↔ ự t ữỡ ợ t ❜➔✐ t♦→♥ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ♠ët sè ❦✐➳♥ t❤ù❝ ❣✐↔✐ t➼❝❤ ❤➔♠ ❝ơ♥❣ ♥❤÷ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ t t ú ởt ữỡ ✷✿ ✣↕♦ ❤➔♠ s✉② rë♥❣ ❦✐➸✉ ①➜♣ ①➾ ❝➜♣ ♠ët ữỡ tố ÷✉ ❝➛♥ ❝➜♣ ❤❛✐✳ • ❈❤÷ì♥❣ ✹✿ ❈→❝ ✤✐➲✉ ❦✐➺♥ tố ữ ữỡ ợ t ❜➔✐ t♦→♥ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ♠ët sè ❦✐➳♥ t❤ù❝ ❣✐↔✐ t➼❝❤ ❤➔♠ ❝ơ♥❣ ♥❤÷ ♠ët sè ❦❤→✐ ♥✐➯♠ ✈➲ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ ❈❤♦ X, Z, W ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ Y ❧➔ ❦❤æ♥❣ C Y õ ỗ õ K Z t ỗ f : X → Y ✱ g : X → Z ✱ ✈➔ h : X → W ❧➔ ❝→❝ →♥❤ ①↕✳ ❈❤ó♥❣ tỉ✐ ①➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì s❛✉ ✤➙②✿ ✭P✮ ♠✐♥C f (x)✱ s❛♦ ❝❤♦ g(x) ∈ −K, h(x) = ú tổ ũ ỵ ỡ N = {1, 2, , n, } ✈➔ R ❧➔ t➟♣ ❤đ♣ ❝→❝ sè t❤ü❝✳ ❱ỵ✐ ❦❤ỉ♥❣ ❣✐❛♥ ✤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}❀ BX (x, r) = {y ∈ X : x − y < r}✱ SX = {y ∈ X : y = 1} ✈➔ ✤è✐ ✈ỵ✐ BX (0, 1) t❛ ✈✐➳t ✤ì♥ ❣✐↔♥ ❧➔ BX ✳ L(X, Y ) ỵ ổ t t➼♥❤ ❜à ❝❤➦♥ tø X ✈➔♦ Y ✈➔ B(X, X, Y ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ →♥❤ ①↕ s♦♥❣ t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥ tø X × X ✈➔♦ Y ✱ tr♦♥❣ ✤â X ✈➔ Y ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✳ ❱ỵ✐ Pn ✱ P tr♦♥❣ L(X, Y )✱ t❛ ✈✐➳t ♣ Pn − → P ❤❛② P = ♣✲lim Pn Pn tử P ỵ ❤✐➺✉ t÷ì♥❣ tü ✤÷đ❝ ❞ò♥❣ ❝❤♦ Mn , M ∈ B(X, X, Y )✳ ❱ỵ✐ ♥â♥ C ⊂ X ✱ ỵ C = {c X : c∗ , c ≥ 0, ∀c ∈ C} ❧➔ ♥â♥ ố ỹ ữỡ C ợ A X ỵ rA tA A A A A ✈➔ A(x) ❧➛♥ ❧÷đt ❧➔ ♣❤➛♥ tr♦♥❣ t÷ì♥❣ ✤è✐✱ ♣❤➛♥ tr õ õ ỗ A ✈➔ ❜❛♦ ♥â♥ ❝õ❛ ♣❤➛♥ ❞à❝❤ ❝❤✉②➸♥ A + x✳ ❱ỵ✐ t > ✈➔ r ∈ N✱ o(tr ) ỵ ởt tở t s❛♦ ❝❤♦ o(tr )/tr → ❦❤✐ t → 0+ ✳ C 1,1 ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ →♥❤ ①↕ ❦❤↔ ✈✐ ❋r➨❝❤❡t s❛♦ ❝❤♦ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣✳ ❚r♦♥❣ ♣❤➛♥ ♥➔② t❛ ①➨t X ✱ Y ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✈➔ h : X → Y ❧➔ →♥❤ ①↕✳ ❚❛ ♥â✐ h ❧➔ ê♥ ✤à♥❤ t x0 tỗ t ởt U x0 ✈➔ κ > s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ x ∈ U✱ h(x) − h(x0 ) ≤ κ x − x0 ✳ h ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ❝❤➦t t↕✐ x0 ∈ X ♥➳✉ ♥â ❝â ✤↕♦ ❤➔♠ ❋r➨❝❤❡t h (x0 ) t↕✐ x0 ✈➔ limy→x0 ,y →x0 h(y) − h(y ) − h (x0 )(y − y ) = 0✳ y−y ❍✐➸♥ ♥❤✐➯♥ r➡♥❣ ♥➳✉ h ❧➔ ❦❤↔ ✈✐ ❝❤➦t t↕✐ x0 ✱ t❤➻ h ❧➔ ▲✐♣s❝❤✐t③ ❣➛♥ x0 ✳ ❑➳t q✉↔ s❛✉ ✤➙② ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ♠ët ❝→❝❤ t÷ì♥❣ tü ♥❤÷ ❇ê ✤➲ ✸ ❝õ❛ ❬✼❪✳ ▼➺♥❤ ✤➲ ✶✳✶✳ ❈❤♦ h ❧➔ →♥❤ ①↕ ❦❤↔ ✈✐ ❋r➨❝❤❡t q✉❛♥❤ x ∈ X ✈ỵ✐ h ❧➔ ê♥ ✤à♥❤ t↕✐ x0 ✱ ✈➔ u, w ∈ X ✳ ◆➳✉ (tn , rn ) → (0 , )✱ tn /rn → ✱ ✈➔ wn := (xn − x0 − tn u)/ 12 tn rn → w✱ t❤➻ h(xn ) − h(x0 ) − tn h (x0 )u yn := → h (x0 )w✳ tn rn /2 + + + ❚❛ ♥❤ỵ ❧↕✐ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ ♥â♥ t✐➳♣ ①ó❝ ✈➔ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ❤❛✐ s❛✉ ✤➙②✳ ✺ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ❈❤♦ x , u ∈ X ✈➔ S ⊂ X ✳ ✭❛✮ ◆â♥ ❝♦♥t✐♥❣❡♥t ✭❤❛② ❇♦✉❧✐❣❛♥❞✮ ❝õ❛ S t↕✐ x0 ❧➔ T (S, x0 ) = {v ∈ X | ∃tn → 0+ , ∃vn → v, ∀n ∈ N, x0 + tn ∈ S}✳ ✭❜✮ ◆â♥ t✐➳♣ ①ó❝ tr♦♥❣ ✭♥â♥ t✐➳♣ ①ó❝ tr♦♥❣ ❈❧❛r❦❡✱ t÷ì♥❣ ù♥❣✮ ❝õ❛ S t↕✐ x0 ❧➔ IT (S, x0 ) = {v ∈ X | ∀tn → 0+ , ∀vn → v, ∀n ✤õ ❧ỵ♥, x0 + tn ∈ S} ✭ITC (S, x0 ) = {v ∈ X | ∀xn →S x0 , ∀tn → 0+ , ∀vn → v, ∀n ✤õ ❧ỵ♥, xn + tn ∈ S}✮✳ ✭❝✮ ❚➟♣ ❝♦♥t✐♥❣❡♥t ✭t➟♣ ❦➲✱ t÷ì♥❣ ù♥❣✮ ❝➜♣ ❤❛✐ ❝õ❛ S t↕✐ (x0 , u) ❧➔ T (S, x0 , u) = {w ∈ X | ∃tn → 0+ , ∃wn → w, ∀n ∈ N, x0 + tn u + 21 t2n wn ∈ S} ✭A2 (S, x0 , u) = {w ∈ X | ∀tn → 0+ , ∃wn → w, ∀n ∈ N, x0 + tn u + 21 t2n wn ∈ S}✮✳ ✭❞✮ ◆â♥ t✐➳♣ ①ó❝ ✭♥â♥ ❦➲✱ t÷ì♥❣ ù♥❣✮ ❝➜♣ ❤❛✐ t✐➺♠ ❝➟♥ ❝õ❛ S t↕✐ (x0 , u) ❧➔ T (S, x0 , u) = {w ∈ X | ∃(tn , rn ) → (0+ , 0+ ) : tn rn → 0, ∃wn → w✱ ∀n ∈ N, x0 + tn u + 12 tn rn wn ∈ S} ✭A (S, x0 , u) = {w ∈ X | ∀(tn , rn ) → (0+ , 0+ ) : tn rn → 0, ∃wn → w, ∀n ∈ N, x0 + tn u + 12 tn rn wn ∈ S}✮✳ ✭❡✮ ❚➟♣ t✐➳♣ ①ó❝ tr♦♥❣ ❝➜♣ ❤❛✐ ❝õ❛ S t↕✐ (x0 , u) ❧➔ IT (S, x0 , u) = {w ∈ X | ∀tn → 0+ , ∀wn → w, ∀n ✤õ ❧ỵ♥, x0 + tn u + 21 t2n wn ∈ S}✳ ✭❢✮ ◆â♥ t✐➳♣ ①ó❝ tr♦♥❣ ❝➜♣ ❤❛✐ t✐➺♠ ❝➟♥ ❝õ❛ S t↕✐ (x0 , u) ❧➔ IT (S, x0 , u) = {w ∈ X | ∀(tn , rn ) → (0+ , 0+ ) : tn rn → 0, ∀wn → w, ∀n ✤õ ❧ỵ♥, x0 + tn u + 12 tn rn wn ∈ S}✳ ❈→❝ ♥â♥ T (S, x0 )✱ IT (S, x0 ) ✈➔ ITC (S, x0 ) ✈➔ ❝→❝ t➟♣ T (S, x0 , u)✱ A2 (S, x0 , u) ✈➔ IT (S, x0 , u) ✤÷đ❝ ❜✐➳t rã✳ ❈→❝ ♥â♥ A (S, x0 , u) ✈➔ T (S, x0 , u) ✤÷đ❝ P❡♥♦t ❬✷✺✱ ✷✻❪ sû ❞ư♥❣✳ ❈❤ó♥❣ tæ✐ ✤à♥❤ ♥❣❤➽❛ ♥â♥ IT (S, x0 , u) ♠ët tỹ ữ ỵ r x0 clS ✱ t❤➻ t➜t ❝↔ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ð tr➯♥ ❧➔ ré♥❣✳ ❱➻ t❤➳✱ ❝❤ó♥❣ tỉ✐ ❧✉ỉ♥ ①➨t ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝❤➾ t↕✐ ♥❤ú♥❣ ✤✐➸♠ t❤✉ë❝ ❜❛♦ ✤â♥❣ ❝õ❛ t➟♣ ✤❛♥❣ ①➨t✳ ❈❤ó♥❣ tỉ✐ ✤÷❛ r❛ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ ð tr➯♥ tr♦♥❣ ❜❛ ♠➺♥❤ ✤➲ s❛✉ ✤➙②✳ ▼➺♥❤ ✤➲ ✶✳✸✳ ❈❤♦ S ⊂ X ✈➔ x , u ∈ X ✳ ❑❤✐ ✤â✱ ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ✤÷đ❝ ❜✐➳t rã ✭✐✮ IT (S, x0 , u) ⊂ A (S, x0 , u) ⊂ T (S, x0 , u) ⊂ clcone[cone(S − x0 ) − u]❀ 2 ✭✐✐✮ IT (S, x0 , u) = IT (intS, x0 , u) ✈➔ ♥➳✉ u ∈ bd[cone(S −x0 )]✱ t❤➻ ∈ IT (S, x0 , u)❀ ✭✐✐✐✮ ♥➳✉ u ∈ T (S, x0 )✱ t❤➻ T (S, x0 , u) = ∅✳ sỷ t ỳ S ỗ tS = ✈➔ u ∈ T (S, x0 )✳ ❚❛ ❝â ✤✐➲✉ s❛✉ ✭❬✶✶✱ ✷✸✱ ✷✾❪✮✿ ✭✐✈✮ ✐♥t❝♦♥❡(S − x0 ) = IT (intS, x0 ) = ITC (intS, x0 ✮ ✈➔ ❞♦ ✤â ∈ intcone(S − x0 ) ✈ỵ✐ x0 ∈ intS ❀ ✭✈✮ ♥➳✉ A2 (S, x0 , u) = ∅✱ t❤➻ ✻ IT (S, x0 , u) = intA2 (S, x0 , u), clIT (S, x0 , u) = A2 (S, x0 , u); ✭✈✐✮ ♥➳✉ u ∈ cone(S − x0 )✱ t❤➻ ✭❛✮ IT (S, x0 , u) = intcone[cone(S − x0 ) − u]❀ ✭❜✮ A2 (S, x0 , u) = clcone[cone(S − x0 ) − u]✳ ▼➺♥❤ ✤➲ ✶✳✹✳ ❈❤♦ S ⊂ X ✈➔ x , u ∈ X ✳ ✭✐✮ IT (S, x0 , u) ⊂ A (S, x0 , u) ⊂ T (S, x0 , u) ⊂ clcone[cone(S − x0 ) − u]✳ ✭✐✐✮ IT (S, x0 , u) = IT (intS, x0 , u) ✈➔ ♥➳✉ u ∈ bd[cone(S−x0 )]✱ t❤➻ ∈ IT (S, x0 , u)✳ ✭✐✐✐✮ ◆➳✉ u ∈ T (S, x0 )✱ t❤➻ T (S, x0 , u) = ∅✳ ✭✐✈✮ A (S, x0 , u) + ITC (S, x0 ) ⊂ IT (S, x0 , u)✱ ✈➔ ❞♦ ✤â✱ ♥➳✉ ITC (S, x0 ) = ∅ ✈➔ A (S, x0 , u) = ∅✱ t❤➻ IT (S, x0 , u) = intA (S, x0 , u), clIT (S, x0 , u) = A (S, x0 , u) S ỗ x0 ∈ S ✱ t❤➻ A (S, x0 , u) + T (T (S, x0 ), u) ⊂ A (S, x0 , u) ⊂ T (T (S, x0 ), u) ✈➔ ❞♦ ✤â✱ ♥➳✉ A (S, x0 , u) = ∅✱ t❤➻ A (S, x0 , u) = T (T (S, x0 ), u)✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈→❝ ♣❤➛♥ ✭✐✮✲✭✐✐✐✮ ✤÷đ❝ s✉② r❛ tø ❝→❝ ✤à♥❤ ♥❣❤➽❛✳ ❱ỵ✐ ♣❤➛♥ ✭✈✮✱ ①❡♠ ❇ê ✤➲ ✹✳✶ ❝õ❛ ❬✷✽❪✳ ●✐í ✤➙②✱ t❛ ①➨t ♣❤➛♥ ✭✐✈✮✳ ❈❤♦ w ∈ A (S, x0 , u)✱ v ∈ ITC (S, x0 ) ✈➔ z := w + v ✳ ❈❤♦ (tn , rn ) → (0+ , 0+ )✿ tn /rn → 0✱ ✈➔ zn → z õ tỗ t wn w s xn := x0 + tn u + 21 tn rn wn ∈ S ✳ ❱➻ := zn − wn → v ✱ t❛ ❝â z ∈ IT (S, x0 , u) ✈➻✱ ✈ỵ✐ n ❧ỵ♥✱ x0 + tn u + 21 tn rn zn = xn + 12 tn rn ∈ S ✳ ▼➺♥❤ ✤➲ ✶✳✺✳ ●✐↔ sû r➡♥❣ X = R ✈➔ x0 ∈ S ⊂ X ✳ ◆➳✉ xn ∈ S \ {x0 } ❤ë✐ tö x0 t tỗ t u T (S, x0 ) \ {0} ❝â ❝❤✉➞♥ ❜➡♥❣ ♠ët ✈➔ ♠ët ỵ xn s m ✭✐✮ ✭❝ê ✤✐➸♥✮ (xn − x0 )/tn → u✱ tr♦♥❣ ✤â tn = xn − x0 ❀ ✭✐✐✮ ✭❬✶✶❪✮ ❤♦➦❝ z ∈ T (S, x0 , u) ∩ u tỗ t s (xn x0 tn u)/ 12 t2n → z ❤♦➦❝ z ∈ T (S, x0 , u)u \{0} rn 0+ tỗ t s❛♦ ❝❤♦ rtnn → 0+ ✈➔ (xn −x0 −tn u)/ 12 tn rn → z ✱ tr♦♥❣ ✤â u⊥ ❧➔ ♣❤➛♥ ❜ò trü❝ ❣✐❛♦ ❝õ❛ u ∈ Rm ✳ ✼ ✽ ❈❤÷ì♥❣ ✷✿ ✣↕♦ ❤➔♠ s✉② rë♥❣ ❦✐➸✉ ①➜♣ ①➾ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ ✣à♥❤ ♥❣❤➽❛ ✷✳✶ ✭❬✶✱ ✶✸❪✮✳ ❳➨t h : X → Y ❧➔ →♥❤ ①↕✳ ✭✐✮ ❚➟♣ Ah (x0 ) ⊂ L(X, Y ) ✤÷đ❝ ❣å✐ ❧➔ ①➜♣ ①➾ ❝➜♣ ♠ët ❝õ❛ h t↕✐ x0 ♥➳✉✱ ợ x tr ởt x0 tỗ t↕✐ r → 0+ s❛♦ ❝❤♦ r x − x0 −1 → ❦❤✐ x → x0 ✈➔✱ h(x) − h(x0 ) ∈ Ah (x0 )(x − x0 ) + rBY ✳ ✭✐✐✮ ❈➦♣ (Ah (x0 ), Bh (x0 ))✱ ✈ỵ✐ Ah (x0 ) ⊂ L(X, Y ) ✈➔ Bh (x0 ) ⊂ B(X, X, Y )✱ ✤÷đ❝ ❣å✐ ❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ h t↕✐ x0 ♥➳✉ Ah (x0 ) ❧➔ ①➜♣ ①➾ ❝➜♣ ♠ët ❝õ❛ h t↕✐ x0 ✱ ✈➔ ✈ỵ✐ x tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ x0 tỗ t r 0+ s r x − x0 −1 → ❦❤✐ x → x0 ✈➔ h(x) − h(x0 ) ∈ Ah (x0 )(x − x0 ) + Bh (x0 )(x − x0 , x − x0 ) + r2 BY ◆❤➟♥ ①➨t ✷✳✷✳ ✭✐✮ ◆➳✉ h : X → Y ❝â ✤↕♦ ❤➔♠ ❋r➨❝❤❡t ❝➜♣ ❤❛✐ h (x )✱ t❤➻ (h (x ), ❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ h t↕✐ x0 ✳ h (x0 )) ✭✐✐✮ ✭❬✶✱ ✶✸❪✮ ◆➳✉ h : Rn → Rm ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ t↕✐ x0 ✱ t❤➻ ❏❛❝♦❜✐❛♥ ❈❧❛r❦❡ ❬✹❪ ∂C h(x0 ) ❧➔ ①➜♣ ①➾ ❝➜♣ ♠ët ❝õ❛ h t↕✐ x0 ✳ ◆➳✉✱ t❤➯♠ ♥ú❛✱ h t❤✉ë❝ ❧ỵ♣ C 1,1 t↕✐ x0 ✱ t❤➻ (h (x0 ), 12 ∂C2 g(x0 )) ❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ h t↕✐ x0 ✱ tr♦♥❣ ✤â ∂C2 h(x0 ) ❧➔ ❍❡ss✐❛♥ ❈❧❛r❦❡ ❬✽❪ ❝õ❛ h t↕✐ x0 ✳ ✭✐✐✐✮ ✭❬✶✺❪✮ ◆➳✉ h : Rn → Rm ❧➔ ❧✐➯♥ tö❝ ✈➔ ❝â →♥❤ ①↕ tü❛ ❏❛❝♦❜✐❛♥ ❬✾❪ ∂h(.) ❧➔ ♥ú❛ ❧✐➯♥ tö❝ tr➯♥ t↕✐ x0 ✱ t❤➻ ❝♦∂h(x0 ) ❧➔ ①➜♣ ①➾ ❝➜♣ ♠ët ❝õ❛ h t↕✐ x0 ✳ ◆➳✉ h ❧➔ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ❋r➨❝❤❡t tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ x0 ✈➔ ❝â →♥❤ ①↕ tü❛ ❍❡ss✐❛♥ ❬✾❪ ∂ h(.) ❧➔ ♥ú❛ ❧✐➯♥ tö❝ tr➯♥ t↕✐ x0 ✱ t❤➻ (h (x0 ), 12 ❝♦∂ h(x0 )) ❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ h t↕✐ x0 ✳ ❉♦ ✤â✱ ❝→❝ ①➜♣ ①➾ ❧➔ ❝→❝ ✤↕♦ ❤➔♠ s✉② rë♥❣ r➜t tê♥❣ q✉→t✳ ❍ì♥ ♥ú❛✱ ♠é✐ →♥❤ ①↕ h ✤➲✉ ❝â ♠ët ①➜♣ ①➾ t➛♠ t❤÷í♥❣ t↕✐ ❜➜t ❝ù ✤✐➸♠ ♥➔♦✱ ❧➔ t♦➔♥ ❜ë ❦❤ỉ♥❣ ❣✐❛♥ L(X, Y )✳ ❈→❝ ✤↕♦ ❤➔♠ ❦✐➸✉ ①➜♣ ①➾ t✐➺♥ ❧đ✐ ❦❤✐ ❞ò♥❣ ❤ì♥ s♦ ✈ỵ✐ ❝→❝ ✤↕♦ ❤➔♠ s rở õ t tỗ t↕✐ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ ♥❣❛② ❝↔ ❝❤♦ →♥❤ ①↕ ❦❤ỉ♥❣ ❧✐➯♥ tö❝✳ ❱➼ ❞ö ❝❤♦ h : R → R ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐  √  x ♥➳✉ x > 0, h(x) = ♥➳✉ x = 0,  −1 x ♥➳✉ x < ❑❤✐ ✤â h ❧➔ ❦❤ỉ♥❣ ❧✐➯♥ tư❝ t↕✐ ✵ ✈➔ t❛ ❝â t❤➸ ❧➜② Ah (0) = (α, +∞) ✈ỵ✐ ❜➜t ❦ý α > ✈➔ Bh (0) = {0}✱ →♥❤ ①↕ ❦❤æ♥❣ tø R ✈➔♦ R✳ ❚✉② ♥❤✐➯♥✱ t❛ ❦❤æ♥❣ ❝â t➼♥❤ ❞✉② ♥❤➜t ❝❤♦ ❝→❝ ①➜♣ ①➾✳ ✣➦❝ ❜✐➺t✱ ❜➜t ❦ý t➟♣ ♥➔♦ ❝❤ù❛ ♠ët ①➜♣ ①➾ t❤➻ ♥â ❝ô♥❣ ❧➔ ♠ët ữợ ự tọ r ❝→❝ ✤↕♦ ❤➔♠ s✉② rë♥❣ ❝➜♣ ♠ët ð tr➯♥ ❝â t❤➸ ❜➡♥❣ ♥❤❛✉ ❤❛② ❦❤→❝ ♥❤❛✉ ❬✶✺❪✳ ❱➼ ❞ö ✷✳✶✳ ❈❤♦ h : R → R ①→❝ ✤à♥❤ ❜ð✐ x2 sin(1/x) + |y| ♥➳✉ x = 0, h(x, y) = |y| ♥➳✉ x = ❑❤✐ ✤â✱ h ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ t↕✐ ✭✵✱ ✵✮ ✈➔ t❛ ❝â tü❛ ❏❛❝♦❜✐❛♥ ✾ h (x0 )(X) = W ❝â t❤➸ ❜ä t tỗ t c C \ {0} ✈ỵ✐ c∗ , f (x0 )u = s❛♦ ❝❤♦ c , f (x0 )w ữ ỵ r ♥➳✉ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ❙❧❛t❡r ❬✶✹❪ t❤ä❛✱ t❤➻ ❜ð✐ ◆❤➟♥ ①➨t ✻✳✷ ❝õ❛ ❬✶✷❪✱ ✤✐➲✉ ❦✐➺♥ ✭❚❘u ✮ ❝ô♥❣ t❤ä❛✳ ❱➻ t❤➳✱ ❍➺ q✉↔ ✸✳✺ ✭✐✐✲✐✐✐✮ ♠ð rë♥❣ ✣à♥❤ ỵ tr õ Y = R K ❧➔ ♥â♥ ✈➔ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ❙❧❛t❡r ✤÷đ❝ ❞ò♥❣✳ t t ỵ ✸✳✹ ✭✐✐✐✮ ❣✐↔ t❤✐➳t r➡♥❣ (g, h) ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t q✉❛♥❤ x0 ✈➔ (f, g) ❧➔ ê♥ ✤à♥❤ t↕✐ x0 ữợ q tr t ữợ t (x0 , u) ố ợ K ì {0} t ✭✐✐✐✮ trð ♥➯♥ ♠↕♥❤ ❤ì♥ ♥❤÷ 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 ❜ð✐ −K ✱ t❛ ❝â✱ ✈ỵ✐ S = (g, h)−1 (−K × {0}) = G ∩ H ✱ T (S, x0 , u) = {w ∈ X | (g, h) (x0 )w ∈ T (−K × {0}, (g, h)(x0 ), (g, h) (x0 )u)} = {w ∈ X | g (x0 )w ∈ T (−K, g(x0 ), g (x0 )u), h (x0 )w = 0}✳ ✭✐✐✮ ▼➦❝ ❞ò ❜✐➸✉ t❤ù❝ supk∈A2 (−K,g(x0 ),g (x0 )u) k ∗ , k ❧➔ ❦❤ỉ♥❣ ❞÷ì♥❣✱ t❛ ✤÷❛ 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❪✳ ▼➦t ❦❤→❝ ✭①❡♠ ♣❤➛♥ ❝✉è✐ ❝õ❛ ♣❤➨♣ ❝❤ù♥❣ ♠✐♥❤ ỵ k [T (T (−K, g(x0 )), g (x0 )u)]∗ = −[clcone(cone(−K − g(x0 )) − g (x0 )u)]∗ ✣✐➲✉ ♥➔② s✉② r❛ r➡♥❣ ❜✐➸✉ t❤ù❝ ♥â✐ tr➯♥ ❧➔ ❦❤ỉ♥❣ ❞÷ì♥❣✱ ✈➔ t❤➟♠ ❝❤➼ ➙♠ ✭①❡♠ ❝→❝ ✈➼ ❞ư s❛✉✮✱ ❦❤ỉ♥❣ ❣✐è♥❣ ♥❤÷ ❦➳t q✉↔ ❝ê ✤✐➸♥✳ ❙ü ❦✐➺♥ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣✲❧✐❦❡ ❞♦ ❑❛✇❛s❛❦✐ ❬✶✹❪ ❧➔ ♥❣÷í✐ ✤➛✉ t✐➯♥ ♣❤→t ❤✐➺♥✳ ❙✉♣r❡♠✉♠ tr♦♥❣ ❜✐➸✉ t❤ù❝ tr➯♥ tr✐➺t t✐➯✉ ♥➳✉ ∈ A2 (−K, g(x0 ), g (x0 )u)✳ ✣➦❝ ❜✐➺t✱ ✤✐➲✉ ♥➔② ①↔② r❛ ♥➳✉ t❛ ①➨t ❝→❝ ữợ u X ✈ỵ✐ g (x0 )u ∈ ❝♦♥❡(−K − g(x0 )) = −K(g(x0 )) ✭①❡♠ ▼➺♥❤ ✤➲ ✶✳✸ ✭✈✐✮ ✭❜✮✮✱ ♥❤÷ ♥❤✐➲✉ t→❝ ❣✐↔ ✤➣ ❧➔♠✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ❤✐➺♥ tữủ ổ r ỵ t ữợ u ợ g (x0 )u clK(g(x0 )) ◆❣❤➽❛ ❧➔✱ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣✲❧✐❦❡ ①↔② r❛ ❝❤♦ ❝→❝ ✤✐➸♠ u tr♦♥❣ ❧é ❤ê♥❣ ❞÷í♥❣ ♥❤÷ ♥❤ä ❝õ❛ ❜❛♦ ✤â♥❣ −clK(g(x0 ))✳ ❚❛ ♥❤➜♥ ♠↕♥❤ r➡♥❣ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣✲❧✐❦❡ ❦❤ỉ♥❣ ①↔② r❛ ♥➳✉ g (x0 )u ∈ −K(g(x0 ))✱ ♥❣❛② ❝↔ ❦❤✐ ✤✐➸♠ ♥➔② ♥➡♠ tr➯♥ ❜✐➯♥✳ ❱➻ t❤➳✱ ♥➳✉ K ❧➔ ♣♦❧②❤❡❞r❛❧✱ ❤✐➺♥ t÷đ♥❣ ♥➔② ❝ơ♥❣ ❦❤ỉ♥❣ ①↔② r❛ K(g(x0 )) õ ữ ỵ r ũ ①➜♣ ①➾ t❛ ❝â t❤➸ tr→♥❤ ❣✐↔ t❤✐➳t ❦❤↔ ✈✐ ❝➜♣ ♠ët ✭❬✶✺✱ ✶✼✲✶✾❪✮✱ t✉② ♥❤✐➯♥ ✤➸ ❧➔♠ ❝❤♦ ❝→❝ ữợ q tữủ ró ỡ t ❣✐↔ t❤✐➳t ❦❤↔ ✈✐ ❝➜♣ ♠ët ✭♥❤÷♥❣ ❦❤ỉ♥❣ ❦❤↔ ✈✐ tử tr ỵ t q s ❝❤♦ ♠ët ✤➦❝ tr÷♥❣ ❝õ❛ ♥â♥ t✐➳♣ ①ó❝ ❝➜♣ ❤❛✐ t ợ t ữủ S = g −1 (−K)✳ ▼➺♥❤ ✤➲ ✸✳✼✳ ❈❤♦ x , u ∈ X ✈➔ g ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t q✉❛♥❤ x 0 ợ g t x0 ữợ q tr t ữợ t (x0 , u) ố ✈ỵ✐ −K ✳ ❑❤✐ ✤â✱ ✈ỵ✐ S = g −1 (−K) t❛ ❝â T (S, x0 , u) = {w ∈ X | g (x0 )w ∈ T (−K, g(x0 ), g (x0 )u)}✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ w ∈ T (S, x0 , u) õ tỗ t (tn , rn ) → (0+ , 0+ ) : tn /rn → ✈➔ wn → w s❛♦ ❝❤♦ xn := x0 + tn u + 21 tn rn wn ∈ S ✳ ❇ð✐ ▼➺♥❤ ✤➲ ✶✳✶✱ t❛ ❝â ✷✵ g(xn ) − g(x0 ) − tn g (x0 )u → g (x0 )w✳ tn rn /2 ❱➻ g(xn ) ∈ −K ✱ s✉② r❛ r➡♥❣ g (x0 )w ∈ T (−K, g(x0 ), g (x0 )u)✳ ✣è✐ ✈ỵ✐ ♣❤➛♥ ✤↔♦✱ ❣✐↔ sû r➡♥❣ g (x0 )w ∈ T (−K, g(x0 ), g (x0 )u) õ tỗ t (tn , rn ) → (0+ , 0+ ) : tn /rn → 0✱ ✈➔ zn → g (x0 )w s❛♦ ❝❤♦ g(x0 ) + tn g (x0 )u + 21 tn rn zn ∈ −K ✈ỵ✐ ♠å✐ n✳ ❇ð✐ ❣✐↔ t❤✐➳t ữợ q tr ợ n ợ t õ tn ∈ (0, ρ) ✈➔ un := u + 21 rn w ∈ BX (u, ρ) s❛♦ ❝❤♦ d(x0 + tn un , S) ≤ µd(g(x0 + tn un ), −K) ≤ µ g(x0 + tn un ) − g(x0 ) − tn g (x0 )u − 21 tn rn zn ≤ µ( g(x0 + tn un ) − g(x0 ) − g (x0 )(tn un ) + 12 tn rn g (x0 )w − 21 tn rn zn ) ≤ µκ tn un + 12 µtn rn g (x0 )w − zn = µtn rn (2κ(tn /rn ) un + g (x0 )w − zn ) ✭❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ❝ò♥❣ ✤÷đ❝ s✉② r❛ tø ✤à♥❤ ỵ tr tr tt ❝õ❛ g ✮✳ ❱➻ zn → g (x0 )w✱ t❛ ❝â t❤➸ t➻♠ ✤÷đ❝ xn ∈ S s❛♦ ❝❤♦ x0 + tn un − xn / 12 tn rn → 0✳ ❍ì♥ ♥ú❛✱ xn − x0 − tn u xn − x0 − tn un wn := = + w → w tn rn /2 tn rn /2 ❉♦ ✤â✱ w ∈ T (S, x0 , u)✳ ❑➳t q✉↔ s❛✉ q trỹ t ỵ ❝❤♦ tr÷í♥❣ ❤đ♣ h = 0✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② t❛ ✤à♥❤ ♥❣❤➽❛ Λ1 (x0 ) := {(c∗ , k ∗ ) ∈ Y ∗ × Z ∗ | (c∗ , k ∗ ) = (0, 0), c∗ ◦ f (x0 ) + k ∗ ◦ g (x0 ) = 0, c∗ ∈ C ∗ , k ∗ ∈ N (−K, g(x0 ))}✳ ❍➺ q✉↔ ✸✳✽✳ ❱ỵ✐ ❜➔✐ t♦→♥ ✭P✮✱ ❝❤♦ h = 0✳ ◆➳✉ ✐♥tC ✈➔ ✐♥tK ❧➔ ❦❤→❝ ré♥❣ ✈➔ x ▲❲❊(f, S)✱ t❤➻ ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ✤➙② t❤ä❛✳ ∈ ✭✐✮ ❈❤♦ f ✈➔ g ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t t↕✐ x0 ✳ ❑❤✐ ✤â✱ Λ1 (x0 ) = ∅✳ ✭✐✐✮ ❈❤♦ (f, g) ❧➔ ❦❤↔ ✈✐ ❝❤➦t t↕✐ x0 ✱ ((f, g) (x0 ), B(f,g) (x0 )) ❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ (f, g) t↕✐ x0 ✈ỵ✐ B(f,g) (x0 ) ❧➔ ♣✲❝♦♠♣❛❝t t✐➺♠ ❝➟♥✱ ✈➔ u ∈ X ✈ỵ✐ A (−K, g(x0 ), g (x0 )u) = ∅✳ ◆➳✉ (f, g) (x0 )u ∈ −[C × clK(g(x0 )) \ int(C × K(g(x0 )))]✱ t❤➻ ✭❛✮ tỗ t (M, N ) B(f,g) (x0 ) ✈➔ (c∗ , k ∗ ) ∈ Λ1 (x0 ) s❛♦ ❝❤♦ c∗ , M (u, u) + k ∗ , N (u, u) ≥ 21 supk∈A2 (−K,g(x0 ),g (x0 )u) k ∗ , k , ✈➔ c∗ = ♥➳✉✱ t❤➯♠ ♥ú❛✱ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② s❛✉ ✤➙② t❤ä❛ ✭❚❘✶u ✮ g (x0 )X − T (T (−K, g(x0 )), g (x0 )u) = Z tỗ t↕✐ (M, N ) ∈ ♣✲B(f,g) (x0 )∞ \ {0} ✈➔ (c∗ , k ∗ ) ∈ C ∗ × K(g(x0 ))∗ \ {(0, 0)} ✈ỵ✐ c∗ , f (x0 )u = k ∗ , g (x0 )u = s❛♦ ❝❤♦ c∗ , M (u, u) + k ∗ , N (u, u) ≥ 0✳ ✭✐✐✐✮ ❈❤♦ f ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t t↕✐ x0 ✱ (f (x0 ), Bf (x0 )) ❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ f t↕✐ x0 ✈ỵ✐ Bf (x0 ) ❧➔ ♣✲❝♦♠♣❛❝t t✐➺♠ ❝➟♥✱ ✈➔ u ∈ X ✈ỵ✐ f (x0 )u ∈ −bdC ✭✈➔ intK ❦❤ỉ♥❣ ❝➛♥ ❦❤→❝ ✷✶ ré♥❣✮✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ w T (S, x0 , u) tỗ t M ∈ ♣✲Bf (x0 )∞ ✈➔ c∗ ∈ C ∗ \ {0} ✈ỵ✐ c∗ , f (x0 )u = s❛♦ ❝❤♦ c∗ , f (x0 )w + M (u, u) tỗ t M Bf (x0 )∞ \ {0} ✈➔ c∗ ∈ C ∗ \ {0} ✈ỵ✐ c∗ , f (x0 )u = s❛♦ ❝❤♦ c∗ , M (u, u) ≥ 0✳ ❱➼ ❞ö s❛✉ ởt trữớ ủ tr õ ỵ ✭❤❛② ❍➺ q✉↔ ✸✳✽✮ ❜→❝ ❜ä ✤✐➸♠ ♥❣❤✐ ♥❣í ❧➔ ♥❣❤✐➺♠ ②➳✉ ✤à❛ ♣❤÷ì♥❣✱ tr♦♥❣ ❦❤✐ ✤â ❝→❝ ❦➳t q✉↔ ❣➛♥ ✤➙② ❦❤ỉ♥❣ →♣ ❞ư♥❣ ✤÷đ❝✳ ❱➼ ❞ư ✸✳✶✳ ❈❤♦ C = R+ ✱ I = [−1, 1]✱ C(I) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ t❤ü❝ ❧✐➯♥ tư❝ tr➯♥ I ✱ C+ (I) := {z ∈ C(I) | z(t) ≥ 0, ∀t ∈ I}✱ ✈➔ (x0 , y0 ) = (0, 0)✳ ❈❤♦ f : R2 → R ✈➔ g : R2 → C(I) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ f (x, y) = x|x| + y, (g(x, y))(t) = y + 3x2 − 2tx + t2 , ∀t ∈ I ✭g ✤÷đ❝ ❧➜② tø ❱➼ ❞ö ✺✳✶ tr♦♥❣ ❬✶✹❪✮✳ ❑❤✐ ✤â✱ f ∈ C 1,1 t↕✐ (0, 0)✱ f (0, 0) = (0, 1)✱ ±1 Bf (0, 0) = , g ∈ C t↕✐ (0, 0) ✈➔✱ ✈ỵ✐ ♠å✐ u = (x, y) ∈ R2 ✈➔ t ∈ I ✱ 0 (g (0, 0)(u))(t) = −2tx + y ✱ (g (0, 0)(u, u))(t) = 6x2 ✳ ❱➻ t❤➳✱ ((f, g) (0, 0), Bf (0, 0) × { 21 g (0, 0)}) ❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ (f, g) ❛t (0, 0) ✈ỵ✐ B(f,g) (0, 0) := Bf (0, 0) × { 12 g (0, 0)} ❜à ❝❤➦♥ ✈➔ ❞♦ ✤â ♣✲❝♦♠♣❛❝t t✐➺♠ ❝➟♥✳ ❱➻ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ❙❧❛t❡r t❤ä❛✱ ✤✐➲✉ ❦✐➺♥ ✭❚❘u ✮ ✭✈➔ ✭❚❘✶u ✮✮ ❝ô♥❣ t❤ä❛✳ ❚➟♣ ♥❤➙♥ tû ❋r✐t③ ❏♦❤♥ ❧➔ Λ1 (0, 0) = {(c∗ , k ∗ ) ∈ R × C(I)∗ | c∗ = α > 0, k ∗ , z = −αz(0), ∀z ∈ C(I)}✳ ❈❤å♥ u = (1, 0)✳ ❑❤✐ ✤â✱ f (0, 0)u = 0✳ ❇ð✐ ❇ê ✤➲ ✻✳✶ ❝õ❛ ❬✶✹❪✱ v(·) ∈ −clK(g(0, 0)) = −clcone(C+ (I) + (·)2 ) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ v(t) ≥ ✈ỵ✐ ♠å✐ t ∈ Iξ := {t ∈ I|t2 = 0} = {0}✳ ❚❛ ❝â (g (0, 0)u)(t) = −2t ✈➔ ❞♦ ✤â g (0, 0)u ∈ − ❝❧K(g(0, 0))✳ ❍ì♥ ♥ú❛✱ t❤❡♦ ❱➼ ❞ư ✺✳✶ tr♦♥❣ ❬✶✹❪✱ A2 (−K, g(0, 0), g (0, 0)u) = {z ∈ C(I) | z(0) ≥ 2} ❉♦ ✤â✱ (f, g) (0, 0)u ∈ −[C × clK(g(0, 0)) \ int(C × K(g(0, 0)))]✳ ❚❛ t❤➜② r➡♥❣✱ ✈ỵ✐ ♠å✐ (M, N ) ∈ ♣✲❝❧B(f,g) (0, 0) = B(f,g) (0, 0) ✈➔ (c∗ , k ∗ ) ∈ Λ1 (0, 0)✱ c∗ , M (u, u) + k ∗ , N (u, u) ≤ −2α < −α = 12 supk∈A2 (−K,g(0,0),g (0,0)u) k ∗ , k ỵ q (0, 0) ∈ ▲❲❊(f, S)✳ ❱➻ f ❦❤æ♥❣ ❦❤↔ ✈✐ ❋r➨❝❤❡t ❝➜♣ ❤❛✐ t↕✐ (0, 0)✱ ❍➺ q✉↔ ✸✳✺ ✈➔ ✣à♥❤ ỵ ổ ữủ r ❝á♥ ❧↕✐ t❛ ①➨t tr÷í♥❣ ❤đ♣ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ❝õ❛ ✭P✮ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❦❤→❝ ❞ü❛ tr ỵ t s ỵ C , C Rm t ỗ s C1 r õ tỗ t ởt s t C1 C2 r✐➯♥❣ ❜✐➺t ✈➔ ❦❤æ♥❣ ❝❤ù❛ C2 ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ C1 riC2 = ỵ ❱ỵ✐ ❜➔✐ t♦→♥ ✭P✮✱ ❝❤♦ X, Y, Z ✈➔ W ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉✱ ✐♥tC ✈➔ ✐♥tK ❦❤→❝ ré♥❣ ✈➔ x0 ∈ ▲❲❊✭f, S ✮✳ ❑❤✐ ✤â✱ ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ✤➙② t❤ä❛✳ ✭✐✮ ❈❤♦ f, g, h ❧➔ ❦❤↔ rt t x0 h ữợ q tr t ữợ t (x0 , u) ố ✈ỵ✐ T = {0} ❦❤✐ u = 0✳ ❑❤✐ ✤â✱ tỗ t (c , k , h ) Λ(x0 ) s❛♦ ❝❤♦ (c∗ , k ∗ ) = (0, 0)✳ ✭✐✐✮ ❈❤♦ f, g, h ❧➔ ❦❤↔ ✈✐ ❝❤➦t t↕✐ x0 ✱ ((f, g, h) (x0 ), B(f,g,h) (x0 )) ❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ (f, g, h) t↕✐ x0 ✱ ✈➔ u ∈ X ✈ỵ✐ A (−K, g(x0 ), g (x0 )u) = ∅✳ ◆➳✉ h ữợ q tr t ữợ t (x0 , u) ✤è✐ ✈ỵ✐ T = {0} ✈➔ (f, g, h) (x0 )u ∈ −[C × clK(g(x0 )) \ int(C × K(g(x0 )))] ì {0} t tỗ t (M, N, P ) ∈ ❝❧B(f,g,h) (x0 ) ✈➔ (c∗ , k ∗ , h∗ ) ∈ Λ(x0 ) s❛♦ ❝❤♦ c∗ , M (u, u) + k ∗ , N (u, u) + h∗ , P (u, u) ≥ 21 supk∈A2 (−K,g(x0 ),g (x0 )u) k ∗ , k , tr♦♥❣ ✤â (c∗ , k ∗ ) = (0, 0) ♥➳✉ h = 0✱ ✈➔ c∗ = ♥➳✉ ✤✐➲✉ u tọ tỗ t (M, N, P ) ∈ B(f,g,h) (x0 )∞ \ {0} ✈➔ (c∗ , k ∗ , h∗ ) ∈ C ∗ × K(g(x0 )) ì W \ {(0, 0, 0)} ợ c∗ , f (x0 )u = k ∗ , g (x0 )u = s❛♦ ❝❤♦ c∗ , M (u, u) + k ∗ , N (u, u) + h∗ , P (u, u) ≥ ✈➔ ♥➳✉✱ t❤➯♠ ♥ú❛ h = 0✱ t❤➻ (c∗ , k ∗ ) = (0, 0)✳ ✭✐✐✐✮ ❈❤♦ f ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t t↕✐ x0 ✱ (f (x0 ), Bf (x0 )) ❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ f t↕✐ x0 ✱ ✈➔ u ∈ X ✈ỵ✐ f (x0 )u ∈ −bdC ✭✈➔ intK ❦❤ỉ♥❣ ❝➛♥ ❦❤→❝ ré♥❣✮✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ w ∈ T (S, x0 , u) tỗ t M Bf (x0 )∞ ✈➔ c∗ ∈ C ∗ \ {0} ✈ỵ✐ c∗ , f (x0 )u = s❛♦ ❝❤♦ c∗ , f (x0 )w + M (u, u) ≥ tỗ t M Bf (x0 ) \ {0} ✈➔ c∗ ∈ C ∗ \ {0} ✈ỵ✐ c∗ , f (x0 )u = s❛♦ ❝❤♦ c∗ , M (u, u) ≥ 0✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❇ð✐ ✣à♥❤ ỵ ợ C = (f, g, h) (x0 )X ✈➔ C2 = −int[C × K(g(x0 ))] × {0}✱ t❛ ❝â ✤÷đ❝ (c , k , h ) ∈ X × Y × Z ∗ ❦❤→❝ ❦❤æ♥❣ ✈➔ α ∈ R s❛♦ ❝❤♦✱ ∀(y, z, t) ∈ (f, g, h) (x0 )X ✱ ∀(c, k) ∈ −(C × K(g(x0 )))✱ ∗ ∗ ∗ ∗ ∗ c∗ , y + k ∗ , z + h∗ , t ≥ α✱ ✭✶✹✮ c∗ , c + k ∗ , k ≤ α, ✭✶✺✮ ✈➔ s✐➯✉ ♣❤➥♥❣ H := {(y, z, t) ∈ X × Y × Z | c∗ , y + k ∗ , z + h∗ , t = α} ❦❤æ♥❣ ❝❤ù❛ C2 ✳ ❱➻ (f, g, h) (x0 )X ✈➔ 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❛ ✤÷đ❝ c∗ ∈ C ∗ ✳ ✣➦t c = tr♦♥❣ ✭✶✺✮ t❛ ❝â k ∗ ∈ K(g(x0 ))∗ = N (−K, g(x0 ))✳ ❱➻ s✐➯✉ ♣❤➥♥❣ H ❦❤æ♥❣ ❝❤ù❛ C2 ✱ (c∗ , k ∗ ) = (0, 0) ỵ ❞ư♥❣ ❇ê ✤➲ ✸✳✾ ✈ỵ✐ C1 = (f, g, h) (x0 )X + 2(M, N, P )(u, u) ✈➔ C2 = −intcone[C + f (x0 )u] × IT (−K, g(x0 ), g (x0 )u) ì {0} t õ ữủ (c∗ , k ∗ , h∗ ) ∈ X ∗ × Y ∗ × Z ∗ ❦❤→❝ ❦❤æ♥❣ ✈➔ α ∈ R s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ (y, z, t) ∈ (f, g, h) (x0 )X ✱ c ∈ −✐♥t❝♦♥❡[C + f (x0 )u] ✈➔ k ∈ IT (−K, g(x0 ), g (x0 )u)✱ c∗ , y + k ∗ , z + h∗ , t + c∗ , M (u, u) + k ∗ , N (u, u) + h∗ , P (u, u) ≥ α✱ ✭✶✻✮ c∗ , c + k ∗ , k ≤ α ✱ ✭✶✼✮ ✈➔ H ❦❤ỉ♥❣ ❝❤ó❛ C2 ✳ ❱➻ (f, g, h) (x0 )X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥✱ tø ✭✶✻✮ t❛ ❝â✱ ✈ỵ✐ ♠å✐ (y, z, t) ∈ ✷✸ (f, g, h) (x0 )X ✱ c∗ , y + k ∗ , z + h∗ , t = 0✱ ✤✐➲✉ ♥➔② ❞➝♥ ✤➳♥ c∗ ◦ f (x0 ) + k ∗ ◦ g (x0 ) + h∗ ◦ h (x0 ) = ✈➔ ✷ c∗ , M (u, u) +2 k ∗ , N (u, u) +2 h∗ , P (u, u) ≥ α✳ ✭✶✽✮ ❱➻ −✐♥t❝♦♥❡[C + f (x0 )u] ❧➔ ♥â♥✱ ✭✶✼✮ s✉② r❛ r➡♥❣ c∗ ∈ C ∗ ✈➔ c∗ , f (x0 )u = 0✳ ❈ô♥❣ tø ✭✶✼✮✱ t❛ ❝â k ∗ , k ≤ α✱ ✈ỵ✐ ♠å✐ k ∈ IT (−K, g(x0 ), g (x0 )u)✳ ❚ø ✤✐➲✉ ♥➔② ✈➔ ✭✶✽✮✱ ❧➟♣ tữỡ tỹ ữ tr ự ỵ ✸✳✹ ✭✐✐✮ ✭❛✮✱ t❛ ❝â ❦➳t ❧✉➟♥✳ ❇➙② ❣✐í✱ ❣✐↔ sû h = 0✳ ◆➳✉ (c∗ , k ∗ ) = (0, 0)✱ t❤➻ ❜ð✐ ✭✶✻✮ ✈➔ ✭✶✼✮✱ α = ✈➔ ❞♦ ✤â H ❝❤ù❛ C2 ✱ ♠ët ✤✐➲✉ t ỵ ❞ư♥❣ ❇ê ✤➲ ✸✳✾ ✈ỵ✐ C1 = {(M, N, P )(u, u)} ✈➔ C2 = −intcone[C + f (x0 )u] × IT (−K, g(x0 ), g (x0 )u) × {0}✱ t❛ ♥❤➟♥ ✤÷đ❝ (c∗ , k ∗ , h∗ ) ∈ X ∗ × Y ∗ × Z ∗ ❦❤→❝ ❦❤ỉ♥❣ ✈➔ α ∈ R s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ c ∈ −✐♥t❝♦♥❡[C + f (x0 )u] ✈➔ k ∈ IT (−K, g(x0 ), g (x0 )u)✱ c∗ , M (u, u) + k ∗ , N (u, u) + h∗ , P (u, u) ≥ α✱ c∗ , c + k ∗ , k ≤ α ✈➔ H ❦❤æ♥❣ ❝❤ù❛ C2 ✳ ❚ø ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥✱ ❧➟♣ ❧✉➟♥ tữỡ tỹ ữ tr ự ỵ ✭✐✐✮ ✭❜✮✱ t❛ ❝â ❦➳t q✉↔✳ ❇➙② ❣✐í✱ ✈ỵ✐ h = 0✱ ♥➳✉ (c∗ , k ∗ ) = (0, 0)✱ t❤➻ ❜ð✐ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥✱ α = ✈➔ t❛ ❝â ✤✐➲✉ ♠➙✉ t❤✉➝♥ t÷ì♥❣ tü ❧➔ H ❝❤ù❛ C2 ✳ ✭✐✐✐✮ ❑➳t q✉↔ ✤÷đ❝ s✉② r❛ tứ ỵ ỵ t tổ tữớ q ữợ ữủ s r trỹ t tứ ỵ ũ ss s✉② rë♥❣ ❈❧❛r❦❡ ✈➔ tü❛ ❍❡ss✐❛♥ ❏❡②❛❦✉♠❛r✲▲✉❝✱ t÷ì♥❣ ù♥❣✳ ❍➺ q✉↔ ✸✳✶✶✳ ❱ỵ✐ ❜➔✐ t♦→♥ ✭P✮✱ ❝❤♦ X, Y, Z ✈➔ W ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉✱ f, g, h t❤✉ë❝ ❧ỵ♣ C 1,1 t↕✐ x0 ∈ X ✱ ✐♥tC ✈➔ ✐♥tK ❦❤→❝ ré♥❣✱ ✈➔ x0 ∈ ▲❲❊(f, S)✳ ❑❤✐ ✤â✱ ♥❤ú♥❣ ❦❤➥♥❣ ✤à♥❤ s❛✉ ✤➙② t❤ä❛✳ ✭✐✮ ❈❤♦ h ❧➔ ữợ q tr t ữợ t (x0 , u) ✤è✐ ✈ỵ✐ T = {0} ❦❤✐ u = 0✳ ❑❤✐ õ tỗ t (c , k , h ) ∈ Λ(x0 ) s❛♦ ❝❤♦ (c∗ , k ∗ ) = (0, 0)✳ ✭✐✐✮ ❈❤♦ u ∈ X ✳ ◆➳✉ h ữợ q tr t ữợ t (x0 , u) ✤è✐ ✈ỵ✐ T = {0} ✈➔ (f, g, h) (x0 )u ∈ −[C × clK(g(x0 )) \ int(C ì K(g(x0 )))] ì {0} t tỗ t (M, N, P ) ∈ ∂C2 (f, g, h)(x0 ) ✈➔ (c∗ , k ∗ , h∗ ) ∈ Λ(x0 ) s❛♦ ❝❤♦ c∗ , M (u, u) + k ∗ , N (u, u) + h∗ , P (u, u) ≥ supk∈A2 (−K,g(x0 ),g (x0 )u) k ∗ , k ✱ ✭✶✾✮ tr♦♥❣ ✤â (c∗ , k ∗ ) = (0, 0) ♥➳✉ h = 0✱ ✈➔ c∗ = ♥➳✉ ✭❚❘u ✮ t❤ä❛✳ ✭✐✐✐✮ ❈❤♦ u ∈ X ✈ỵ✐ f (x0 )u ∈ −bdC ✭✈➔ intK ❦❤æ♥❣ ❝➛♥ ❦❤→❝ ré♥❣✮✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ w ∈ T (S, x0 , u) tỗ t c C \ {0} ợ c∗ , f (x0 )u = s❛♦ ❝❤♦ c∗ , f (x0 )w ≥ 0✳ ❍➺ q✉↔ ✸✳✶✶ ✭✐✐✮ ❝↔✐ t❤✐➺♥ ❍➺ q✉↔ ✹ ❝õ❛ ❬✼❪✱ tr♦♥❣ ✤â h ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ❝➜♣ ❤❛✐ t↕✐ x0 ✳ ❍➺ q✉↔ ✸✳✶✷✳ ❱ỵ✐ ❜➔✐ t♦→♥ ✭P✮✱ ❝❤♦ X, Y, Z ✈➔ W ✷✹ ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉✱ f, g, h t❤✉ë❝ ❧ỵ♣ C t↕✐ x0 ∈ X ✱ ✐♥tC ✈➔ ✐♥tK ❦❤→❝ ré♥❣✱ ✈➔ x0 ∈ ▲❲❊(f, S)✳ ✭✐✮ ❑❤➥♥❣ ✤à♥❤ ✭✐✮ ❝õ❛ ❍➺ q✉↔ ✸✳✶✶ t❤ä❛✳ ✭✐✐✮ ❈❤♦ (f, g, h) ❝â →♥❤ ①↕ tü❛ ❍❡ss✐❛♥ ∂ (f, g, h)(.) ❧➔ ♥ú❛ ❧✐➯♥ tö❝ tr➯♥ t↕✐ x0 ✱ ✈➔ u ∈ X ✈ỵ✐ A (−K, g(x0 ), g (x0 )u) = h ữợ q tr t ữợ t (x0 , u) ố ợ T = {0} ✈➔ (f, g, h) (x0 )u ∈ −[C × clK(g(x0 )) \ int(C × K(g(x0 )))] × {0} t tỗ t (M, N, P ) ∈ ❝❧❝♦∂ (f, g, h)(x0 ) ✈➔ (c∗ , k ∗ , h∗ ) ∈ Λ(x0 ) s❛♦ ❝❤♦ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✾✮ t❤ä❛✱ (c∗ , k ∗ ) = (0, 0) ♥➳✉ h = 0✱ ✈➔ c∗ = ♥➳✉ t❤➯♠ ♥ú❛ ✤✐➲✉ ❦✐➺♥ ✭❚❘u ✮ t❤ä❛❀ ✭❜✮ tỗ t (M, N, P ) (f, g, h)(x0 )∞ \{0} ✈➔ (c∗ , k ∗ , h∗ ) ∈ C ∗ ×K(g(x0 ))∗ × W ∗ \ {(0, 0, 0)} ✈ỵ✐ c∗ , f (x0 )u = k ∗ , g (x0 )u = s❛♦ ❝❤♦ c∗ , M (u, u) + k ∗ , N (u, u) + h∗ , P (u, u) ≥ 0✱ ✈➔ ♥➳✉✱ t❤➯♠ ♥ú❛✱ h = 0✱ t❤➻ (c∗ , k ∗ ) = (0, 0)✳ ✭✐✐✐✮ ❈❤♦ f ❝â →♥❤ ①↕ tü❛ ❍❡ss✐❛♥ ∂ f (.) ❧➔ ♥ú❛ ❧✐➯♥ tö❝ tr➯♥ t↕✐ x0 ✱ ✈➔ u ∈ X ✈ỵ✐ f (x0 )u ∈ −bdC ✭✈➔ intK ❦❤ỉ♥❣ ❝➛♥ ❦❤→❝ ré♥❣✮✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ w ∈ T (S, x0 , u) tỗ t M ❝♦∂ f (x0 )∞ ✈➔ c∗ ∈ C ∗ \ {0} ✈ỵ✐ c∗ , f (x0 )u = s❛♦ ❝❤♦ c∗ , f (x0 )w + M (u, u) tỗ t M f (x0 )∞ \ {0} ✈➔ c∗ ∈ C ∗ \ {0} ✈ỵ✐ c∗ , f (x0 )u = s❛♦ ❝❤♦ c∗ , M (u, u) ≥ 0✳ ❚r♦♥❣ t➻♥❤ ❤✉è♥❣ s❛✉ ✤➙② ✈ỵ✐ ❝→❝ ❤➔♠ ♥❣♦➔✐ ❧ỵ♣ C ỵ q ❜ä ✤✐➸♠ ♥❣❤✐ ♥❣í ❧➔ ♥❣❤✐➺♠ ②➳✉ ✤à❛ ♣❤÷ì♥❣✱ tr♦♥❣ ❦❤✐ ✤â ❝→❝ ❦➳t q✉↔ ❣➛♥ ✤➙② ❦❤ỉ♥❣ →♣ ❞ư♥❣ ✤÷đ❝✳ ❱➼ ❞ư ✸✳✷✳ ❈❤♦ ϕ : [0, +∞) → R ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐  ✐❢ s > 1,  1/(q + 1) ✐❢ 1/(q + 1) < s ≤ 1/q, q ∈ N, ϕ(s) =  ✐❢ s = ❑❤✐ ✤â✱ ϕ ❧➔ ❦❤æ♥❣ ❣✐↔♠ tr➯♥ [0, +∞) ✈➔ ✈➻ t❤➳ ❤➔♠ θ(x) = |x| (s)ds ữủ ợ x R ✈➔ θ ✤÷đ❝ ❧➜② tø ❬✷❪✮✳ ❚❛ ❝â ❝→❝ ✤↕♦ ❤➔♠ tr→✐ ✈➔ ♣❤↔✐✱ ✈ỵ✐ q ∈ N✱ θ+ (1/q) = 1/q, θ− (1/q) = 1/(q+1)✳ ✭✷✵✮ ❈❤♦ C = R+ ✱ K = {(k1 , k2 , k3 ) ∈ R3 | k2 k3 ≥ 2k12 , k2 ≤ 0, k3 ≤ 0}✱ (x0 , y0 ) = (0, 0) ✭K ✤÷đ❝ ❧➜② tø ❱➼ ❞ư ✶ tr♦♥❣ ❬✼❪✮✱ ✈➔ f : R2 → R ✈➔ g : R2 → R3 ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ f (x, y) = −|x| + θ(x) + y ✱ g(x, y) = (x, x2 + 1, |x| − θ(x) + y)✳ ❑❤✐ ✤â✱ (f, g) ❦❤ỉ♥❣ t❤✉ë❝ ❧ỵ♣ C t↕✐ (0, 0)✱ ❦❤æ♥❣ ❦❤↔ ✈✐ tr♦♥❣ U \ {(0, 0)} ✈ỵ✐ ❜➜t ❦ý ❧➙♥ ❝➟♥ U ❝õ❛ (0, 0)✱ ❜ð✐ ✈➻ ✭✷✵✮✱ ✈➔ t❛ ❝â ✷✺   f (0, 0) = (0, 1)✱ g (0, 0) ❂ 0 0✱ g(0, 0) = (0, 1, 0) ∈ −K ✳ ❍ì♥ ♥ú❛✱ θ ❧➔ l✲ê♥ ✤à♥❤ ✈➔ ❞♦ ✤â✱ ❦❤↔ ✈✐ ❝❤➦t t↕✐ ✭✈➲ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ l✲ê♥ ✤à♥❤✱ ①❡♠ ❬✷❪✮✳ ❱➻ t❤➳✱ (f, g) ❧➔ ❦❤↔ ✈✐ ❝❤➦t t↕✐ (0, 0)✳ ◗✉❛♥ s→t r K õ ỗ õ ợ tK = ✈➔ K(g(0, 0)) = {(k1 , k2 , k3 ) ∈ R3 | k3 < 0} ∪ {(0, k2 , 0)| k2 ∈ R}, T (−K, g(0, 0)) = {(k1 , k2 , k3 ) ∈ R3 | k3 ≥ 0}, N (−K, g(0, 0) = {λ(0, 0, −1)| λ ∈ R}, Λ1 (0, 0) = {(c∗ , k ∗ ) ∈ R × R3 | c∗ = α > 0, k ∗ = α(0, 0, −1)} ❚❛ ❝â ①➜♣ ①➾ ❝➜♣ ❤❛✐ B(f,g) (0, 0) = {(Mβ , Nβ )| β < −2}✱ tr♦♥❣ ✤â Mβ ❂ β 0 ✈➔ Nβ : R2 × R2 → R3 ❧➔ ♠❛ tr➟♥ × × Nβ ❂ 0 −β ✱ 0 0 0 tù❝ ❧➔✱ Nβ (u, v) = (0, u1 v1 , −βu1 v1 )✱ ✈ỵ✐ u = (u1 , u2 ) ✈➔ v = (v1 , v2 ) tr♦♥❣ R2 ✳ ❱➻ t❤➳✱ ❝❧B(f,g) (0, 0) = {(Mβ , Nβ )| β ≤ −2}, B(f,g) (0, 0)∞ = {(Mβ , Nβ )| β ≤ 0}✱ tr♦♥❣ ✤â Nβ ❂ 0 0 −β ✳ 0 0 0 ❈❤å♥ u = (1, 0)✱ t❛ ❝â (f, g) (0, 0)u ∈ −[C×❝❧K(g(0, 0))\ ✐♥t(C × K(g(0, 0))]✱ A2 (−K, g(0, 0), g (0, 0)u) = {(k1 , k2 , k3 ) ∈ R3 | k3 ≥ 4}✳ ❈❤♦ (c∗ , k ∗ ) = (α, 0, 0, −α) ∈ Λ1 (0, 0) ✈ỵ✐ α > 0✱ t❛ t❤➜② r➡♥❣ supk∈A2 (−K,g(0,0),g (0,0)u) k ∗ , k = −4α✱ ✈➔✱ ✈ỵ✐ ♠å✐ (Mβ , Nβ ) ∈ ❝❧B(f,g) (0, 0)✱ c∗ , Mβ (u, u) + k ∗ , Nβ (u, u) = 2αβ ≤ −4α < 12 supk∈A2 (−K,g(0,0),g (0,0)u) k ∗ , k ✳ ❱ỵ✐ ♠å✐ (c∗ , k ∗ ) ∈ C ∗ × K(g(0, 0))∗ \ {(0, 0)} ✈ỵ✐ c∗ , f (0, 0)u = k ∗ , g (0, 0)u = 0✱ tù❝ ❧➔✱ c∗ = α ✈➔ k ∗ = γ(0, 0, −1) ✈ỵ✐ α ≥ 0, γ ≥ ✈➔ (α, γ) = (0, 0) ✈➔ ♠å✐ (Mβ , Nβ ) ∈ B(f,g) (0, 0)∞ \ {0}✱ t❛ ❝â c∗ , Mβ (u, u) + k ∗ , Nβ (u, u) = β(α + γ) < 0✳ ❉♦ ✤â✱ t ỵ q (0, 0) ∈ ▲❲❊(f, S)✳ ❱➻ (f, g) ∈ C t↕✐ (0, 0)✱ ❝→❝ ❍➺ q✉↔ ✸✳✶✶ ✈➔ ✸✳✶✷ ❦❤æ♥❣ →♣ ❞ư♥❣ ✤÷đ❝✳ ❍ì♥ ♥ú❛✱ ✈➻ d2 (f, g)((0, 0), u) = ∅ ✭✈➲ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ d2 ✱ ①❡♠ ❬✼✱ ỵ ụ ổ ✤÷đ❝✳ ✷✻ ❈❤÷ì♥❣ ✹✿ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ ❝➜♣ ❤❛✐ ❚r♦♥❣ ♣❤➛♥ ♥➔②✱ t❛ ①➨t ❜➔✐ t♦→♥ ✭P✮ ✈ỵ✐ h = 0✱ ✈➔ C ✱ K ❧➔ tê♥❣ qt õ t ổ ỗ ợ tr rộ (f, g) ổ t ỵ ❱ỵ✐ ❜➔✐ t♦→♥ ✭P✮ ✈ỵ✐ h = 0✱ ❣✐↔ sû r➡♥❣ X ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✈➔ f ✈➔ g ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t t↕✐ x0 ∈ S ✳ ●✐↔ sû ❤ì♥ ♥ú❛ (f (x0 ), Bf (x0 )) ✈➔ (g (x0 ), Bg (x0 )) ❧➔ ❝→❝ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ f ✈➔ g ✱ t÷ì♥❣ ÷♥❣✱ t↕✐ x0 ✱ ✈ỵ✐ t❤➔♥❤ ♣❤➛♥ t❤ù ❤❛✐ ❧➔ ♣✲❝♦♠♣❛❝t t✐➺♠ ❝➟♥✳ ❑❤✐ ✤â✱ ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ ✤õ ❝❤♦ x0 ∈ ▲❋❊(2, f, S)✳ ✭✐✮ ∀u ∈ SX ✱ ∃(c∗ , k ∗ ) ∈ C ∗ × K(g(x0 ))∗ ✱ c∗ , f (x0 )u + k ∗ , g (x0 )u > 0✳ ✭✐✐✮ ❱ỵ✐ ♠å✐ u ∈ SX ∩ T (S, x0 ) t❤ä❛ f (x0 )u ∈ −C ✱ t❛ ❝â ✭❛✮ ∀w ∈ T (S, x0 , u)∩u⊥ ✱ ∀(M, N ) ∈ ♣✲❝❧B(f,g) (x0 )✿ g (x0 )w+2N (u, u) ∈ T (−K, g(x0 ), g (x0 )u)✱ ∃(c∗ , k ∗ ) ∈ Λ1 (x0 )✱ c∗ , 2M (u, u) + k ∗ , 2N (u, u) > k ∗ , g (x0 )w + 2N (u, u) ✱ ✈➔ ∀(M, N ) ∈ ♣✲B(f,g) (x0 )∞ \ {0}✿ N (u, u) ∈ T (−K, g(x0 ), g (x0 )u)✱ ∃c∗ ∈ C ∗ \ {0}✿ c∗ , f (x0 )u = 0✱ c∗ , M (u, u) > 0❀ ✭❜✮ ∀w ∈ T (S, x0 , u) ∩ u⊥ \ {0}✱ ∀M ∈ ♣✲Bf (x0 )∞ ✱ ∃c∗ ∈ C ∗ \ {0}✱ c∗ , f (x0 )u = 0, c∗ , f (x0 )w + M (u, u) > 0, ✈➔ ∀M ∈ ♣✲Bf (x0 )∞ \ {0}✱ ∃c∗ ∈ C ∗ \ {0}✱ c∗ , f (x0 )u = 0, c∗ , M (u, u) > 0✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❑➳t q✉↔ ✤÷đ❝ s✉② r tứ ỵ sỷ ự r tỗ t xn S BX (x0 , n1 ) \ {x0 } ✈➔ cn ∈ C s❛♦ ❝❤♦ f (xn )−f (x0 )+cn ∈ BY (0, n1 t2n )✱ ✭✷✶✮ tr♦♥❣ ✤â tn = xn − x0 → 0+ ✳ ❚❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ (xn − x0 )/tn → u ∈ T (S, x0 ) ∩ SX ✳ ❈❤✐❛ ✭✷✶✮ ❜ð✐ tn ✈➔ ❝❤✉②➸♥ q✉❛ ❣✐ỵ✐ ❤↕♥ t❛ ❝â f (x0 )u ∈ −C ✳ ▼➦t ❦❤→❝✱ ❜ð✐ ▼➺♥❤ ✤➲ ✶✳✺✱ ❝❤➾ ❝➛♥ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣ s❛✉ ✤➙② ❧➔ ✤õ ✭❞ò♥❣ ❝→❝ rữớ ủ ởt ỗ t w ∈ T (S, x0 , u) ∩ u⊥ ✈ỵ✐ wn := (xn − x0 − tn u)/ 21 t2n w ợ n ợ tỗ t (Mn , Nn ) ∈ B(f,g) (x0 ) s❛♦ ❝❤♦ (f, g)(xn ) − (f, g)(x0 ) = (f, g) (x0 )(xn − x0 ) + (Mn , Nn )(xn − x0 , xn − x0 ) + o(t2n )✳ ♣ ◆➳✉ {(Mn , Nn )} ❜à ❝❤➦♥✱ ❣✐↔ sû r➡♥❣ (Mn , Nn ) − → (M, N ) ∈ ♣✲❝❧B(f,g) (x0 )✳ ❑❤✐ ✤â✱ (f, g)(xn ) − (f, g)(x0 ) − tn (f, g) (x0 )u → (f, g) (x0 )w + 2(M, N )(u, u)✱ t2n /2 ❱➻ g(xn ) ∈ −K ✱ ✤✐➲✉ ♥➔② ❞➝♥ ✤➳♥ g (x0 )w + 2N (u, u) ∈ T (−K, g(x0 ), g (x0 )u) ✈➔ ❞♦ ✤â✱ g (x0 )u ∈ T (−K, g(x0 ))✳ ❇ð✐ ❣✐↔ t❤✐➳t ✭✐✐✮ ✭❛✮✱ tỗ t (c , k ) (x0 ) t❤ä❛ c∗ , 2M (u, u) + k ∗ , 2N (u, u) > k ∗ , g (x0 )w + 2N (u, u) ✱ ✷✼ ✈➔ ✈➻ t❤➳ c∗ , y > 0✳ ▼➦t ❦❤→❝✱ tø ✭✷✶✮ s✉② r❛ r➡♥❣ f (xn ) − f (x0 ) − tn f (x0 )u cn + tn f (x0 )u + → 0✳ t2n /2 t2n /2 ❉♦ ✤â✱ y ∈ − ❝❧❝♦♥❡(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✱ ✈➔ ✈➻ ✈➟② c∗ ∈ ❬❝❧❝♦♥❡✭C + f (x0 )u)]∗ ✳ ◆❤ö t❤➳✱ c∗ , y ≤ 0✱ ♠ët ✤✐➲✉ ♠➙✉ t❤✉➝♥✳ ♣ ◆➳✉ {(Mn , Nn )} ❦❤æ♥❣ ❜à ❝❤➦♥✱ ❣✐↔ sû r➡♥❣ αn := (Mn , Nn ) → ∞ ✈➔ (Mn , Nn ) − → αn (M, N ) ∈ ♣✲B(f,g) (x0 )∞ \ {0}✳ ❉♦ ✤â✱ (f, g)(xn ) − (f, g)(x0 ) − tn (f, g) (x0 )u → (M, N )(u, u)✳ αn t2n ✭✷✷✮ ◆➳✉ M (u, u) = ✈➔ N (u, u) = ∈ T (−K, g(x0 ), g (x0 )u)✱ tt tỗ t c C ∗ \ {0} ✈ỵ✐ c∗ , f (x0 )u = t❤ä❛ c∗ , M (u, u) > 0✱ ♠ët ✤✐➲✉ ♠➙✉ t❤✉➝♥✳ ◆➳✉ (M, N )(u, u) = 0✱ t❤➻ αn tn → 0+ ✳ ❇ð✐ ✭✷✷✮✱ t❛ ❝â N (u, u) ∈ T (−K, g(x0 ), g (x0 )u) tt ỳ tỗ t c∗ ∈ C ∗ \ {0} ✈ỵ✐ c∗ , f (x0 )u = 0✱ tù❝ ❧➔✱ c∗ ∈ ❬❝❧❝♦♥❡✭C + f (x0 )u)]∗ ✱ t❤ä❛ c∗ , M (u, u) > 0✳ ▼➦t ❦❤→❝✱ ✭✷✶✮ ❞➝♥ ✤➳♥ f (xn ) − f (x0 ) − tn f (x0 )u cn + tn f (x0 )u + → 0✱ αn t2n αn t2n ✈➔ ✈➻ t❤➳ M (u, u) ∈ −❝❧❝♦♥❡(C + f (x0 )u)✳ ❉♦ ✤â✱ c∗ , M (u, u) ≤ 0✱ ♠ët ✤✐➲✉ ♠➙✉ t❤✉➝♥✳ ❚r÷í♥❣ ❤đ♣ ❤❛✐✿ ỗ t rn 0+ s tn /rn ✈➔ xn − x0 − tn u wn := → w ∈ T (S, x0 , u) ∩ u⊥ \ {0}✳ tn rn /2 ❇ð✐ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ Bf (x0 ) ợ n ợ tỗ t Mn Bf (x0 ) t❤ä❛ 2tn o(t2n ) f (xn ) − f (x0 ) − tn f (x0 )u = f (x0 )wn +( )Mn (u+ 12 rn wn , u+ 21 rn wn )+ ✳ ✭✷✸✮ tn rn /2 rn tn rn /2 ❇➡♥❣ ❝→❝❤ ❞ò♥❣ ❝→❝ ❞➣② ❝♦♥ ♥➳✉ t t trữớ ủ s ( 2trnn )Mn → 0✳ ❑❤✐ ✤â✱ ✭✷✸✮ s✉② r❛ r➡♥❣ f (xn ) − f (x0 ) − tn f (x0 )u → f (x0 )w✳ tn rn /2 ❚ø ✭✷✶✮ t❛ ❝â f (xn ) − f (x0 ) − tn f (x0 )u cn + tn f (x0 )u + → 0✱ tn rn tn rn ✈➔ ❞♦ ✤â f (x0 )w ∈ −❝❧❝♦♥❡(C + f (x0 )u)✱ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t ✭✐✐✮ ✭❜✮ ✈ỵ✐ M = 0✳ ♣ • ( 2trnn )Mn → a > 0✳ ❑❤✐ ✤â✱ Mn → ∞ ✈➔ tn Mn → 0✳ ❉♦ ✤â✱ Mn / Mn → M ∈ ♣✲Bf (x0 )∞ \ {0}✱ ✈➔ t❛ ❝â − ✷✽ a(f (xn ) − f (x0 ) − tn f (x0 )u) → f (x0 )w + aM (u, u)✳ t2n Mn ❚÷ì♥❣ tü ♥❤÷ tr➯♥✱ ✭✷✶✮ ❞➝♥ ✤➳♥ ✤✐➲✉ ♠➙✉ t❤✉➝♥ f (x0 )w + aM (u, u) ∈ −❝❧❝♦♥❡(C + f (x0 )u)✳ Mn ♣ • ( 2trnn )Mn → ∞✳ ❑❤✐ ✤â✱ Mn → ∞✱ → M ∈ ♣✲Bf (x0 )∞ \ {0}✱ ✈➔ t❛ ✤÷đ❝ − Mn f (xn ) − f (x0 ) − tn f (x0 )u → M (u, u)✳ t2n Mn ❚÷ì♥❣ tü ữ trữợ t ổ t ữủ M (u, u) ∈ −❝❧❝♦♥❡(C + f (x0 )u)✳ ◆❤➟♥ ①➨t tr ỵ ró r➔♥❣ ✤÷đ❝ s✉② r❛ ❜ð✐ ✤✐➲✉ ❦✐➺♥ s❛✉ ✭❛ ✮ ∀(M, N ) ∈ ♣✲❝❧B(f,g) (x0 )✱ ∃(c∗ , k ∗ ) ∈ Λ1 (x0 )✱ c∗ , M (u, u) + k ∗ , N (u, u) > 21 supk∈T (−K,g(x0 ),g (x0 )u) k ∗ , k ✱ ✈➔ ∀(M, N ) ∈ ♣✲B(f,g) (x0 )∞ \ {0}✱ ∃c∗ ∈ C ∗ \ {0}✿ c∗ , f (x0 )u = 0✱ c∗ , M (u, u) > 0✳ ✭✐✐✮ ●✐↔ sû r➡♥❣ g ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t q✉❛♥❤ x0 ✈ỵ✐ g ❧➔ ê♥ ✤à♥❤ t↕✐ x0 ✳ õ ỵ ụ ú t t w ∈ T (S, x0 , u)∩u⊥ \{0} tr♦♥❣ ✤✐➲✉ ❦✐➺♥ ✭✐✐✮ ✭❜✮ ❜ð✐ w ∈ u⊥ \{0} ✈ỵ✐ g (x0 )w ∈ T (−K, g(x0 ), g (x0 )u) ✭❤❛② g (x0 )w ∈ ❝❧❝♦♥❡❬❝♦♥❡(−K−g(x0 ))−g (x0 )u]✮✳ ❚❤➟t ✈➟②✱ ♥➳✉ w ∈ T (S, x0 , u)✱ t❤➻ ❜ð✐ ▼➺♥❤ ✤➲ ✶✳✶✱ g (x0 )w ∈ T (−K, g(x0 ), g (x0 )u) ⊂ ❝❧❝♦♥❡❬❝♦♥❡(−K − g(x0 )) − g (x0 )u]✳ ❍➺ q✉↔ s❛✉ ✤➙② ✤÷đ❝ s✉② r trỹ t tứ ỵ ợ (f, g) ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ❝➜♣ ❤❛✐ t↕✐ x0 ✳ ❍➺ q✉↔ ✹✳✸✳ ❱ỵ✐ ❜➔✐ t♦→♥ ✭P✮ ✈ỵ✐ h = 0✱ ❣✐↔ sû r➡♥❣ X ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✈➔ f ✈➔ g ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ❝➜♣ ❤❛✐ t↕✐ x0 ∈ S ✳ ❑❤✐ ✤â✱ ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ ✤õ ❝❤♦ x0 ∈ ▲❋❊(2, f, S)✳ ✭✐✮ ợ u SX tỗ t (c , k ∗ ) ∈ C ∗ × K(g(x0 ))∗ s❛♦ ❝❤♦ c∗ , f (x0 )u + k ∗ , g (x0 )u > 0✳ ✭✐✐✮ ❱ỵ✐ ♠å✐ u ∈ SX ∩ T (S, x0 ) ✈ỵ✐ f (x0 )u ∈ −C ✱ t❛ ❝â ✭❛✮ ∀w ∈ T (S, x0 , u)∩u⊥ ✿ g (x0 )w+g (x0 )(u, u) ∈ T (−K, g(x0 ), g (x0 )u)✱ ∃(c∗ , k ∗ ) ∈ Λ1 (x0 )✱ c∗ , f (x0 )w + f (x0 )(u, u) > 0❀ ✭❜✮ ∀w ∈ T (S, x0 , u) ∩ u⊥ \ {0}✱ ∃c∗ ∈ C ∗ \ {0}✿ c∗ , f (x0 )u = 0✱ c∗ , f (x0 )w > 0✳ ❍➺ q✉↔ ✹✳✸ ✭✐✐✮ ♠ð rë♥❣ ✣à♥❤ ỵ tr õ Y = R ❍➺ q✉↔ ✼ ❝õ❛ ❬✼❪✱ tr♦♥❣ ✤â Y ✈➔ Z ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉✳ ❈→❝ ❍➺ q✉↔ ✹✳✹ ✈➔ ✹✳✺ ữợ ữủ s r tự tứ ỵ ũ ss s rở r tỹ ss r tữỡ ự q ợ t P ✈ỵ✐ h = 0✱ ❣✐↔ sû r➡♥❣ X, Y, Z (f, g) t❤✉ë❝ ❧ỵ♣ C 1,1 ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✈➔ t↕✐ x0 ∈ S ✳ ❑❤✐ ✤â✱ ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ ✤õ ❝❤♦ x0 ∈ ✷✾ ▲❋❊(2, f, S)✳ ✭✐✮ ❱ỵ✐ ♠å✐ u ∈ SX ✱ tỗ t (c , k ) C × K(g(x0 ))∗ s❛♦ ❝❤♦ c∗ , f (x0 )u + k ∗ , g (x0 )u > 0✳ ✭✐✐✮ ❱ỵ✐ ♠å✐ u ∈ SX ∩ T (S, x0 ) ✈ỵ✐ f (x0 )u ∈ −C ✱ t❛ ❝â ✭❛✮ ∀w ∈ T (S, x0 , u)∩u⊥ ✱ ∀(M, N ) ∈ ∂C2 (f, g)(x0 )✿ g (x0 )w+N (u, u) ∈ T (−K, g(x0 ), g (x0 )u)✱ ∃(c∗ , k ∗ ) ∈ Λ1 (x0 )✱ c∗ , M (u, u) + k ∗ , N (u, u) > k ∗ , g (x0 )w + N (u, u) ; ✭❜✮ ∀w ∈ T (S, x0 , u) ∩ u⊥ \ {0}✱ ∃c∗ ∈ C ∗ \ {0}✿ c∗ , f (x0 )u = 0✱ c∗ , f (x0 )w > 0✳ ❍➺ q✉↔ ✹✳✹ ✭✐✐✮ ♠ð rë♥❣ ❍➺ q✉↔ ✽ ❝õ❛ ❬✼❪✳ ❍➺ q✉↔ ✹✳✺✳ ❱ỵ✐ ❜➔✐ t♦→♥ ✭P✮ ✈ỵ✐ h = 0✱ ❣✐↔ sû r➡♥❣ X, Y, Z ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✈➔ (f, g) t❤✉ë❝ ❧ỵ♣ C t↕✐ x0 ∈ S ✳ ●✐↔ sû ❤ì♥ ♥ú❛ f ✈➔ g ❝â ❝→❝ →♥❤ ①↕ tü❛ ❍❡ss✐❛♥ ∂ f (.) ✈➔ ∂ g(.)✱ t÷ì♥❣ ù♥❣✱ ❧➔ ♥ú❛ ❧✐➯♥ tư❝ tr➯♥ t↕✐ x0 ✳ ❑❤✐ ✤â✱ ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ❧➔ ✤õ ❝❤♦ x0 ∈ ▲❋❊(2, f, S)✳ ✭✐✮ ❱ỵ✐ ♠å✐ u ∈ SX ✱ ∃(c∗ , k ∗ ) ∈ C ∗ × K(g(x0 ))∗ ✱ c∗ , f (x0 )u + k ∗ , g (x0 )u > 0✳ ✭✐✐✮ ❱ỵ✐ ♠å✐ u ∈ SX ∩ T (S, x0 ) ✈ỵ✐ f (x0 )u ∈ −C ✱ t❛ ❝â ✭❛✮ ∀w ∈ T (S, x0 , u)∩u⊥ ✱ ∀(M, N ) ∈ ❝❧❝♦∂ (f, g)(x0 )✿ g (x0 )w+N (u, u) ∈ T (−K, g(x0 ), g (x0 )u)✱ ∃(c∗ , k ∗ ) ∈ Λ1 (x0 )✱ c∗ , M (u, u) + k ∗ , N (u, u) > k ∗ , g (x0 )w + N (u, u) , ✈➔ ∀(M, N ) ∈ ❝♦∂ (f, g)(x0 )∞ \ {0}✿ N (u, u) ∈ T (−K, g(x0 ), g (x0 )u)✱ ∃c∗ ∈ C ∗ \ {0}✿ c∗ , f (x0 )u = 0✱ c∗ , M (u, u) > 0❀ ✭❜✮ ∀w ∈ T (S, x0 , u) ∩ u⊥ \ {0}✱ ∀M ∈ ❝♦∂ f (x0 )∞ ✱ ∃c∗ ∈ C ∗ \ {0}✿ c∗ , f (x0 )u = 0✱ c∗ , f (x0 )w + M (u, u) > ✈➔ ∀M ∈ ❝♦∂ f (x0 )∞ \ {0}✱ ∃c∗ ∈ C ∗ \ {0}✿ c∗ , f (x0 )u = 0✱ c∗ , M (u, u) > 0✳ ❚r♦♥❣ ✈➼ ❞ö s❛✉ ✤➙②✱ ✣à♥❤ ỵ t r ữủ tr ✤â ❝→❝ ❦➳t q✉↔ ❣➛♥ ✤➙② t❤➻ ❦❤ỉ♥❣✳ ❱➼ ❞ư ✹✳✶✳ ❈❤♦ C = R +✱ K = {(k1 , k2 , k3 ) ∈ R3 |k2 k3 ≥ 2k12 , k2 ≤ 0, k3 ≤ 0}✱ (x0 , y0 ) = (0, 0)✱ ✈➔ f : R2 → R ✈➔ g : R2 → R3 ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐  ✐❢ x ≥ 0, y ≥ 0,  x2 + y f (x, y) = −x2 + y ✐❢ x ≥ 0, y < 0,  3θ(x) + y ✐❢ x < 0, g(x, y) = (x, x2 + 1, y)✱ tr♦♥❣ ✤â θ ✤÷đ❝ ①→❝ ✤à♥❤ tr♦♥❣ ❱➼ ❞ö ✸✳✷✳ ❑❤✐ ✤â✱ t↕✐ (0, 0)✱ f ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ♥❤÷♥❣ ❦❤ỉ♥❣ ❦❤↔ ✈✐ ❝❤➦t✱ ✈➔ t❛ ❝â ✸✵   f (0, 0) = (0, 1)✱ g (0, 0) ❂ 0 0✱ g(0, 0) = (0, 1, 0) ∈ −K ✱ T (−K, g(0, 0)) = {(k1 , k2 , k3 ) ∈ R3 | k3 ≥ 0}, N (−K, g(0, 0)) = {λ(0, 0, −1)| λ ∈ R}, Λ1 (0, 0) = {(c∗ , k ∗ ) ∈ R × R3 | c∗ = α > 0, k ∗ = α(0, 0, −1)} ❚❛ ❝â t❤➸ ❧➜② ❝→❝ ①➜♣ ①➾ ❝➜♣ ❤❛✐ Bf (0, 0) = {Mβ | β ∈ {−1} ∪ (1, ∞)}✱ B(f,g) (0, 0) = {(Mβ , N )| β ∈ {−1} ∪ (1, ∞)}✱ tr♦♥❣ ✤â Mβ ❂ β 0 ✈➔ N : R2 × R2 → R3 ❧➔ ♠❛ tr➟♥ × × N ❂ 0 0 ✱ 0 0 0 tù❝ ❧➔✱ N (u, v) = (0, u1 v1 , 0)✱ ✈ỵ✐ u, v ∈ R2 ✳ ❱➻ t❤➳✱ ❝❧Bf (0, 0) = {Mβ | β ∈ {−1} ∪ [1, ∞)}✱ ❝❧B(f,g) (0, 0) = {(Mβ , N )| β ∈ {−1} ∪ [1, ∞)}✱ Bf (0, 0)∞ = {Mβ | β ≥ 0}✱ B(f,g) (0, 0)∞ = {(Mβ , 02×2×3 )| β ≥ 0}✳ ❈❤å♥ u = (1, 0) ∈ S2 ✳ ❚❛ ❝â✱ ✈ỵ✐ ♠å✐ (c∗ , k ∗ ) ∈ C ∗ × K(g(0, 0))∗ ✱ c∗ , f (0, 0)u + k ∗ , g (0, 0)u = 0✳ ❉♦ õ tr ỵ ổ tọ ❈❤♦ u = (u1 , u2 ) ∈ S2 s❛♦ ❝❤♦ (f, g) (0, 0)u ∈ −[C×❝❧K(g(0, 0))]✳ ❑❤✐ ✤â✱ u = (u1 , 0) ✈ỵ✐ u1 = ±1✳ ❚❛ ❝â T (−K, g(0, 0), g (0, 0)u) = A2 (−K, g(0, 0), g (0, 0)u)✱ ✈➔ ❞♦ ✤â✱ ✈ỵ✐ k ∗ = (0, 0, −1) ∈ N (−K, g(0, 0))✱ supk∈T (−K,g(0,0),g (0,0)u) k ∗ , k = −4 ✭q✉❛♥ s→t r➡♥❣ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ①↔② r❛✮✳ ❇➙② ❣✐í✱ ✈ỵ✐ ♠å✐ (Mβ , N ) ∈ ❝❧B(f,g) (0, 0) tỗ t (c , k ) = (1, 0, 0, −1) ∈ Λ1 (0, 0) t❤ä❛ c∗ , 2Mβ (u, u) + k ∗ , 2N (u, u) = 2β > supk∈T (−K,g(0,0),g (0,0)u) k ∗ , k ✈➔✱ ✈ỵ✐ ♠å✐ (Mβ , N ) ∈ B(f,g) (0, 0) \ {0} tỗ t c = ∈ C ∗ \ {0} ✈ỵ✐ c∗ , f (0, 0)u = t❤ä❛ c∗ , Mβ (u, u) = β > 0✳ ❱➻ t❤➳✱ ✭❛ ✮ ❝õ❛ ◆❤➟♥ ①➨t õ tr ỵ t❤ä❛✳ ❍ì♥ ♥ú❛✱ ❝❤♦ w = (w1 , w2 ) ∈ v ⊥ \ {(0, 0)}✱ tù❝ ❧➔✱ w1 = ✈➔ w2 = 0✱ ♥➳✉ g (0, 0)w = (0, 0, w2 ) ∈ ❝❧❝♦♥❡❬❝♦♥❡(−K − g(0, 0)) − g (0, 0)u] = {(k1 , k2 , k3 ) ∈ R3 | k3 ≥ 0}✱ t❤➻ w2 > 0✳ ❱➻ t❤➳✱ ✈ỵ✐ ♠å✐ Mβ ∈ Bf (0, 0)∞ ✱ tỗ t c = C \ {0} ✈ỵ✐ c∗ , f (0, 0)u = t❤ä❛ c∗ , f (0, 0)w + Mβ (u, u) = w2 + β > 0✱ ✈➔✱ ✈ỵ✐ ♠å✐ Mβ ∈ Bf (0, 0) \ {0} tỗ t c = C ∗ \ {0} ✈ỵ✐ c∗ , f (0, 0)u = t❤ä❛ c∗ , Mβ (u, u) = β > 0✳ ❱➻ t❤➳✱ ❜ð✐ ◆❤➟♥ ①➨t ✹✳✷ ✭✐✐✮✱ ✤✐➲✉ ỵ tọ q ❧➔✱ (0, 0) ∈ ▲❋❊(2, f, S)✳ ❱➻ f ∈ C t↕✐ (0, 0)✱ ❝→❝ ❍➺ q✉↔ ✼✱ ✽ ỵ q✉↔ ✹✳✹ ✈➔ ✹✳✺ ð tr➯♥ ❦❤ỉ♥❣ →♣ ❞ư♥❣ ✤÷đ❝✳ ❍ì♥ ♥ú❛✱ ✈➻ d2 (f, g)((0, 0), u) = ∅✱ ỵ ụ ổ ữủ t ữợ ự rở t➔✐ ❚r♦♥❣ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ♥➔②✱ ✤➛✉ t✐➯♥✱ ❝❤ó♥❣ tỉ✐ ❣✐ỵ✐ t❤✐➺✉ ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ ✈➔ ❦❤↔♦ s→t ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ ❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ tỉ✐ ✤➲ ①✉➜t ❦❤→✐ ♥✐➺♠ ✤↕♦ ❤➔♠ s✉② rë♥❣ ❦✐➸✉ ①➜♣ ①➾ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ ✈➔ ✤÷❛ r❛ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ ❈✉è✐ ❝ò♥❣✱ ❞ò♥❣ ❝→❝ ✤↕♦ ❤➔♠ s✉② rë♥❣ ữợ tt t ✭tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥✮ ❤❛② ❦❤↔ ✈✐ ✭tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ✮✱ ❝❤ó♥❣ tỉ✐ t❤✐➳t tố ữ ợ ❝→❝ ♥❣❤✐➺♠ ②➳✉ ✤à❛ ♣❤÷ì♥❣ ✈➔ ❝→❝ ♥❣❤✐➺♠ ❝❤➢❝ ❝❤➢♥ ữỡ ợ t t ữủ ró ỡ ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ❦❤ỉ♥❣ trì♥ tr♦♥❣ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✈æ ❤↕♥ ❝❤✐➲✉ ✭P✮✳ ❚r♦♥❣ ❦➳ ❤♦↕❝❤ ♥❣❤✐➯♥ ự tữỡ ú tổ s rở ữợ ❝ù✉ ❝õ❛ ✤➲ t➔✐ ❜➡♥❣ ❝→❝❤ ①➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ❦❤ỉ♥❣ trì♥ ✈ỵ✐ r➔♥❣ ❜✉ë❝ ❜❛♦ ❤➔♠ t❤ù❝ ❦❤→ tê♥❣ q✉→t s❛✉ ✤➙②✿ ✭P✶✮ ♠✐♥C f (x)✱ s❛♦ ❝❤♦ x ∈ S ✱ ∈ F (x)✱ tr♦♥❣ ✤â f : X → Y ❧➔ →♥❤ ①↕ ✤ì♥ trà ✈➔ F : X → 2Z ❧➔ →♥❤ ①↕ ✤❛ trà✱ X ✈➔ Z ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ Y ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ S ⊂ X ✱ C Y õ ỗ õ ú tổ s➩ t❤✐➳t ❧➟♣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥ ✈➔ ✤õ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ ❝❤♦ ❝→❝ ♥❣❤✐➺♠ ②➳✉ ✈➔ ♥❣❤✐➺♠ ❝❤➢❝ ❝❤➢♥ ❝õ❛ ❜➔✐ t♦→♥ ✭P✶✮ ❜➡♥❣ ❝→❝ q✉② t➢❝ ♥❤➙♥ tû ❋r✐t③✲ ❏♦❤♥✲▲❛❣r❛♥❣❡✳ ❈❤ó♥❣ tỉ✐ ❞ò♥❣ ❝→❝ ✤↕♦ ❤➔♠ s✉② rë♥❣ ❦✐➸✉ ①➜♣ ①➾ ❝❤♦ f ✱ t ữợ tr F ❝→❝ ♥â♥ t✐➳♣ ①ó❝ ✈➔ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ữợ tt ữủ ✸✷ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❆❧❧❛❧✐✱ ❑✳✱ ❆♠❛❤r♦q✱ ❚✳✿ ❙❡❝♦♥❞✲♦r❞❡r ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ ♣r✐♠❛❧ ❛♥❞ ❞✉❛❧ ♥❡❝❡ss❛r② ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s✱ ❖♣t✐♠✐③❛t✐♦♥ ✭✶✾✾✼✮ ✷✷✾✲✷✹✻✳ ❬✷❪ ❇❡❞♥❛✞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✐♠✳ ✭✶✾✾✵✮ ✷✻✺✲✷✽✼✳ ❬✻❪ ❉♦♥t❝❤❡✈✱ ❆✳ ▲✳✱ ❘♦❝❦❛❢❡❧❧❛r✱ ❘✳ ❚✳✿ ❘❡❣✉❧❛r✐t② ❛♥❞ ❝♦♥❞✐t✐♦♥✐♥❣ ♦❢ s♦❧✉t✐♦♥ ♠❛♣♣✐♥❣s ✐♥ ✈❛r✐❛t✐♦♥❛❧ ❛♥❛❧②s✐s✱ ❙❡t✲✈❛❧✉❡❞ ❆♥❛❧✳ ✭✷✵✵✹✮ ✼✾✲✶✵✾✳ ❬✼❪ ●✉t✐➨rr❡③✱ ❈✳✱ ❏✐♠➨♥❡③✱ ❇✳✱ ◆♦✈♦✱ ❱✳✿ ❖♥ s❡❝♦♥❞ ♦r❞❡r ❋r✐t③ ❏♦❤♥ t②♣❡ ♦♣t✐♠❛❧✐t② ❝♦♥✲ ❞✐t✐♦♥s ✐♥ ♥♦♥s♠♦♦t❤ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♣r♦❣r❛♠♠✐♥❣✱ ▼❛t❤✳ Pr♦❣r❛♠✳ ✭❙❡r✳ ❇✮ ✭✷✵✶✵✮ ✶✾✾✲✷✷✸✳ ❬✽❪ ❍✐r✐❛rt✲❯rr✉t②✱ ❏✳ ❇✳✱ ❙tr♦❞✐♦t✱ ❏✳ ❏✳✱ ◆❣✉②❡♥✱ ❱✳ ❍✳✿ ●❡♥❡r❛❧✐③❡❞ ❍❡ss✐❛♥ ♠❛tr✐① ❛♥❞ s❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ♣r♦❜❧❡♠s ✇✐t❤ C 1,1 ❞❛t❛✱ ❆♣♣❧✳ ▼❛t❤✳ ❖♣t✐♠✳ ✭✶✾✽✹✮ ✹✸✲✺✻✳ ❬✾❪ ❏❡②❛❦✉♠❛r✱ ❱✳✱ ▲✉❝✱ ❉✳ ❚✳✿ ◆♦♥s♠♦♦t❤ ❱❡❝t♦r ❋✉♥❝t✐♦♥s ❛♥❞ ❈♦♥t✐♥✉♦✉s ❖♣t✐♠✐③❛✲ t✐♦♥✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥ ✭✷✵✵✽✮✳ ❬✶✵❪ ❏✐♠➨♥❡③✱ ❇✳✱ ◆♦✈♦✱ ❱✳✿ ❙❡❝♦♥❞ ♦r❞❡r ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ✐♥ s❡t ❝♦♥str❛✐♥❡❞ ❞✐❢❢❡r✲ ❡♥t✐❛❜❧❡ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥✱ ▼❛t❤✳ ▼❡t❤✳ ❖♣❡r✳ ❘❡s✳ ✭✷✵✵✸✮ ✷✾✾✲✸✶✼✳ ❬✶✶❪ ❏✐♠➨♥❡③✱ ❇✳✱ ◆♦✈♦✱ ❱✳✿ ❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✐♥ ❞✐❢❢❡r❡♥t✐❛❜❧❡ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ✈✐❛ s❡❝♦♥❞✲♦r❞❡r t❛♥❣❡♥t s❡ts✱ ❆♣♣❧✳ ▼❛t❤✳ ❖♣t✐♠✳ ✭✷✵✵✹✮ ✶✷✸✲✶✹✹✳ ❬✶✷❪ ❏♦✉r❛♥✐✱ ❆✳✿ ▼❡tr✐❝ r❡❣✉❧❛r✐t② ❛♥❞ s❡❝♦♥❞✲♦r❞❡r ♥❡❝❡ss❛r② ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ✉♥❞❡r ✐♥❝❧✉s✐♦♥ ❝♦♥str❛✐♥ts✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✭✶✾✾✹✮ ✾✼✲✶✷✵✳ ❬✶✸❪ ❏♦✉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✐♥❣ ❛♣✲ ♣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✳ ◗✳✱ ❚✉❛♥✱ ◆✳ ❉✳✿ ❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✉s✐♥❣ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r ♥♦♥✲ s♠♦♦t❤ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ✉♥❞❡r ❣❡♥❡r❛❧ ✐♥❡q✉❛❧✐t② ❝♦♥str❛✐♥ts✱ ❏✳ ❈♦♥✈❡① ❆♥❛❧✳ ✭✷✵✵✾✮ ✶✻✾✲✶✽✻✳ ❬✶✾❪ ❑❤❛♥❤✱ P✳ ◗✳✱ ❚✉❛♥✱ ◆✳ ❉✳✿ ❈♦rr✐❣❡♥❞✉♠ t♦ ✏❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✉s✐♥❣ ❛♣♣r♦①✐♠❛✲ t✐♦♥s ❢♦r ♥♦♥s♠♦♦t❤ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ✉♥❞❡r ❣❡♥❡r❛❧ ✐♥❡q✉❛❧✐t② ❝♦♥str❛✐♥ts✧✱ ❏✳ ❈♦♥✈❡① ❆♥❛❧✳ ✭✷✵✶✶✮ ✽✾✼✲✾✵✶✳ ❬✷✵❪ ❑❤❛♥❤✱ 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✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✭✷✵✶✸✮ ✻✾✺✲✼✵✷✳ ❬✷✶❪ ❑❤❛♥❤✱ P✳ ◗✳✱ ❚✉❛♥✱ ◆✳ ❉✳✿ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✇✐t❤ t❤❡ ❡♥✈❡❧♦♣❡✲ ❧✐❦❡ ❡❢❢❡❝t ✐♥ ♥♦♥s♠♦♦t❤ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠♠✐♥❣✱ ■■✿ ❖♣t✐♠❛❧✐t② ❝♦♥✲ ❞✐t✐♦♥s✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✭✷✵✶✸✮ ✼✵✸✲✼✶✹✳ ❬✷✷❪ ❑❤❛♥❤✱ P✳ ◗✳✱ ❚✉❛♥✱ ◆✳ ❉✳✿ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✇✐t❤ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ❡❢✲ ❢❡❝t ❢♦r ♥♦♥s♠♦♦t❤ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ✐♥ ✐♥❢✐♥✐t❡ ❞✐♠❡♥s✐♦♥s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✭✷✵✶✸✮ ✶✸✵✲✶✹✽✳ ❬✷✸❪ ▼❛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❡♥♦t✱ ❏✳ P✳✿ ❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠♠✐♥❣ ❛♥❞ ❝♦♠♣♦s✐t❡ ♦♣t✐♠✐③❛t✐♦♥✱ ▼❛t❤✳ Pr♦❣r❛♠✳ ✭✶✾✾✹✮ ✷✷✺✲✷✹✺✳ ❬✷✺❪ P❡♥♦t✱ ❏✳ P✳✿ ❙❡❝♦♥❞ ♦r❞❡r ❝♦♥❞✐t✐♦♥s ❢♦r ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ✇✐t❤ ❝♦♥str❛✐♥ts✱ ❙■❆▼ ❏✳ ❈♦♥tr♦❧ ❖♣t✐♠✳ ✭✶✾✾✽✮ ✸✵✸✲✸✶✽✳ ❬✷✻❪ P❡♥♦t✱ ❏✳ P✳✿ ❘❡❝❡♥t ❛❞✈❛♥❝❡s ♦♥ s❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s✱ ✐♥ ❖♣t✐♠✐③❛✲ t✐♦♥✱ ❱✳ ❍✳ ◆❣✉②❡♥✱ ❏✳ ❏✳ ❙tr♦❞✐♦t✱ P✳ ❚♦ss✐♥❣s ❡❞s✳✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥✱ ✭✷✵✵✵✮ ✸✺✼✲✸✽✵✳ ❬✷✼❪ ❘♦❝❦❛❢❡❧❧❛r✱ ❘✳ ❚✳✿ ❈♦♥✈❡① ❆♥❛❧②s✐s✱ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✱ Pr✐♥❝❡t♦♥✱ ◆❡✇ ❏❡rs❡② ✭✶✾✼✵✮✳ ❬✷✽❪ ❚❛❛✱ ❆✳✿ ❙❡❝♦♥❞✲♦r❞❡r ❝♦♥❞✐t✐♦♥s ❢♦r ♥♦♥s♠♦♦t❤ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜✲ ❧❡♠s ✇✐t❤ ✐♥❝❧✉s✐♦♥ ❝♦♥str❛✐♥ts✱ ❏✳ ●❧♦❜❛❧ ❖♣t✐♠✳ ✭✷✵✶✶✮ ✷✼✶✲✷✾✶✳ ❬✷✾❪ ❲❛r❞✱ ❉✳ ❊✳✿ ❈❛❧❝✉❧✉s ❢♦r ♣❛r❛❜♦❧✐❝ s❡❝♦♥❞✲♦r❞❡r ❞❡r✐✈❛t✐✈❡s✱ ❙❡t ❱❛❧✉❡❞ ❆♥❛❧✳ ✭✶✾✾✸✮ ✷✶✸✲✷✹✻✳ ✶✻ ✶✽ ✹✵✸ ✹✵✸ ✼✼ ✶✺ ✻✼ ✸✼ ✺✵ ✸✹ ✶