ĐẠI HỌC THÁI NGUYÊN TRƢỜNG ĐẠI HỌC KHOA HỌC  - PHẠM LỆ QUYÊN VỀ PHƢƠNG PHÁP LỒI LÔGARIT VÀ MỘT VÀI ỨNG DỤNG LUẬN VĂN THẠC SĨ TOÁN HỌC THÁI NGUYÊN - 2019 ĐẠI HỌC THÁI NGUYÊN TRƢỜNG ĐẠI HỌC KHOA HỌC  - PHẠM LỆ QUYÊN VỀ PHƢƠNG PHÁP LỒI LÔGARIT VÀ MỘT VÀI ỨNG DỤNG Chuyên ngành: Toán Ứng Dụng Mã số: 46 01 12 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC TS Bùi Việt Hƣơng THÁI NGUYÊN - 2019 ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✶ ✶ ❑■➌◆ ❚❍Ù❈ ỗ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷✳✶✳ P❤➙♥ ❧♦↕✐ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✷✳✷✳ ▼➦t ✤➦❝ tr÷♥❣✳ ❇➔✐ t♦→♥ ❈❛✉❝❤② ✈ỵ✐ ❞ú ❦✐➺♥ ❝❤♦ tr➯♥ ♠➦t ✤➦❝ tr÷♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷✳✸✳ ❙ü ♣❤ö t❤✉ë❝ ❧✐➯♥ tö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Pữỡ ỗ ❧æ❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷ ▼❐❚ ❱⑨■ Ù◆● ❉Ư◆● ❈Õ❆ P❍×❒◆● P❍⑩P ▲➬■ ▲➷●❆❘■❚ ✷✳✶✳ Ù♥❣ ❞ư♥❣ tr♦♥❣ ❜➔✐ t♦→♥ ❈❛✉❝❤② ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ ♥❣÷đ❝ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✶✳ P❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ ♥❣÷đ❝ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✷✳ ✣→♥❤ ❣✐→ ê♥ ✤à♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳ Ù♥❣ ❞ư♥❣ tr♦♥❣ ❜➔✐ t♦→♥ ❈❛✉❝❤② ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✶✳ P❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✷✳ ✣→♥❤ ❣✐→ ê♥ ✤à♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✷✵ ✷✵ ✷✵ ✷✹ ✷✽ ✷✽ ✷✾ ✹✵ ▼Ð ✣❺❯ ❇➔✐ t♦→♥ ✤➦t ❦❤æ♥❣ ❝❤➾♥❤ ①✉➜t ❤✐➺♥ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ù♥❣ ❞ö♥❣✳ ❇➔✐ t♦→♥ ♥➔② ❝â ❧✐➯♥ q t ỵ t ỵ s t♦→♥ ✈➲ ❧➽♥❤ ✈ü❝ ✤✐➺♥ s✐♥❤ ❤å❝✳✳✳ ❚r♦♥❣ ♠ët ❜➔✐ ❜→♦ ♥ê✐ t✐➳♥❣ ❝õ❛ ❍❛❞❛♠❛r❞✱ ❜➔✐ t♦→♥ ♥➔② ❧➛♥ ✤➛✉ t ữủ ợ t ữ ởt ✤✐➸♥ ✈➲ ❜➔✐ t♦→♥ ✤➦t ❦❤æ♥❣ ❝❤➾♥❤✳ ✣➦❝ ✤✐➸♠ ♥ê✐ ❜➟t ❝õ❛ ❜➔✐ t♦→♥ ♥➔② ❧➔ ♠ët t❤❛② ✤ê✐ ♥❤ä tr♦♥❣ ❞ú ❦✐➺♥ ❝ô♥❣ ❝â t❤➸ ❞➝♥ ✤➳♥ ♠ët s❛✐ ❧➺❝❤ ❧ỵ♥ ✈➲ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥✳ ❍❛❞❛♠❛r❞ ❝❤♦ r➡♥❣ t t ổ ổ õ ỵ ✈➟t ❧➼✳ ❈❤➼♥❤ ✈➻ ✈➟②✱ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❜➔✐ t♦→♥ ✤➦t ❦❤æ♥❣ ❝❤➾♥❤ ✤➸ t➻♠ r❛ ❝→❝ ✤→♥❤ ❣✐→ ê♥ ✤à♥❤ ✈➔ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❝❤➾♥❤ ❤â❛ ❧➔ ♠ët q trồ Pữỡ ỗ ổrt ởt tr ♥❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❞ò♥❣ ✤➸ ê♥ ✤à♥❤ ❤â❛ ❝→❝ ❜➔✐ t♦→♥ ✤➦t ❦❤ỉ♥❣ ❝❤➾♥❤ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✳ P❤÷ì♥❣ ♣❤→♣ ♥➔② ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐ P✉❝❝✐ ✭✶✾✺✺✮✱ ❏♦❤♥ ✭✶✾✺✺✱ ✶✾✻✵✮✱ ▲❛✈r❡♥t✐❡✈ ✭✶✾✺✻✮ ❛♥❞ P❛②♥❡ ✭✶✾✻✵✮✱ ✣✐♥❤ ◆❤♦ ❍➔♦ ✈➔ ◆❣✉②➵♥ ❱➠♥ ✣ù❝ ✭✷✵✵✾✱ ✷✵✶✵✱ ✷✵✶✶✮✳ ✣➙② ❧➔ ❦➽ t❤✉➟t ✤→♥❤ ❣✐→ ❞ü❛ tr➯♥ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❜➟❝ ữ r ợ tr ợ ữợ ởt ỗ ổrt ❧➔ ♠ët ❤➔♠ ❝õ❛ ♥❣❤✐➺♠✳ ❈→❝ ✤→♥❤ ❣✐→ ✤â ✤÷đ❝ ❞ò♥❣ ✤➸ t❤✐➳t ❧➟♣ t➼♥❤ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✈➔ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ sü ♣❤ư t❤✉ë❝ ❧✐➯♥ tö❝ ❝õ❛ ♥❣❤✐➺♠ ✈➔♦ ❞ú ❦✐➺♥ ✤➣ ❝❤♦ t❤❡♦ ♠ët ♥❣❤➽❛ ♥➔♦ ✤â✳ ▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ✈➲ ữỡ ỗ ổrt ởt số ự ♣❤÷ì♥❣ ♣❤→♣ ✤➸ ê♥ ✤à♥❤ ❤â❛ ❜➔✐ t♦→♥ ✤➦t ❦❤ỉ♥❣ ❝❤➾♥❤ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✳ ❈ư t❤➸✱ ❧✉➟♥ ỗ ữỡ ữỡ t tr ỗ ởt tự ỡ ữỡ tr r ữỡ ỗ ổrt ❈❤÷ì♥❣ ✷✱ t→❝ ❣✐↔ tr➻♥❤ ❜➔② ❤❛✐ ❜➔✐ t♦→♥ ♠✐♥❤ ❤å❛ ❝❤♦ ♣❤÷ì♥❣ ✶ ♣❤→♣ ♥➔②✱ ✤â ❧➔ ❜➔✐ t♦→♥ ❈❛✉❝❤② ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ ♥❣÷đ❝ t❤í✐ ❣✐❛♥ ✈➔ ❜➔✐ t♦→♥ ❈❛✉❝❤② ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡✳ ✣➙② ❧➔ ❝→❝ ❜➔✐ t♦→♥ ✤➦t ❦❤ỉ♥❣ ❝❤➾♥❤ ✈➔ t→❝ ❣✐↔ ✤➣ sû ❞ư♥❣ ữỡ ỗ ổrt ữ r ✤à♥❤ ❝❤♦ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ❜➔✐ t♦→♥ ♥➔② ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✤÷đ❝ ❜ê s✉♥❣✳ P❤➛♥ ❝✉è✐ ❈❤÷ì♥❣ ✷✱ t→❝ ❣✐↔ ❝â tr➻♥❤ ❜➔② t❤➯♠ ♠ët ❜➔✐ t♦→♥ ❝â t❤➸ ①❡♠ ♥❤÷ ♠ð rë♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤② ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ữủ t ữợ sỹ ữợ ũ t ữỡ ổ t t ữợ ❞➝♥✱ ❝❤➾ ❜↔♦ ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ❊♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ❈ỉ✳ ❊♠ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ tr➙♥ t❤➔♥❤ tỵ✐ ❚❤➛② ❈ỉ ❣✐→♦ ❦❤♦❛ ❚♦→♥ ✲ ❚✐♥✱ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t➟♥ t➻♥❤ ❣✐↔♥❣ ❞↕② ✈➔ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ tr♦♥❣ q✉→ tr➻♥❤ ❡♠ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ t↕✐ tr÷í♥❣✳ ❊♠ ①✐♥ tr➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❚❙✳ ▼❛✐ ❱✐➳t ❚❤✉➟♥ ✈➔ ❚❙✳ ❚r÷ì♥❣ ▼✐♥❤ ❚✉②➯♥ ✤➣ ❞➔♥❤ sü q✉❛♥ t➙♠ ✈➔ ❝â ♥❤ú♥❣ ❧í✐ ✤ë♥❣ ✈✐➯♥ ❦à♣ t❤í✐ ✤➸ ❡♠ ❝è ❣➢♥❣ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❈✉è✐ ❝ò♥❣ ❡♠ ①✐♥ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ỗ ổ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ✷ ❈❤÷ì♥❣ ✶ ❑■➌◆ ỗ ỗ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠✱ ✤à♥❤ ♥❣❤➽❛ ✈➔ ❦➳t q tt q ỗ t ỗ ữủ t tứ ỗ a, b ∈ Rn✳ ✐✮ ✣÷í♥❣ t❤➥♥❣ ✤✐ q✉❛ ❤❛✐ ✤✐➸♠ a ✈➔ b ❧➔ t➟♣ ❤ñ♣ ❝â ❞↕♥❣ {x ∈ Rn |x = λa + (1 − λ)b, λ ∈ R} ✐✐✮ ✣♦↕♥ t❤➥♥❣ ✤✐ q✉❛ ❤❛✐ ✤✐➸♠ a ✈➔ b ❧➔ t➟♣ ❤ñ♣ ❝â ❞↕♥❣ {x ∈ Rn |x = λa + (1 − λ)b, λ ∈ [0, 1]} ✣à♥❤ ♥❣❤➽❛ ✶✳✷ ❚➟♣ C ⊂ Rn ✤÷đ❝ ❣å✐ ❧➔ t ỗ C ự t ố ✤✐➸♠ ❜➜t ❦ý ❝õ❛ ♥â✱ tù❝ ❧➔ ∀x, y ∈ C, ∀λ ∈ [0, 1], t❛ ❝â λx + (1 − λ)y ∈ C ✣à♥❤ ♥❣❤➽❛ ✶✳✸ ✐✮ ❚❛ ♥â✐ x tờ ủ ỗ tỡ x1, x2, · · · , xk ♥➳✉ k k λj x ✈ỵ✐ λj > 0, ∀j = 1, 2, · · · , k ✈➔ j x= j=1 λj = j=1 ✸ ✐✐✮ ❚❛ ♥â✐ x ❧➔ tê ❤ñ♣ ❛❢❢✐♥❡ ❝õ❛ ❝→❝ ✤✐➸♠ ✭✈❡❝tì✮ x1 , x2 , · · · , xk ♥➳✉ k k λj x ✈ỵ✐ j x= j=1 λj = j=1 ▼➺♥❤ ✤➲ ✶✳✶ ủ C ỗ õ ự tờ ủ ỗ õ tù❝ ❧➔ ✈ỵ✐ ♠å✐ k ∈ N✱ ✈ỵ✐ ♠å✐ λ1 , λ2 , · · · , λk > s❛♦ ❝❤♦ k λj = j=1 ✈➔ ✈ỵ✐ ♠å✐ x1 , x2 , · · · , xk ∈ C t❛ ❝â k λj xj ∈ C j=1 ✣à♥❤ ♥❣❤➽❛ ✶✳✹ ▼ët t➟♣ C ✤÷đ❝ ❣å✐ ❧➔ ♥â♥ ♥➳✉ ✈ỵ✐ ♠å✐ λ > 0✱ ✈ỵ✐ ♠å✐ x ∈ C t❛ ❝â λx ∈ C ✳ ✐✮ ▼ët ♥â♥ ✤÷đ❝ õ ỗ õ t ỗ ởt õ ỗ ữủ õ õ ❦❤ỉ♥❣ ❝❤ù❛ ✤÷í♥❣ t❤➥♥❣✱ ❦❤✐ ✤â t❛ ♥â✐ ❧➔ õ õ ởt t ỗ t t õ õ õ ỗ ❞✐➺♥✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✺ ❈❤♦ C ⊂ Rn ❧➔ ♠ët t ỗ x C NC (x) = {w : w, y − x ≤ 0, ∀y ∈ C}, ✤÷đ❝ ❣å✐ ❧➔ ♥â♥ ♣❤→♣ t✉②➳♥ ✭♥❣♦➔✐✮ ❝õ❛ C t↕✐ x✳ ✐✐✮ ❚➟♣ −NC (x) = {w : w, y − x ≥ 0, ∀y ∈ C}, ✤÷đ❝ ❣å✐ ❧➔ ♥â♥ ♣❤→♣ t✉②➳♥ ✭tr♦♥❣✮ ❝õ❛ C t↕✐ x ỵ ỵ t t t ỗ t ỗ õ rộ ổ trò♥❣ ✈ỵ✐ t♦➔♥ ❜ë ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉ ❧➔ ❣✐❛♦ ❝õ❛ t➜t ❝↔ ❝→❝ ♥û❛ ❦❤æ♥❣ ❣✐❛♥ tü❛ ❝õ❛ ♥â✳ ✹ ✣à♥❤ ♥❣❤➽❛ ✶✳✻ ❈❤♦ ❤❛✐ t➟♣ C ✈➔ D ❦❤→❝ ré♥❣✱ t❛ ♥â✐ s✐➯✉ ♣❤➥♥❣ aT x = α t→❝❤ C ✈➔ D ♥➳✉ aT x ≤ α ≤ aT y, ∀a ∈ C, y ∈ D ❚❛ ♥â✐ s✐➯✉ ♣❤➥♥❣ aT x = α t→❝❤ ❝❤➦t C ✈➔ D ♥➳✉ aT x < α < aT y, ∀a ∈ C, y ∈ D ❚❛ ♥â✐ s✐➯✉ ♣❤➥♥❣ aT x = α t→❝❤ ♠↕♥❤ C ✈➔ D ♥➳✉ sup aT x < α < inf aT y, ∀a ∈ C, y D xC yD ỵ ỵ t C D t ỗ ❦❤→❝ ré♥❣ tr♦♥❣ Rn s❛♦ ❝❤♦ C ∩ D = ∅✳ ❑❤✐ ✤â ❝â ♠ët s✐➯✉ ♣❤➥♥❣ t→❝❤ C ✈➔ D ỵ ỵ t C D t ỗ õ rộ tr Rn s❛♦ ❝❤♦ C ∩ D = ∅✳ ●✐↔ sû ➼t ♥❤➜t ♠ët tr♦♥❣ ❤❛✐ t➟♣ ❧➔ t➟♣ ❝♦♠♣❛❝t✳ ❑❤✐ ✤â✱ ❤❛✐ t➟♣ ♥➔② ❝â t❤➸ t→❝❤ ♠↕♥❤ ✤÷đ❝ ❜ð✐ ởt s ỗ C Rn t ỗ f : C R ❦➼ ❤✐➺✉ ❞♦♠f = {x ∈ C : f (x) < +∞}, ❡♣✐f = {(x, α) ∈ C × R : f (x) ≤ α} ✣à♥❤ ♥❣❤➽❛ ✶✳✼ ❚➟♣ ❞♦♠f ✤÷đ❝ ❣å✐ ❧➔ ♠✐➲♥ ❤ú✉ ❤✐➺✉ ❝õ❛ f ✳ ❚➟♣ f ữủ tr ỗ t f ❇➡♥❣ ❝→❝❤ ✤➦t f (x) = +∞ ♥➳✉ x ∈ / C ✱ t❛ ❝â t❤➸ ❝♦✐ f ①→❝ ✤à♥❤ tr➯♥ t♦➔♥ ❦❤æ♥❣ ❣✐❛♥✳ ❑❤✐ ✤â✱ t❛ ❝â ❞♦♠f = {x ∈ Rn : f (x) ≤ +∞}, ❡♣✐f = {(x, α) ∈ Rn × R : f (x) ≤ α} ✺ ✣à♥❤ ♥❣❤➽❛ ✶✳✽ ❈❤♦ C ⊂ Rn✱ C = t ỗ f : C [, +] õ f ỗ tr C f t ỗ tr Rn+1 tr tữỡ ữỡ ợ x, y C, (0, 1) t❛ ❝â f [λx + (1 − λ)y] ≤ λf (x) + (1 − λ)f (y) ◆❤➟♥ ①➨t ✶✳✶ ❱➲ ♠➦t ❤➻♥❤ ❤å❝✱ ✤÷í♥❣ ❝♦♥❣ ❜✐➸✉ ❞✐➵♥ ♠ët ỗ tọ t t s ❦❤æ♥❣ ♥➡♠ tr➯♥ ✤♦↕♥ t❤➥♥❣ ♥è✐ ❜➜t ❦ý ❤❛✐ ✤✐➸♠ tở ữớ ổ ữợ t t t↕✐ ❜➜t ❦ý ✤✐➸♠ ♥➔♦ t❤✉ë❝ ✤÷í♥❣ ❝♦♥❣✳ ❱➲ ♠➦t t t tr õ t ữợ ❞↕♥❣ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ f (a) + f (a)(x − a) ≤ f (x) ≤ f (a) + f (b) − f (a) (x − a) b−a ✭✶✳✶✮ ✣à♥❤ ♥❣❤➽❛ ✶✳✾ ❈❤♦ C ⊂ Rn✱ C = ∅ ❧➔ t ỗ f : Rn [, +] ữủ ỗ t tr C f [x + (1 − λ)y] < λf (x) + (1 − λ)f (y), ∀x, y ∈ C, ∀λ ∈ (0, 1) ✐✐✮ ❍➔♠ f : Rn → [−∞, +∞] ✤÷đ❝ ❣å✐ ỗ tr C ợ số > ♥➳✉ ✈ỵ✐ ♠å✐ x, y ∈ C, ✈ỵ✐ ♠å✐ λ ∈ (0, 1) f [λx + (1 − λ)y] ≤ λf (x) + (1 − λ)f (y) − ηλ(1 − λ) x − y ✻ ✐✐✐✮ ❍➔♠ f : Rn → [−∞, +∞] ✤÷đ❝ ❣å✐ ó tr C f ỗ tr➯♥ C ✳ ▼➺♥❤ ✤➲ ✶✳✷ ▼ët ❤➔♠ f : C R ỗ tr C ❝❤➾ ❦❤✐ ✈ỵ✐ ♠å✐ x, y ∈ C ✱ ✈ỵ✐ ♠å✐ α, β t❤ä❛ ♠➣♥ f (x) < α, f (y) < β ✱ ✈ỵ✐ ♠å✐ sè λ ∈ [0, 1] t❛ ❝â f [λx + (1 − λ)y] ≤ λα + (1 − λ)β ❱➼ ❞ö ✶✳✶ ▼ët sè ỗ ||x|| ởt ỗ tr Rn tr õ x Rn C Rn t ỗ ❦❤→❝ ré♥❣✱ ❤➔♠ ❝❤➾ ❝õ❛ C ✱ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛   0 ♥➳✉ x ∈ C δC (x) := + x /C ởt ỗ C Rn t ỗ rộ ❤➔♠ tü❛ ❝õ❛ C ✱ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ SC (x) := sup y, x yC ởt ỗ C Rn t ỗ rộ ❦❤♦↔♥❣ ❝→❝❤ ✤➳♥ t➟♣ C ✱ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ dC (x) := x − y y∈C ❧➔ ♠ët ❤➔♠ ỗ f ữủ ❝❤➼♥❤ t❤÷í♥❣ ♥➳✉ ❞♦♠f = ∅ ✈➔ f (x) > ợ x f ữủ ❣å✐ ❧➔ ❤➔♠ ✤â♥❣ ♥➳✉ ❡♣✐f ❧➔ t➟♣ ✤â♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Rn+1 ✳ ✼ ❤❛② F (t) ❧➔ ❤➔♠ ỗ ổrt ổ tự tr t ❝â ✤→♥❤ ❣✐→ F (t) ≤ F (0) ❚❤❛② F (t) = u(t) T −t T t · F (T ) T , ∀t ∈ [0, T ) tr♦♥❣ ✭✷✳✷✵✮ ✈➔ sû ❞ö♥❣ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ ✭✷✳✶✺✮ ✈➔ ✤✐➲✉ ❦✐➺♥ ❝✉è✐ ✭✷✳✶✻✮ t❛ ♥❤➟♥ ✤÷đ❝ ✤→♥❤ ❣✐→ s❛✉ u(t) ≤ u0 (x) 2(1−t/T ) · u(T ) 2t/T , ∀t ∈ [0, T ) ✭✷✳✷✹✮ ❚➼♥❤ ❝❤➜t ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❣✐→ trà ❜✐➯♥ ❜❛♥ ✤➛✉ ✭✷✳✶✸✮✕✭✷✳✶✻✮ ❝â t❤➸ ✤÷đ❝ s✉② r❛ tø ✤→♥❤ ❣✐→ ✭✷✳✷✸✮ ✈➔ rữợ t ú t õ t t r ♥➳✉ ❤➔♠ F (t) t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✷✸✮ ✈➔ F ❜à tr✐➺t t✐➯✉ t↕✐ ♠ët ✤✐➸♠ ♥➔♦ ✤â t1 ∈ [0, T ] t❤➻ tø t➼♥❤ ❧✐➯♥ tö❝ F t õ t s r F (t) ỗ ♥❤➜t ❜➡♥❣ ✈ỵ✐ ♠å✐ t ∈ [0, T ]✳ ❚ø ✤â✱ t❛ s✉② r❛ t➼♥❤ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤② t✉②➳♥ t➼♥❤ ✭✷✳✶✸✮✕✭✷✳✶✻✮✳ ✣➸ tr↔ ❧í✐ ❝❤♦ ❝➙✉ ❤ä✐ ✈➲ t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ♥❣❤✐➺♠ t❛ ❣✐↔ sû u1 (x, t) ✈➔ u2 (x, t) ❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ s❛✉ uit = uixx , ≤ x ≤ 1, < t ≤ T, ui (0, t) = ui (1, t) = 0, < t ≤ T, ui (x, 0) = ui0 (x), ≤ x ≤ 1, ui (x, T ) = uiT (x), i = 1, ✣➦t u = u1 − u2 t❤➻ u s➩ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✸✮✕✭✷✳✶✻✮ ợ tữỡ ự u0 (x) = u10 (x) − u20 (x)✳ ❑❤✐ ✤â ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✷✹✮ s➩ ❝❤♦ t❛ ❦❤æ♥❣ ❣✐❛♥ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✤➸ s❛♦ ❝❤♦ u0 ♥❤ä s➩ ❦➨♦ t❤❡♦ u(t) ❝ô♥❣ ♥❤ä ✈ỵ✐ t ∈ [0, T ) ❤ú✉ ❤↕♥✳ ❚✉② ♥❤✐➯♥✱ ✤✐➲✉ ❦✐➺♥ u0 (x) ♥❤ä ❧➔ ❦❤æ♥❣ ✤õ ✤➸ t➼❝❤ u0 (x) 2(1−t/T ) · u(T ) 2t/T s➩ ♥❤ä ✈ỵ✐ t ∈ [0, T )✳ ❉♦ ✤â✱ ✤➸ ❝â t➼♥❤ ♣❤ö t❤✉ë❝ ❧✐➯♥ tö❝ ❝õ❛ ♥❣❤✐➺♠ ✈➔♦ ❞ú ❦✐➺♥ ❜❛♥ ✤➛✉ ❝❤ó♥❣ t❛ ❝➛♥ ♠ët ❤↕♥ ❝❤➳ ❝❤♦ ợ t t t ỵ ❤✐➺✉ M ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ❤➔♠ ϕ(x, t)✱ ❧✐➯♥ tư❝ tr♦♥❣ Ω = [0, 1] × [0, T ] ✈➔ ✈ỵ✐ ♠é✐ t ∈ (0, T ) ❝è ✤à♥❤ t❤➻ ϕ(x, t) ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ❤❛✐ ❧➛♥ t❤❡♦ ❜✐➳♥ x✱ ✈ỵ✐ ♠é✐ x ∈ [0, 1] t❤➻ ϕ(x, t) ❦❤↔ ✈✐ ❧✐➯♥ tö❝ t❤❡♦ ❜✐➳♥ t ∈ (0, T ) ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❜à ❝❤➦♥ ϕ(T ) ✭✷✳✷✺✮ ≤ M 2, ✈ỵ✐ M ❧➔ ❤➡♥❣ sè✳ ❚❛ t❤➜② r➡♥❣✱ tr♦♥❣ ❧ỵ♣ ❤➔♠ u ∈ M t❤➻ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✸✮✕✭✷✳✶✻✮ s➩ ♣❤ö t❤✉ë❝ tử t oăr ỳ tr L2 (0, 1) ✈ỵ✐ t ∈ [0, T )✳ ❑❤✐ õ t õ t q s ỵ ởt ♥❣❤✐➺♠ ❜➜t ❦ý ❝õ❛ ❜➔✐ t♦→♥ ❣✐→ trà ❜✐➯♥ ❜❛♥ ✤➛✉ ✭✷✳✶✸✮✕✭✷✳✶✻✮ t❤✉ë❝ ❧ỵ♣ M ✤➲✉ t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ u(t) ≤ M 2t/T u0 (x) 2(1−t/T ) ỵ r t q tr õ ỵ tỹ t số M õ t ữủ t t tứ ỳ t ỵ ❝õ❛ ❜➔✐ t♦→♥ ❦❤✐ t❛ ✤✐ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♠æ ❤➻♥❤ t♦→♥ ❤å❝ ❝õ❛ ♥â✳ ❚r♦♥❣ ♥❤✐➲✉ ❜➔✐ t♦→♥ ✈➟t ỵ t t õ t ữ r số M ✱ ✈➼ ❞ư ♥❤÷ ♥➳✉ ♥❣❤✐➺♠ u ❜✐➸✉ ❞✐➵♥ ♥❤✐➺t ✤ë ❝õ❛ ❝õ❛ ♠ët ❜➔✐ t♦→♥ t❤ü❝ t➳ t❤➻ t❛ õ t ữ r ợ tr u ❦❤✐ ✤â✱ t❛ ❦❤ỉ♥❣ ♥❤➜t t❤✐➳t ♣❤↔✐ ❝â ❞↕♥❣ t÷í♥❣ ♠✐♥❤ ❝õ❛ ♥❣❤✐➺♠✳ ▼➦t ❦❤→❝✱ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ❞→♥❣ ✤✐➺✉ ❝õ❛ ♥❣❤✐➺♠✱ tø ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✷✸✮ t❛ ❝â ❣✐ỵ✐ ữợ F (t) ữủ F (t) ≥ F (0) ❡①♣ tF (0) F (0) ✭✷✳✷✼✮ ❚❤❛② ❝→❝ ❞ú ❦✐➺♥ ❝â ✤÷đ❝ tø ♣❤➛♥ tr➯♥ t❛ ♥❤➟♥ ✤÷đ❝ u(t) ≥ u0 (x) ❡①♣ 2t ux u0 ✭✷✳✷✽✮ ❱➻ ✈➟②✱ ♥➳✉ u(t) ①→❝ ✤à♥❤ tr➯♥ [0, ∞) t❤➻ u(t) s➩ t➠♥❣ t❤❡♦ ❤➔♠ sè ♠ô✳ ❱➔ ❞♦ ✤â✱ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✷✽✮ ❝❤♦ t❛ ✤→♥❤ ❣✐→ ✈➲ tè❝ ✤ë t➠♥❣ ❝õ❛ ♥❣❤✐➺♠ u(x, t)✳ ✷✼ ✷✳✷✳ Ù♥❣ ❞ö♥❣ tr♦♥❣ ❜➔✐ t♦→♥ ❈❛✉❝❤② ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ✷✳✷✳✶✳ P❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ❚r♦♥❣ ♠ư❝ ♥➔② ❝❤ó♥❣ tỉ✐ ①➨t ❜➔✐ t♦→♥ ❈❛✉❝❤② ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡✳ ✣➸ ✤ì♥ ❣✐↔♥✱ ❝❤ó♥❣ tỉ✐ ①➨t ❜➔✐ t♦→♥ tr♦♥❣ ♠✐➲♥ t❤✉ë❝ ❦❤æ♥❣ ❣✐❛♥ R2 ✭①❡♠ ❬✹❪✮✳ ❈❤♦ R2 ữủ ợ < x < X, < y < 1✱ ✈ỵ✐ X < + số ố trữợ t t ❜✐➯♥ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ tr♦♥❣ ♠✐➲♥ Ω uxx + uyy = 0, ✭✷✳✷✾✮ (x, y) ∈ Ω ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❉✐r✐❝❤❧❡t u(x, 0) = u(x, 1) = 0, 0 ợ E ữủ ❜ð✐ ✭✷✳✸✻✮✳ ❚r♦♥❣ ♣❤➛♥ ❝✉è✐ ❝õ❛ ♠ư❝✱ ❝❤ó♥❣ tỉ✐ t❤↔♦ ❧✉➟♥ ✈➲ ♠ët ❦➳t q✉↔ t÷ì♥❣ tü ❝â t❤➸ ♥❤➟♥ ✤÷đ❝ ❝❤♦ ♠ët ❞↕♥❣ ❜➔✐ t♦→♥ tê♥❣ q✉→t ❤ì♥ ✭①❡♠ ❬✺❪✮✳ ✸✷ ❈❤♦ H ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ✈ỵ✐ t ổ ữợ , ã sỷ D ⊂ H ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ trò ♠➟t tr♦♥❣ H ❀ M ✈➔ N ❧➔ ❝→❝ t♦→♥ tû t✉②➳♥ t➼♥❤ tø D ✈➔ H ✳ ❳➨t ❜➔✐ t♦→♥ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ M d2 u + N u = 0, dx2 ✭✷✳✹✸✮ x ∈ [0, X] ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ du (0) = v0 dx u(0) = u0 , ✭✷✳✹✹✮ ✣➸ ✤ì♥ ❣✐↔♥ t❛ ❣✐↔ sû r➡♥❣ M ✈➔ N ❧➔ ❝→❝ t♦→♥ tû ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ t❤❛♠ sè x ✈➔ t❤ä❛ ♠➣♥ ❝→❝ ❣✐↔ t❤✐➳t s❛✉ ✐✮ M ❧➔ t♦→♥ tû ✤è✐ ①ù♥❣✱ ①→❝ ✤à♥❤ ❞÷ì♥❣❀ ✐✐✮ N ❧➔ t♦→♥ tû ✤è✐ ①ù♥❣❀ ✐✐✐✮ ♥❣❤✐➺♠ u ❝õ❛ ❜➔✐ t♦→♥ t❤✉ë❝ ❧ỵ♣ C ([0, X], H)✳ ✣➸ t❤✉➟♥ t✐➺♥ t❛ ✤✐ ①➨t ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ❜➔✐ t♦→♥ tr➯♥ ❜➡♥❣ ❝→❝❤ ♥❤➙♥ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✹✸✮ ✈ỵ✐ ❤➔♠ t❤û ϕ ∈ C ([0, X], H) t❛ ❝â ϕ(x), M d2 u + ϕ(x), N u = ϕ, dx2 ❱➻ M ❧➔ t♦→♥ tû ✤è✐ ①ù♥❣ ♥➯♥ d2 u M ϕ(x), + ϕ(x), N u = ϕ, dx ▲➜② t➼❝❤ ♣❤➙♥ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ t❛ ✤÷đ❝ x d2 u M ϕ(η), dη + dη x ϕ(η), N u dη = 0 ❙û ❞ö♥❣ ❝æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ ✈➔ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ x du dx (0) = v0 t❛ ❝â x d2 u du x du M ϕ(η), dη = M ϕ(η), d(M ϕ(η))dη − dη dη η=0 dη x du du dϕ du = M ϕ(x), (x) − M ϕ(0), (0) − M , dη dx dx dη dη x du dϕ du = M ϕ(x), (x) − M ϕ(0), v0 − M , dη dx dη dη ✸✸ ❚❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ t❛ ✤÷đ❝ du M ϕ(x), (x) = M ϕ(0), v0 + dx x M dϕ du , − ϕ, N u dη dη dη ✭✷✳✹✺✮ ✣➦t E(x) = M du du , + u(x), N u(x) dx dx ❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ♥➠♥❣ ❧÷đ♥❣ t❛ ❝â E(x) = M du du , + u, N u dx dx ≤ E(0) ✭✷✳✹✻✮ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ ♥❣❤✐➺♠ ❝ê ✤✐➸♥ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✹✺✮ ❝ơ♥❣ s➩ t❤ä❛ ♠➣♥ ✭✷✳✹✻✮ ✈ỵ✐ ❞➜✉ ❜➡♥❣✳ ❈❤ó♥❣ t❛ s➩ sû ởt t tự ỗ ổrt ✤➸ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❞✉② ♥❤➜t ✈➔ t➼♥❤ ♣❤ö t❤✉ë❝ ❧✐➯♥ tö❝ ✈➔♦ ❞ú ❦✐➺♥ ❜❛♥ ✤➛✉ ❝õ❛ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✤❛♥❣ ①➨t✳ ✣➸ ✤↕t ✤÷đ❝ ♠ư❝ ✤➼❝❤ ✤â✱ t❛ ①➨t ❤➔♠ G(x) = u, M u + β(x + x0 )2 , ✭✷✳✹✼✮ tr♦♥❣ ✤â β ✈➔ x0 ❧➔ ❝→❝ ❤➡♥❣ sè ❦❤ỉ♥❣ ➙♠ ✤÷đ❝ ❝❤å♥ ♣❤ò ❤đ♣✳ ✣↕♦ ❤➔♠ ❤❛✐ ✈➳ ❝õ❛ ✭✷✳✹✼✮ t❛ ♥❤➟♥ ✤÷đ❝ G (x) = M u, ux + 2β(x + x0 ) ⑩♣ ❞ö♥❣ ✭✷✳✹✺✮✱ t❤❛② ϕ(x) = u(x) ✈➔ sû ❞ư♥❣ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ u(0) = u0 t❛ ✤÷đ❝ G (x) = M u, ux + 2β(x + x0 ) x = M u0 , v0 + M du du , − u, N u dη + 2β(x + x0 ) dη dη ✭✷✳✹✽✮ ❚✐➳♣ tư❝ ✤↕♦ ❤➔♠ ❤❛✐ ✈➳ ❝õ❛ ✭✷✳✹✽✮ t❛ ♥❤➟♥ ✤÷đ❝ G (x) = M du du , − u, N u dx dx ✸✹ + 2β ✭✷✳✹✾✮ ❇➡♥❣ ❝→❝❤ sû ❞ư♥❣ ✭✷✳✹✻✮ ❦❤✐ u(x) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✹✺✮ ✭tù❝ ❧➔ ❦❤✐ ✭✷✳✹✻✮ ①↔② r❛ ❞➜✉ ❜➡♥❣✮ t❛ õ t t ữợ du du , − u, N u + 2β dx dx du du du du =4 M , + 4β − 2β + M , + u, N u dx dx dx dx G (x) = M ✭✷✳✺✵✮ = M ux , ux + 4β − (2β + 4E(0)) ❉♦ ✤â✱ t❛ ❝â GG − (G )2 = u, M u + β(x + x0 )2 M ux , ux + 4β − (2β + 4E(0)) − M u, ux + 2β(x + x0 ) = M u, u + β(x + x0 )2 M ux , ux + β − M u, ux + β(x + x0 ) ✭✷✳✺✶✮ − 2β + 4E(0) G(x) ❱➻ M ①→❝ ✤à♥❤ ❞÷ì♥❣ ✈➔ t❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ❙❝❤✇❛r③✱ t❛ ❝â M u, M +β(x+x0 )2 M ux , ux +β −4 M u, ux +β(x+x0 ) ❚❤➟t ✈➟②✱ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ tr t ỵ u, v M t tỷ ✤è✐ ①ù♥❣✱ ①→❝ ✤à♥❤ ❞÷ì♥❣ ♥➯♥ , ∗ ∗ ≥ ✭✷✳✺✷✮ := M u, v ✳ ❱➻ ❧➔ ♠ët t ổ ữợ tr ổ rt H →♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❙❝❤✇❛r③✱ t❛ ❝❤å♥ u ˆ ❧➔ ✈❡❝tì trü❝ ❣✐❛♦ ✈ỵ✐ ❤➺ ✈❡❝tì {u, ux } ✈➔ u ˆ = 1✳ ✣➦t ❝→❝ ✈❡❝tì u (x) = u(x) + v (x) = ux (x) + β(x + x0 )ˆ u; β uˆ ❱➻ ❤➺ ✈❡❝tì u ˆ(x) trü❝ ❣✐❛♦ ✈ỵ✐ ❤➺ {ut (x), uˆ(x)} ✈➔ uˆ = ♥➯♥ t❛ ❝â u ,u ∗ = u, u v ,v ∗ = ux , ux u ,v ∗ = u, ux ∗ + β(x + x0 )2 = M u, u + β(x + x0 )2 ; ∗ ∗ + β = M ux , ux + β; + β(x + x0 ) = M u, ux + β(x + x0 ) ✸✺ ▼➔ t❛ ❧↕✐ ❝â u ,u ∗ · v ,v ∗ ≥ | u ,v ∗| ❉♦ ✤â✱ t❛ ❝â ✭✷✳✺✷✮✳ ❱➟②✱ s✉② r❛ GG − (G )2 ≥ − 2β + 4E(0) G(x) ✭✷✳✺✸✮ ❚ø ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✺✸✮ ❝❤ó♥❣ t❛ ❝â t❤➸ ❝❤➾ r❛ ❝→❝ ❦➳t q✉↔ ✈➲ sü ❞✉② ♥❤➜t✱ ❞→♥❣ ✤✐➺✉ ✈➔ sü ♣❤ö t❤✉ë❝ ❧✐➯♥ tö❝ ❝õ❛ ♥❣❤✐➺♠✳ ✶✳ ❙ü ❞✉② ♥❤➜t ♥❣❤✐➺♠✳ Ð ✤➙②✱ ❝❤ó♥❣ t❛ ❝➛♥ ❝❤➾ r❛ r➡♥❣ tø u0 = v0 = ❦➨♦ t❤❡♦ u(x) ≡ 0✳ ❚❤➟t ✈➟②✱ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② t❛ s➩ ❝❤å♥ β = 0✳ ❚ø E(0) = t❛ ❝â ✭✷✳✺✹✮ GG − (G )2 ≥ 0, ✈➔ tø ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✸✮✱ t❛ s✉② r❛ G(x) ≤ G(0)1−x/X G(X)x/X ✭✷✳✺✺✮ ❱➻ G(0) = ♥➯♥ G(x) ♣❤↔✐ ❜➡♥❣ ✈ỵ✐ ♠å✐ x ∈ [0, X]✳ ❱➻ M ❧➔ t♦→♥ tû ①→❝ ✤à♥❤ ❞÷ì♥❣ ♥➯♥ t❛ s✉② r❛ u ≡ tr♦♥❣ ✤♦↕♥ [0, X]✳ ✷✳ ❉→♥❣ ✤✐➺✉ ❝õ❛ ♥❣❤✐➺♠✳ ●✐↔ sû X = ∞ ✈➔ ❝❤ó♥❣ t❛ ✤✐ ♥❣❤✐➯♥ ❝ù✉ ❞→♥❣ ✤✐➺✉ ❝õ❛ ♥❣❤✐➺♠ ❦❤✐ x → ∞✳ ❑❤✐ ✤â t❛ s➩ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣ E(0) < ✈➔ E(0) > 0✱ tr÷í♥❣ ❤đ♣ E(0) = ❦❤→ ✤➦❝ ❜✐➺t✱ ❜↕♥ ✤å❝ ❝â t❤➸ t❤❛♠ ❦❤↔♦ tr♦♥❣ ♠ët ❜➔✐ ❜→♦ ❝õ❛ ❘✳❏✳ ❑♥♦♣s ✈➔ ▲✳❊✳P❛②♥❡ ✤÷đ❝ ✤➠♥❣ tr➯♥ t↕♣ ❝❤➼ ❆r❝❤✳ ❘❛t✐♦♥❛❧ ▼❡❝❤✳ ❆♥❛❧✳ ♥➠♠ ✶✾✼✶✳ ❚r♦♥❣ ❦❤✉ỉ♥ ❦❤ê ❝õ❛ ❧✉➟♥ ✈➠♥✱ ❝❤ó♥❣ tỉ✐ ❦❤ỉ♥❣ ①➨t ✤➳♥ tr÷í♥❣ ❤đ♣ ♥➔②✳ ✭✐✮ E(0) < 0✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② t❛ ❝❤å♥ β = −2E(0) ✈➔ t❛ ❝ô♥❣ ❞➝♥ ✤➳♥ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✺✹✮✳ ❉♦ ✤â✱ tø ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✷✮ t❛ ❝ô♥❣ ❝â ✤→♥❤ ❣✐→ ữợ G(x) ữ s G(x) G(0) ❡①♣ G (0) x , G(0) ✭✷✳✺✻✮ ❤❛② u, M u + β(x + x0 ) ≥ u0 , M u0 + βx20 ✸✻ ❡①♣ 2x M u0 , v0 + βx0 u0 , M u0 + βx20 ✭✷✳✺✼✮ ❚❛ t❤➜② r➡♥❣✱ ♥➳✉ M u0 , v0 ❜à ❝❤➦♥ t❤➻ t❛ ❧✉ỉ♥ ❝â t❤➸ ❝❤å♥ ✤÷đ❝ x0 ✤õ ợ số ụ tr số ữỡ õ t õ t q s ỵ ởt t ý t tỗ t ✈ỵ✐ ♠å✐ x ✈➔ ♥➳✉ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ t❤ä❛ ♠➣♥ E(0) < t❤➻ ♥❣❤✐➺♠ ✤â ♣❤↔✐ t➠♥❣ t❤❡♦ ❤➔♠ sè ♠ơ ✭t❤❡♦ ❝❤✉➞♥✮ ❦❤✐ x ❞➛♥ tỵ✐ ∞✳ ❇➯♥ ❝↕♥❤ ✤â✱ ❦❤✐ ❝â ✭✷✳✺✺✮ ✈ỵ✐ β = −2E(0) t❤➻ ♥❣❤✐➺♠ u(x) s➩ t❤✉ë❝ ❧ỵ♣ ❝→❝ ❤➔♠ ♠➔ ❝❤✉➞♥ ❝õ❛ ♥â ❜à ❝❤➦♥ t↕✐ x = x ˆ✳ ❑❤✐ ✤â✱ tr♦♥❣ ❧ỵ♣ ❤➔♠ ♥➔② t❤➻ ♥❣❤✐➺♠ u(x) ♣❤ư t❤✉ë❝ tử oăr t ỳ t♦→♥ ✈ỵ✐ ≤ x ≤ x ˆ✳ ✭✐✐✮ E(0) > 0✳ ❚ø ✭✷✳✺✸✮✱ t❛ s✉② r❛ GG − (G )2 ≥ − [2β + 4E(0)]G2 β(x + x0 )2 ✭✷✳✺✽✮ ❈❤♦ β r➜t ❧ỵ♥✱ ✤➦t ε = 4E(0)/β t❤➻ t❛ ❝â t❤➸ ✈✐➳t ❧↕✐ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ữợ GG (G )2 (2 + )(x + x0 )−2 G2 , ✭✷✳✺✾✮ ❤❛② G(x) (x + x0 )2+ε ❧♦❣ ✭✷✳✻✵✮ ≤ ❑❤✐ ✤â✱ t❛ t❤✉ ✤÷đ❝ ❤❛✐ ❦➳t q✉↔ s❛✉ G(x)(x + x0 ) −(2+ε) ≤ −(2+ε) G(0)x0 −(2+ε) G(x)(x + x0 )−(2+ε) ≥ G(0)x0 1−x/x∗ ∗ ∗ G(x )(x + x0 ) ❡①♣ x x/x∗ −(2+ε) G (0) (2 + ε) − G(0) x0 , , ✭✷✳✻✶✮ ✭✷✳✻✷✮ ð ✤➙②✱ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✻✶✮ ❝❤➾ ✤ó♥❣ ❦❤✐ ≤ x ≤ x∗ ✳ ❚ø ✭✷✳✻✶✮ t❛ t❤➜② r➡♥❣ ♥➳✉ lim ❧♦❣ x→∞ G(x) (x∗ )−1 = 0, 2+ε (x + x0 ) ✭✷✳✻✸✮ t❤➻ G(x) ≤ G(0) (x + x0 )2+ε x2+ε ❍❛② ♥â✐ ❝→❝❤ ❦❤→❝✱ t❛ ❝â ❦➳t q✉↔ s❛✉ ✸✼ ỵ ởt t ý t tỗ t ợ x ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ t❤ä❛ ♠➣♥ E(0) > t❤➻ ❤♦➦❝ ❧➔ ♥❣❤✐➺♠ ♣❤↔✐ ♣❤→t tr✐➸♥ t❤❡♦ ❤➔♠ sè ♠ô ✭t❤❡♦ ❝❤✉➞♥✮ ❦❤✐ x ❞➛♥ tỵ✐ ∞ ❤♦➦❝ ❧➔ u t➠♥❣ ❦❤ỉ♥❣ ♥❤❛♥❤ ❤ì♥ O(t1+ε ) ✈ỵ✐ ε ♥❤ä tò② þ✳ ❈ơ♥❣ ♥❤÷ tr÷í♥❣ ❤đ♣ ✭✐✮✱ t❛ ♥❤➟♥ t❤➜② r➡♥❣ tø ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✻✶✮ t❤➻ tr♦♥❣ ❧ỵ♣ ♥❣❤✐➺♠ ♠➔ ❤➔♠ G(x) ❜à ❝❤➦♥ t❤➻ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ❜➔✐ t♦→♥ tở tử oăr t ỳ ❦✐➺♥ ❝õ❛ ❜➔✐ t♦→♥ ✈ỵ✐ ≤ x ≤ x∗ ✳ ✸✽ ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❧↕✐ ♣❤÷ì♥❣ ỗ ổrt ởt ữỡ ữủ sỷ ❞ö♥❣ ♥❤✐➲✉ tr♦♥❣ ✈✐➺❝ ê♥ ✤à♥❤ ❤â❛ ❝→❝ ❜➔✐ t♦→♥ ✤➦t ❦❤ỉ♥❣ ❝❤➾♥❤ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✳ ❈ư t❤➸ ✲ ❚r♦♥❣ ❝❤÷ì♥❣ ✶✱ t→❝ ❣✐↔ tr➻♥❤ ❜➔② ❧↕✐ ữỡ ỗ ổrt r ữỡ ✷✱ t→❝ ❣✐↔ tr➻♥❤ ❜➔② ❧↕✐ ❤❛✐ ✈➼ ❞ö sû ữỡ ỗ ổrt õ t♦→♥✱ ✤â ❧➔ ❜➔✐ t♦→♥ ❈❛✉❝❤② ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ ♥❣÷đ❝ t❤í✐ ❣✐❛♥ ✈➔ ❜➔✐ t♦→♥ ❈❛✉❝❤② ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡✳ ✸✾ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❆✳ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ ❚❤ø❛ ❍đ♣ ✭✷✵✵✻✮✱ ●✐→♦ tr➻♥❤ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❬✷❪ ❚r➛♥ ❱ô ❚❤✐➺✉✱ ◆❣✉②➵♥ ❚❤à ❚❤✉ ❚❤õ② ✭✷✵✶✶✮✱ ●✐→♦ tr➻♥❤ tè✐ ÷✉ ♣❤✐ t✉②➳♥✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❇✳ ❚✐➳♥❣ ❆♥❤ ❬✸❪ ❑✳ ❆✳ ❆♠❡s ❛♥❞ ❇✳ ❙tr❛✉❣❤❛♥ ✭✶✾✾✼✮✱ ◆♦♥✕st❛♥❞❛r❞ ❛♥❞ ■♠♣r♦♣❡r❧② P♦s❡❞ Pr♦❜❧❡♠s✱ ❆❝❛❞❡♠✐❝ Pr❡ss✳ ❬✹❪ ❏✳ ◆✳ ❋❧❛✈✐♥ ❛♥❞ ❙✳ ❘✐♦♥❡r♦ ✭✶✾✾✻✮✱ ◗✉❛❧✐t❛t✐✈❡ ❊st✐♠❛t❡s ❢♦r P❛rt✐❛❧ ❉✐❢✲ ❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✿ ❆♥ ■♥tr♦❞✉❝t✐♦♥✱ ❈❘❈ Pr❡ss✳ ❬✺❪ ▲✳ ❊✳ P❛②♥❡ ✭✶✾✼✺✮✱ ■♠♣r♦♣❡r❧② P♦s❡❞ Pr♦❜❧❡♠s ✐♥ P❛rt✐❛❧ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛✲ t✐♦♥s✱ ❙♦❝✐❡t② ♦❢ ■♥❞✉str✐❛❧ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ P❤✐❧❛❞❡♥♣❤✐❛✱ P❡♥♥✲ s②❧✈❛♥✐❛✳ ✹✵ ... NGUYÊN TRƢỜNG ĐẠI HỌC KHOA HỌC  - PHẠM LỆ QUYÊN VỀ PHƢƠNG PHÁP LỒI LƠGARIT VÀ MỘT VÀI ỨNG DỤNG Chun ngành: Tốn Ứng Dụng Mã số: 46 01 12 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN