LUẬN văn THẠC sĩ HAY về toán tử chiếu metric lên tập lồi đóng và ứng dụng vào bài toán bất đẳng thức biến phân

ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - VÀNG VĂN HÀ VỀ TOÁN TỬ CHIẾU METRIC LÊN TẬP LỒI ĐÓNG VÀ ỨNG DỤNG VÀO BÀI TOÁN BẤT ĐẲNG THỨC BIẾN PHÂN LUẬN VĂN THẠC SĨ TOÁN HỌC THÁI NGUYÊN - 2020 LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - VÀNG VĂN HÀ VỀ TOÁN TỬ CHIẾU METRIC LÊN TẬP LỒI ĐĨNG VÀ ỨNG DỤNG VÀO BÀI TỐN BẤT ĐẲNG THỨC BIẾN PHÂN Chuyên ngành: Toán ứng dụng Mã số : 46 01 12 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC GS.TSKH Lê Dũng Mưu THÁI NGUYÊN - 2020 LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ử ỵ ì♥ ▲í✐ ♥â✐ ✤➛✉ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ỗ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❚♦→♥ tû ❝❤✐➳✉ ❦❤♦↔♥❣ ❝→❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ❈❤÷ì♥❣ ✷✳ Ù♥❣ ❞ö♥❣ ✈➔♦ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✷✷ ✷✳✶ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ▼ët t❤✉➟t t♦→♥ ❝❤✐➳✉ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ♣❛r❛✲✤ì♥ ✤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✷✷ ✷✺ ✸✼ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ỵ R t sè t❤ü❝ Rn ❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ ∅ t➟♣ ré♥❣ ∀x ợ x tỗ t n x x x tỡ x x, y t ổ ữợ ✈➨❝✲tì x ❝❤✉➞♥ ❝õ❛ ✈➨❝✲tì x ✈➔ y x V IP (F ; C) ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ S(F ; C) t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ V IP (F ; C) LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✐✐ ▲í✐ ❝↔♠ ì♥ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ữợ sỹ ữợ ụ ▼÷✉✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s tợ ữớ tớ t t ữợ ú ù t tr sốt q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tợ ổ tr trữớ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t t rữớ ỗ tớ tổ ụ ỷ ỡ tợ ỗ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ♥❤➜t ❝❤♦ tỉ✐ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ ✈➔ tr♦♥❣ q✉→ tr➻♥❤ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✷✵✳ ❚→❝ ❣✐↔ ❱➔♥❣ ❱➠♥ ❍➔ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✶ ▲í✐ ♥â✐ ✤➛✉ ❚r♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ♣❤ê t❤ỉ♥❣✱ ❝❤ó♥❣ t❛ ✤➣ ❧➔♠ q✉❡♥ ✈ỵ✐ ♣❤➨♣ ❝❤✐➳✉ ✈✉ỉ♥❣ ❣â❝ ①✉è♥❣ ♠ët ♠➦t ♣❤➥♥❣ tr♦♥❣ ❦❤✐ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ❤➻♥❤ ❤å❝ ✈➔ ❧÷đ♥❣ ❣✐→❝✳ ❑❤→✐ ♥✐➺♠ ♥➔② ✤➣ ✤÷đ❝ ♠ð rë♥❣ ❧➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ♥❤✐➲✉ ❝❤✐➲✉✱ t❤➟♠ ❝❤➼ ✈ỉ ❤↕♥ ❝❤✐➲✉ ❝ị♥❣ ✈ỵ✐ t t ởt t ỗ õ ✈ỵ✐ ♠ët ❦❤♦↔♥❣ ❝→❝❤ ✭♠❡tr✐❝✮ ❦❤ỉ♥❣ ♥❤➜t t❤✐➳t ❧➔ ❦❤♦↔♥❣ ❝→❝❤ ❒✲❝ì✲❧✐t✳ ⑩♥❤ ①↕ ❝❤✉②➸♥ ♠ët ✤✐➸♠ ❜➜t ❦ý ❝❤♦ trữợ tr ổ ởt tr ởt t trữợ ợ ọ t ữủ t♦→♥ tû ❝❤✐➳✉ ❧➯♥ t➟♣ ✤â✳ ◆❣÷í✐ t❛ ✤➣ ❝❤➾ r❛ r➡♥❣✱ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✱ t♦→♥ tû ❝❤✐➳✉ ởt t ỗ õ ữủ t tỷ t ỗ õ õ ✤➦❝ tr÷♥❣ t❤ó ✈à✱ ❞♦ ✤â ♥â ❝â ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ♥❤✐➲✉ ✈➜♥ ✤➲ ❝õ❛ t♦→♥ ❤å❝ ✈➔ tỹ t ữ tr ỵ tt tố ữ ❤â❛✱ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✱ ❝➙♥ ❜➡♥❣ ✈➔ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❦❤→❝✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❜↔♥ ❧✉➟♥ ✈➠♥ ❜❛♦ ỗ tự ỡ t t ỗ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❒✲❝ì✲❧✐t Rn ✱ ❝→❝ ❦➳t q✉↔ ✈➲ t tỷ t ỗ õ t✐➳♣ t❤❡♦ ❧✐➯♥ q✉❛♥ ✤➳♥ ✈✐➺❝ →♣ ❞ö♥❣ t♦→♥ tû ❝❤✐➳✉ ✈➔♦ ✈✐➺❝ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ♣❛r❛✲✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ Rn ✳ ◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✷ ❝ù✉ tr♦♥❣ ❜↔♥ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② t ữỡ ợ t ữỡ ởt số ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❈❤÷ì♥❣ ✷✿ Ù♥❣ ❞ư♥❣ ✈➔♦ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❝→❝ ❝❤÷ì♥❣ ♥❤÷ s❛✉✿ ❚r♦♥❣ ❝❤÷ì♥❣ ✶✱ tỉ✐ tr➻♥❤ ❜➔② t ỗ ởt số t t ỡ t ỗ ỗ t tr ỵ t t ỗ ởt ữỡ tr➻♥❤ ❜➔② ✈➲ ✤à♥❤ ♥❣❤➽❛ t♦→♥ tû ❝❤✐➳✉✱ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ t♦→♥ tû ❝❤✐➳✉✳ ❈❤÷ì♥❣ ✷ ❝õ❛ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ù♥❣ ❞ö♥❣ ❝õ❛ t♦→♥ tû tr ởt t ỗ õ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✳ ❇➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❧➔ ♠ët ❧ỵ♣ ❜➔✐ t♦→♥ q✉❛♥ trå♥❣ ❝õ❛ ●✐↔✐ t➼❝❤ ù♥❣ ❞ư♥❣✳ ❇➔✐ t♦→♥ ♥➔② ❧➔ ♠ët ❧ỵ♣ t tờ qt t q ỗ ❤ì♥ ♥ú❛ ♥❤✐➲✉ ❜➔✐ t♦→♥ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ r õ t ổ t ữợ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✳ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✸ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ tr➻♥❤ ❜➔② ởt số t t ỡ ỵ q t ỗ ỗ ởt ữỡ tr ự sỹ tỗ t t t ởt t ỗ õ s→t ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ t♦→♥ tû ❝❤✐➳✉✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ ❝❤õ ②➳✉ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶✱ ✸❪✳ ✶✳✶ ❚➟♣ ỗ ỗ rữợ t ú tổ ợ t t ỗ ởt số t t ❝➛♥ t❤✐➳t✳ ◆❤➢❝ ❧↕✐ r➡♥❣✱ ♠ët Rn ✤÷í♥❣ t❤➥♥❣ ❧➔ t➟♣ ❤đ♣ t➜t ❝↔ ❝→❝ ✈➨❝✲tì ♥è✐ ❤❛✐ ✤✐➸♠ ✭❤❛✐ ✈➨❝✲tì✮ x ∈ Rn a, b tr♦♥❣ ❝â ❞↕♥❣ {x ∈ Rn |x = αa + βb, α, β ∈ Rn , α + β = 1} ✣♦↕♥ t❤➥♥❣ ♥è✐ ❤❛✐ ✤✐➸♠ a ✈➔ b tr♦♥❣ Rn ❧➔ t➟♣ ❤ñ♣ ❝→❝ ✈➨❝✲tì x ❝â ❞↕♥❣ {x ∈ Rn |x = αa + βb, α ≥ 0, β ≥ 0, α + β = 1} ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ▼ët t➟♣ C ⊆ Rn ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♠å✐ ✤♦↕♥ t❤➥♥❣ ✤✐ q✉❛ ❤❛✐ ✤✐➸♠ ❜➜t ❦ý ❝õ❛ ♥â✳ ❚ù❝ ❧➔ t➟♣ ỗ C ự C ỗ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✹ ∀x, y ∈ C, ∀λ ∈ [0, 1] ⇒ λx + (1 − λ)y ∈ C ❱➼ ❞ö ✶✳✶✳ ❛✮ ❚➟♣ t Rn t ỗ AB Rn ởt t ỗ trỏ ỗ ởt t ỗ ✈➻ ✤♦↕♥ t❤➥♥❣ ♥è✐ ❤❛✐ ✤✐➸♠ X, Y tr♦♥❣ ❤➻♥❤ trá♥ ♥➡♠ trå♥ ✈➭♥ tr♦♥❣ ❤➻♥❤ trá♥✳ ❍➻♥❤ ✶✳✶✿ ❚➟♣ ỗ ữợ t ổ ỗ ✈➻ ❝→❝ ✤÷í♥❣ ♥➨t ✤ùt ❝❤ù❛ ♥❤✐➲✉ ✤✐➸♠ ❦❤ỉ♥❣ ♥➡♠ tr t õ ổ ỗ õ x tờ ủ ỗ ❝→❝ ✤✐➸♠ x1, , xk ♥➳✉ k k j λj x , λj > ∀j = 1, , k, x= j=1 λj = j=1 LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ủ C ỗ õ ự tờ ủ ỗ õ ự C ỗ ❝❤➾ ❦❤✐ k k ∀k ∈ N, ∀λ1 , , λk > : k λj = 1, ∀x , , x ∈ C ⇒ j=1 λj xj ∈ C j=1 ❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ ✤õ ❧➔ ❤✐➸♥ ♥❤✐➯♥ tø ✤à♥❤ ♥❣❤➽❛✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❜➡♥❣ q✉② ♥↕♣ t❤❡♦ sè ✤✐➸♠✳ ❱ỵ✐ k = 2✱ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ s✉② r tứ t ỗ tờ ủ ỗ sỷ ú ợ sỷ x k−1 ✤✐➸♠✳ ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ✈ỵ✐ ❧➔ tê ủ ỗ k k x1 , , x k ∈ C ✳ k k j λj x , λj > ∀j = 1, , k, x= ❚ù❝ ❧➔ j=1 λj = j=1 ✣➦t k−1 λj ξ= j=1 ❑❤✐ ✤â 0 x∗ = x∗ ∈ (1; 2) ✭✷✳✷✮ t❤➻ ✭✷✳✷✮ ❝❤➾ t❤➻ ✭✷✳✷✮ ❝❤➾ t❤ä❛ ♠➣♥ ✈ỵ✐ ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t✳ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❝â ❝→❝ tr÷í♥❣ ❤đ♣ r✐➯♥❣ q✉❛♥ trồ t ỹ t ỗ tr t ỗ t ũ t C ởt t ỗ õ rộ tr R n ởt ỗ F ❧➔ ♠ët →♥❤ ①↕ ✤✐ tø t➟♣ C ✈➔♦ Rn ✈➔ F (x) = f (x)✳ ❑❤✐ ✤â ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ V IP (F ; C) tữỡ ữỡ ợ t ỹ tr P x ∈ C t❤ä❛ ♠➣♥ f (x∗) ≤ f (y) ∀y ∈ C ✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû x∗ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ V IP (F ; C)✱ tù❝ ❧➔ f :C→R F (x∗ ), y − x∗ ≥ ∀y ∈ C LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com f ỗ ♥➯♥ t❛ ❝â f (y) − f (x∗ ) ≥ ▼➔ F (x) = f (x) ♥➯♥ f (x∗ ), y − x∗ , ∀y ∈ C f (x∗ ) ≤ f (y) ∀y ∈ C ✱ ✤✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔ x∗ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭❖P✮✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû x∗ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭❖P✮✱ t tố ữ ỗ t õ 0∈ ❚ø ✤â✱ t❛ s✉② r❛ − f (x∗ ) + NC (x∗ ) f (x∗ ) ∈ NC (x∗ ) ❤❛② −F (x∗ ) ∈ NC (x∗ )✳ ❚ù❝ ❧➔ −F (x∗ ), y − x∗ ≤ 0, ∀y ∈ C ⇔ F (x∗ ), y − x∗ ≥ 0, ∀y ∈ C ❱➟② x∗ ❑❤✐ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ C V IP (F ; C)✳ ❧➔ ởt õ ỗ tr ổ Rn t t V IP (F ; C) trð t❤➔♥❤ ❜➔✐ t♦→♥ ❜ò✿ ✭❈P✮ ❚➻♠ tr♦♥❣ ✤â x∗ ∈ C ✱ F (x∗ ) ∈ C s❛♦ ❝❤♦ C := {y ∈ C : x, y ≥ 0, ∀x ∈ C} F (x∗ ), x∗ = ❧➔ ♥â♥ ✤è✐ ♥❣➝✉ ❝õ❛ C✳ ❚❛ ❝â ♠➺♥❤ ✤➲ s❛✉✿ ▼➺♥❤ ✤➲ ✷✳✷✳ ◆➳✉ C ởt õ ỗ t tr ổ R n t t ũ P tữỡ ữỡ ợ t ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ V IP (F ; C)✱ ♥❣❤➽❛ ❧➔ t➟♣ ♥❣❤✐➺♠ ❤❛✐ ❜➔✐ t♦→♥ ♥➔② trò♥❣ ♥❤❛✉✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû x∗ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ V IP (F ; C)✱ tù❝ ❧➔ F (x∗ ), y − x∗ > ✭✷✳✸✮ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✷✺ C õ ỗ x C y + x∗ ∈ C ✱ ∀y ∈ C ✳ ❚❤❛② y = y + x∗ ✈➔♦ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✸✮ t❛ ✤÷đ❝ F (x∗ ), y + x∗ − x∗ ≥ 0, ∀y ∈ C ⇔ F (x∗ ), y ≥ 0, ∀y ∈ C F (x∗ ) t❤✉ë❝ ♥â♥ ✤è✐ ♥❣➝✉ C ✳ y = x∗ ✈➔♦ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✸✮ ❙✉② r❛ ❚❤❛② t❛ ✤÷đ❝ F (x∗ ), x∗ ≤ 0, ∀y ∈ C ❙✉② r❛ F (x∗ ), x∗ = ❤❛② x∗ ∈ C ✱ F (x∗ ) ∈ C ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❜ị ♣❤✐ t✉②➳♥ ✭❈P✮✳ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ x∗ ∈ C ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❜ò ✭❈P✮ t❤➻ F (x∗ ), x∗ = 0, F (x∗ ) ∈ C ❱➻ F (x∗ ) ∈ C ♥➯♥ F (x∗ ), y ≥ 0, ∀y ∈ C ✳ ❚❛ ❝â F (x∗ ), y − x∗ ≥ 0, ∀y ∈ C ❤❛② x∗ ∈ C ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ V IP (F ; C)✳ ✷✳✷ ▼ët t❤✉➟t t♦→♥ ❝❤✐➳✉ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ♣❛r❛✲✤ì♥ ✤✐➺✉ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳ C R ❈❤♦ F : C → Rn ởt t ỗ tr ổ ❣✐❛♥ ❑❤✐ ✤â✱ →♥❤ ①↕ F n ✈➔ ❧➔ ✣ì♥ tr C tỗ t ởt số γ > s❛♦ ❝❤♦ F (x) − F (y), x − y ≥ γ x − y , ∀x, y ∈ C; ✭✐✐✮ ✣ì♥ ✤✐➺✉ tr➯♥ C ♥➳✉ F (x) − F (y), x − y ≥ 0, ∀x, y ∈ C; LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✷✻ ✭✐✐✐✮ ●✐↔ ✤ì♥ ✤✐➺✉ tr➯♥ C ♥➳✉ F (x), y − x ≥ ⇒ F (y), y − x ≥ 0, ∀x, y ∈ C ✭✐✈✮ P❛r❛✲✤ì♥ ✤✐➺✉ tr➯♥ C ♥➳✉ x∗ ∈ S(F, C), x ∈ C, F (x), x∗ −x = 0, F (x∗ ), x−x∗ = ⇒ x ∈ S(F, C) ◆❤➟♥ ①➨t ✷✳✶✳ ❱➼ ❞ö ✷✳✷✳ ❚❛ ❝â ✭✐✮ F ❛✮ ❈❤♦ →♥❤ ①↕ ⇒ ✭✐✐✮✱ ✭✐✐✮ ⇒ ✤ì♥ trà ①→❝ ✤à♥❤ tr➯♥ ✭✐✐✐✮ ❧➔ ❤✐➸♥ ♥❤✐➯♥✳ R ♥❤÷ s❛✉✿ F (x) = 2x, ∀x ∈ R ✈ỵ✐ F (x) ❧➔ ỗ t r F ❧➔ ✤ì♥ ✤✐➺✉ tr➯♥ tr➟♥ ✈✉ỉ♥❣ ❝ï n × n✳ ❦❤æ♥❣ ❣✐❛♥ ❦❤✐ Q Q R✳ ❜✮ ❈❤♦ f ①→❝ ✤à♥❤ tr➯♥ F (x) = Qx✱ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛✱ t❛ t❤➜② F R✳ ❑❤✐ ✤â ❞➵ tr♦♥❣ ✤â Q ❧➔ ♠❛ ❧➔ ✤ì♥ ✤✐➺✉ tr➯♥ t♦➔♥ ❧➔ ♠❛ tr➟♥ ✈✉ỉ♥❣✱ ✤è✐ ①ù♥❣✱ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣✳ ◆➳✉ ❧➔ ✤è✐ ①ù♥❣✱ ữỡ t x2 ởt ỗ tr➯♥ C t❤➻ f F ✤ì♥ ✤✐➺✉ ♠↕♥❤✳ ❚ê♥❣ q✉→t ỡ ỡ tr C ú ỵ ổ ♠å✐ t♦→♥ tû ✤ì♥ ✤✐➺✉ ✤➲✉ ❧➔ ✤↕♦ ❤➔♠ ❝õ❛ ỗ õ s s ✤➸ sü ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tö ❝õ❛ t❤✉➟t t♦→♥✳ ❇ê ✤➲ ✷✳✶✳ ●✐↔ sû {νk } ✈➔ {δk } ❧➔ ❤❛✐ ❞➣② sè t❤ü❝ ❦❤æ♥❣ ➙♠ t❤ä❛ ♠➣♥ νk+1 ≤ νk + δk ✈ỵ✐ +∞ k=1 δk < +∞✳ ❑❤✐ ✤â ❞➣② {νk } ❤ë✐ tö✳ ❇ê ✤➲ ✷✳✷✳ ❬✹❪ ●✐↔ sû θ✱ β ✈➔ ξ ❧➔ ❝→❝ sè t❤ü❝ ❦❤æ♥❣ ➙♠ t❤ä❛ ♠➣♥ θ2 − βθ − ξ ≤ 0✱ ❦❤✐ ✤â ❬✹❪ βθ ≤ β + ξ ✭✷✳✹✮ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✷✼ ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ❤➔♠ sè ❜➟❝ ❤❛✐ s(θ) = θ2 − βθ − ξ ✱ ❦❤✐ ✤â s(θ) ≤ s✉② r❛ θ≤ ✈➻ β+ β + 4ξ , θ > β ◆❤➙♥ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✈ỵ✐ ✈➔ →♣ ❞ư♥❣ t➼♥❤ ❝❤➜t a2 + b ab ≤ t❛ ✤÷đ❝ βθ ≤ 2−1 β + β ≤2 −1 β + 4ξ β + β + 4ξ β + 2 = 2−1 β + β + 2ξ = β + ξ ❙✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tö t❛ s➩ ❣✐↔ sû t➟♣ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✶✮ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ s❛✉ ❚➻♠ x∗ ∈ C s❛♦ ❝❤♦ F (y), x∗ − y ≤ 0, ∀y ∈ C ❚➟♣ t ữủ ỵ Sd (F ; C) ❚❤✉➟t t♦→♥ ✤÷đ❝ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉✳ ❈❤♦ t❤❛♠ sè ❞÷ì♥❣ ρ ✈➔ ❝→❝ ❞➣② sè t❤ü❝ {ρk } ✈➔ {βk } t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ρk > ρ, βk = +∞, ρk βk > 0, ∀k ∈ N, ✭✷✳✻✮ βk2 < +∞, ✭✷✳✼✮ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✷✽ ❱➼ ❞ư t❛ ❧➜② ρk = ✈ỵ✐ ♠å✐ k ✱ k = Pữỡ ữợ ữợ ữợ ✷✿ ♥❣❤➽❛ ❧➔ ❈❤å♥ x0 ∈ C ✳ ●✐↔ sû xk ∈ C ✳ βk γk ✈ỵ✐ m > 0✳ k = 0✳ ▲➜② g k = F (xk )✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ γk = max{ρk , g k } ✭✷✳✽✮ αk g k + xk+1 − xk , x − xk+1 ≥ 0, ∀x ∈ C, ✭✷✳✾✮ αk = ❚➼♥❤ ✣➦t m k+1 xk+1 ∈ C tr♦♥❣ ✤â s❛♦ ❝❤♦ xk+1 = PC (xk − αk g k )✳ ứ t t s ứ t ữợ k✱ ♥➳✉ gk = ❤❛② xk+1 = xk ✳ õ sỡ ỗ tt t s ỡ ỗ tt t LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com tt t ữợ ✤↕♦ ❤➔♠ s✐♥❤ r❛ ♠ët ❞➣② ❤ú✉ ❤↕♥ t❤➻ ✤✐➸♠ ❝✉è✐ ❝ò♥❣ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ V IP (F ; C)✳ ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ gk ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ =0 F (xk ), y − xk = t❤➻ ✈ỵ✐ ♠å✐ xk+1 = PC (xk − αk F (xk ))✱ t❤➻ tø ✈➟② xk ❧➔ V IP (F ; C)✳ ❇➙② ❣✐í ❣✐↔ sû t❤✉➟t t♦→♥ ❦➳t t❤ó❝ t ữợ xk = xk+1 y xk = xk+1 ✳ ◆➳✉ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ♣❤➨♣ ❝❤✐➳✉✱ t❛ ❝â xk+1 − (xk − αk F (xk )), y − xk ≤ 0, ∀y ∈ C ❉♦ xk = xk+1 ✈➔ αk > 0✱ ❜➜t ✤➥♥❣ t❤ù❝ ❝✉è✐ ❝ò♥❣ trð t❤➔♥❤ F (xk )), y − xk ≥ ∀y ∈ C, ♥❣❤➽❛ ❧➔ xk ❧➔ ♠ët ♥❣❤✐➺♠✳ ❚ø ❣✐í trð ✤✐✱ ❝❤ó♥❣ t❛ ❣✐↔ sû t❤✉➟t t♦→♥ s r ởt ổ ữủ ỵ {xk }✳ ❚❛ ❝â t➼♥❤ ❝❤➜t s❛✉✳ ❇ê ✤➲ ✷✳✸✳ ❱ỵ✐ ♠é✐ k✱ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ✤ó♥❣ ✭✐✮ ✭✐✐✮ αk g k ≤ βk ❀ βk xk+1 − xk ≤ βk2 ✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❚ø ✭✷✳✽✮ t❛ ❝â αk g ✭✐✐✮ ❇➡♥❣ ❝→❝❤ ❧➜② x = xk k βk g k = ≤ βk max{ρk , g k } ✭✷✳✶✵✮ tr♦♥❣ ✭✷✳✾✮ t❛ ✤÷đ❝ xk+1 − xk ≤ αk g k , xk − xk+1 LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✸✵ ≤ αk g k xk+1 − xk ✭✷✳✶✶✮ ≤ βk xk+1 − xk ❉♦ ✤â✱ tø ❇ê ✤➲ ✷✳✷ ✈ỵ✐ k∈N θ = xk+1 − xk , β = βk ξ = 0✱ ✈➔ ✈ỵ✐ ♠é✐ t❛ s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ●✐↔ t❤✐➳t t✐➳♣ t❤❡♦ s➩ ✤÷đ❝ sû ❞ö♥❣ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ s❛✉ ♥➔②✳ ❆✶✳ ❚➟♣ ♥❣❤✐➺♠ S(F ; C) ❦❤→❝ ré♥❣❀ ▼➺♥❤ ✤➲ ✷✳✹✳ ●✐↔ sû ❆✶ t❤ä❛ ♠➣♥✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ x ∗ ∈ S(F ; C) ✈ỵ✐ ♠é✐ k✱ t❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ xk+1 − x∗ ≤ xk − x∗ + 2αk F (xk ), x∗ − xk + δk , ✈➔ ✭✷✳✶✷✮ tr♦♥❣ ✤â δk = 2βk2✳ ❈❤ù♥❣ ♠✐♥❤✳ ❇➡♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ✤ì♥ ❣✐↔♥✱ t❛ ❝â xk+1 − x∗ = xk − x∗ − xk+1 − xk ≤ xk − x∗ + xk − xk+1 , x∗ − xk+1 ❑➳t ❤ñ♣ ✭✷✳✶✸✮ ✈➔ ✭✷✳✾✮ ✈ỵ✐ xk+1 − x∗ x = x∗ + xk − xk+1 , x∗ − xk+1 ✭✷✳✶✸✮ t❛ s✉② r❛ ≤ xk − x∗ + αk g k , x∗ − xk+1 = xk − x∗ + αk g k , x∗ − xk ✭✷✳✶✹✮ + αk g k , xk − xk+1 ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤② ✲ ❙❝❤✇❛r③ ✈➔ ❇ê ✤➲ ✷✳✸ ✭✐✮✱ s✉② r❛ xk+1 − x∗ ≤ xk − x∗ + 2αk g k , x∗ − xk + 2βk xk − xk+1 (2.15) ❚❤❡♦ ✭✷✳✶✺✮ ✈➔ ❇ê ✤➲ ✷✳✸ ✭✐✐✮✱ t❛ ❝â xk+1 − x∗ ≤ xk − x∗ + 2αk F (xk ), x∗ − xk + 2βk2 ✭✷✳✶✻✮ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✸✶ ❉♦ ✤â✱ ✈➻ αk > ♥➯♥ t❛ s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ s❛✉ ✤÷đ❝ ❞ị♥❣ ✤➸ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ❜à ❝❤➦♥ ❝õ❛ ❞➣② {xk } ✤÷đ❝ s✐♥❤ ❜ð✐ t❤✉➟t t♦→♥✳ ❆✷✳ S(F ; C) ⊆ Sd (F ; C) ú ỵ r F tử ✈➔ ❣✐↔ ✤ì♥ ✤✐➺✉✱ t❤➻ ❣✐↔ t❤✐➳t ❆✷ ✤ó♥❣✳ ✣à♥❤ ỵ sỷ tọ ❑❤✐ ✤â✱ ❧➔ ❞➣② ❤ë✐ tư ✈ỵ✐ ♠å✐ x∗ ∈ S(F ; C)❀ k ✭✐✐✮ {x } ❧➔ ❞➣② ❜à ❝❤➦♥✳ ✭✐✮ { xk − x∗ } ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ●✐↔ sû x∗ ∈ S(F ; C) ✈➔ k ∈ N✳ ❚❤❡♦ ❆✷ t❛ ❝â xk ≤ F (xk ), x∗ − ❝ị♥❣ ✈ỵ✐ ▼➺♥❤ ✤➲ ✷✳✹ s✉② r❛ xk+1 − x∗ tr♦♥❣ ✤â ≤ xk − x∗ + δk , ✭✷✳✶✼✮ δk = 2βk2 ❉♦ ✤â✱ t❤❡♦ ✭✷✳✼✮ ✈➔ ✭✷✳✽✮ t❛ ❝â +∞ δk < +∞ ✭✷✳✶✽✮ k=0 ❉♦ ✤â✱ tø ✭✷✳✶✼✮✱ ✭✷✳✶✽✮ ✈➔ ❇ê ✤➲ ✷✳✸ s✉② r❛ { xk − x∗ } ❧➔ ♠ët ❞➣② ❤ë✐ tö✳ ✭✐✐✮ ❙✉② r❛ tø ỵ sỷ F tử ❝→❝ ❣✐↔ t❤✐➳t ❆✶✱ ❆✷ ✤➲✉ t❤ä❛ ♠➣♥✳ ❑❤✐ ✤â✱ t❛ ❝â lim sup F (xk ), x∗ − xk = ∀x∗ ∈ S(F ; C) k→+∞ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✸✷ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû x∗ ∈ S(F ; C)✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✹ ✈➔ ❆✷ s✉② r❛ ≤ 2αk [− F (xk ), x∗ − xk ] ≤ xk − x∗ − xk+1 − x∗ + δk ✭✷✳✶✾✮ ❉♦ ✤â✱ m αk [− F (xk ), x∗ − xk ] 0≤2 k=0 m ≤ x −x ∗ − x m+1 −x ∗ δk + ✭✷✳✷✵✮ k=0 m ≤ x0 − x∗ δk + k=0 ❑❤✐ m → +∞ t❛ ❝â +∞ +∞ k ∗ k ∗ αk [− F (x ), x − x ] ≤ x − x 0≤2 + δk , ✭✷✳✷✶✮ k=0 k=0 ❦➳t ❤đ♣ ✈ỵ✐ ✭✷✳✶✽✮ s✉② r❛ +∞ αk [− F (xk ), x∗ − xk ] < +∞ 0≤ ✭✷✳✷✷✮ k=0 ▼➦t ❦❤→❝✱ t❛ ❝â { gk } ❧➔ ❞➣② ❜à ❝❤➦♥✳ ❑❤✐ ✤â✱ γk L = max{1, ρ−1 gk } ≤ k ρk ρ ∀k ∈ N ❉♦ ✤â αk = βk ρ βk ≥ γk L ρk ∀k ∈ N ✭✷✳✷✸✮ ❚ø ✭✷✳✷✷✮ ✈➔ ✭✷✳✷✸✮✱ t❛ ❝â +∞ k=0 βk [− F (xk ), x∗ − xk ] < +∞ ρk ✭✷✳✷✹✮ ❱➟②✱ tø ✭✷✳✷✹✮ ✈➔ ✭✷✳✼✮ s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✸✸ ✣➸ ❝â ✤÷đ❝ sü ❤ë✐ tư ❝õ❛ ❝↔ ❞➣② ❝❤ó♥❣ t❛ ✤÷❛ r❛ ❣✐↔ t❤✐➳t s❛✉✳ ❆✸✳ ●✐↔ sû x∗ ∈ S(F ; C) ✈➔ x¯ ∈ C ✳ ◆➳✉ F (¯ x), x∗ − x¯ = F (x∗ ), x¯ − x∗ = t❤➻ x¯ ∈ S(F ; C)❀ ✣à♥❤ ỵ sỷ tọ ♠➣♥✳ ❑❤✐ ✤â✱ ❞➣② {x } ❤ë✐ k tö ✤➳♥ ♠ët ♥❣❤✐➺♠ ❝õ❛ V IP (F ; C)✳ ❈❤ù♥❣ ♠✐♥❤✳ sỷ x S(F ; C) ỵ tỗ t ởt {xkj } {xk } s❛♦ ❝❤♦ lim sup F (xk ), x∗ − xk = lim F (xkj ), x∗ − xkj r ỵ t õ {xkj } {xkj } x¯ ∈ C ❧➔ ❞➣② ❜à ❝❤➦♥✳ ❱➟②✱ ❝â ❦❤ỉ♥❣ ♠➜t tê♥❣ q✉→t✱ ❝ư t❤➸ ❧➔ {xkj } lim xkj = x¯ F ✈➔ ♠ët ❞➣② s❛♦ ❝❤♦ j→+∞ ❉♦ ✭✷✳✷✺✮ j→+∞ k→+∞ ✭✷✳✷✻✮ ❧✐➯♥ tö❝✱ ♥➯♥ F (¯ x), x∗ − x¯ = lim F (xkj ), x∗ − xkj j→+∞ ✭✷✳✷✼✮ = ❚ø ❣✐↔ t❤✐➳t ❆✷ t❛ ❝â F (¯ x), x∗ − x¯ ≤ 0✱ ❞â ✤â t❛ ❝â F (¯ x), x∗ − x¯ = ❉♦ ✤â✱ t❛ s✉② r❛ { xk − x¯ } x¯ ∈ S(F ; C)✳ ✭✷✳✷✽✮ ỵ ỳ t ữủ ❤ë✐ tư✱ ❦➳t ❤đ♣ ✈ỵ✐ ✭✷✳✷✻✮ s✉② r❛ lim xk = x¯, k→+∞ x¯ ∈ S(F ; C) LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✸✹ ❱➼ ❞ö ✷✳✸✳ ❈❤♦  ❈❤♦ F : R2 → R2 ①→❝ ✤à♥❤ ❜ð✐✿ F (x) = Ax ✈ỵ✐ A =  0  ✳ C = {x = (x1 , x2 ) ∈ R2 : x ≤ 1}✳ ❱ỵ✐ ♠å✐ x = (x1 , x2 ) ∈ C ✱ ✈ỵ✐ ♠å✐ y = (y1 , y2 ) ∈ C t❛ ❝â✿ F (x) − F (y), x − y = A(x − y), x − y = 2(x1 − y1 )(x1 − y1 ) + 2(x2 − y2 )(x2 − y2 ) = 2(x1 − y1 )2 + 2(x2 − y2 )2 = x − y ❉♦ ✤â F ❧➔ ✤ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥ C ✈ỵ✐ γ = 2✳ ❚❛ ❝â F (x) − F (y) = A(x − y) = x − y ❉♦ ✤â F ❧➔ ✷ ✲ ❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ tr➯♥ ❚ø t➼♥❤ ✤ì♥ ✤✐➺✉ ♠↕♥❤ ❝õ❛ ❚➻♠ F C✳ s✉② r❛ ❜➔✐ t♦→♥✿ x∗ ∈ C ✿ F (x∗ ), y − x∗ ≥ 0, ∀y ∈ C ❝â ❞✉② ♥❤➜t ♥❣❤✐➺♠✳ ❉➵ t❤➜② r➡♥❣ x∗ = (0, 0) ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥✳ ❈❤å♥ ❝→❝ t❤❛♠ sè t❤ä❛ ♠➣♥ ❣✐↔ t❤✐➳t ρk = 1, βk = , k k+1 Pữỡ ữợ ❝â ❞↕♥❣✿     x∈C    g k = A(xk ) = (2xk1 2xk2 )      xk+1 = PC (xk − αk g k ) LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✸✺ tr♦♥❣ ✤â αk = βk γk ✈ỵ✐ PC (x) = γk = max{ρk , g k }✳ P❤➨♣ ❝❤✐➳✉ tr➯♥    x   0 + (x − 0) x−0 ♥➳✉ x ≤1 ♥➳✉ x > C ❝â ❞↕♥❣ ❍❛② PC (x) = ❱ỵ✐ ✤✐➸♠ ❜❛♥ ✤➛✉ x0 =    x x    x 1 , 2 ♥➳✉ x ≤1 ♥➳✉ x > ✳ ▲➟♣ tr➻♥❤ tr➯♥ ▼❛t❧❛❜ t❛ ❝â ❜↔♥❣ ❦➳t q✉↔ s❛✉✿ k xk1 xk2 xk − x∗ ✶ ✲✵✳✷✵✼✶✵✻✼✽✶ ✲✵✳✷✵✼✶✵✻✼✽✶ ✵✳✷✾✷✽✾✸✷✶✽ ✷ ✵✳✵✻✾✵✸✺✺✾✸ ✵✳✵✻✾✵✸✸✺✺✾✸ ✵✳✵✾✼✻✸✶✵✼✷✾ ✸ ✵ ✵ ✵ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✸✻ ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ ♥❣❤✐➯♥ ❝ù✉ t tỷ tr t ỗ õ ù♥❣ ❞ö♥❣ ✈➔♦ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ♣❛r❛✲✤ì♥ ✤✐➺✉✳ ❈ư t❤➸ ❧➔✿ ✶✳ ◆❤➢❝ ❧↕✐ ♠ët sè t t ỡ t ỗ ỗ ỵ t t ỗ ●✐ỵ✐ t❤✐➺✉ ✤à♥❤ ♥❣❤➽❛ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♣❤➨♣ ởt t ỗ õ ổ tự t ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ ♠ët ✤✐➸♠ ❧➯♥ ❝→❝ t➟♣ ✤➦❝ ❜✐➺t ♥❤÷ ♥û❛ ❦❤ỉ♥❣ ❣✐❛♥✱ ❤➻♥❤ ❝➛✉ ✤â♥❣ ❤❛② s✐➯✉ ❤ë♣✱✳✳✳ ✸✳ ●✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔ ♥➯✉ ♠è✐ ❧✐➯♥ q✉❛♥ ❝õ❛ ❜➔✐ t♦→♥ ♥➔② ✈ỵ✐ t ỹ t ỗ tr t ỗ ❜➔✐ t♦→♥ ❜ị ♣❤✐ t✉②➳♥✳ ✹✳ ❙û ❞ư♥❣ ♣❤➨♣ ❝❤✐➳✉ ỹ tt t ữợ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ♣❛r❛✲✤ì♥ ✤✐➺✉✳ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✸✼ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ ❱➠♥ ❍✐➲♥✱ ▲➯ ❉ô♥❣ ữ ỳ tr t ỗ ự ❞ö♥❣✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ●✐❛ ❍➔ ◆ë✐✳ ●✐→♦ ❚✐➳♥❣ ❆♥❤ ❬✷❪ ■✳ ❑♦♥♥♦✈ ✭✷✵✶✶✮✱ ❈♦♠❜✐♥❡❞ ❘❡❧❛①❛t✐♦♥ ❆❧❣♦r✐t❤♠s ❢♦r ❱❛r✐❛t✐♦♥❛❧ ■♥❡q✉❛❧✐t✐❡s✱ ❙♣r✐♥❣❡r✳ ❬✸❪ ❍♦❛♥❣ ❚✉② ✭✷✵✶✸✮✱ ❈♦♥✈❡① ❆♥❛❧②s✐s ❛♥❞ ●❧♦❜❛❧ ❖♣t✐♠✐③❛t✐♦♥✱ ❙♣r✐♥❣❡r✳ ❬✹❪ P✳ ❙❛♥t♦s ❛♥❞ ❙✳ ❙❝❤❡✐♠❜❡r❣ ✭✷✵✶✶✮✱ ✏❆♥ ✐♥❡①❛❝t s✉❜❣r❛❞✐❡♥t ❛❧❣♦✲ r✐t❤♠ ❢♦r ❡q✉✐❧✐❜r✐✉♠ ♣r♦❜❧❡♠s✑✱ ❈♦♠♣✉t❛t✐♦♥❛❧ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤✲ ❡♠❛t✐❝s✱ ✸✵✱ ♣♣✳ ✾✶✲✶✵✼✳ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ...  - VÀNG VĂN HÀ VỀ TOÁN TỬ CHIẾU METRIC LÊN TẬP LỒI ĐÓNG VÀ ỨNG DỤNG VÀO BÀI TOÁN BẤT ĐẲNG THỨC BIẾN PHÂN Chuyên ngành: Toán ứng dụng Mã số : 46 01 12 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI... ♠✐♥❤ t➼♥❤ ❜à ❝❤➦♥ ❝õ❛ ❞➣② {xk } ✤÷đ❝ s✐♥❤ ❜ð✐ t❤✉➟t t♦→♥✳ ❆✷✳ S(F ; C) ⊆ Sd (F ; C)❀ ú ỵ r F tử ỡ t tt ú ỵ sû ❆✶ ✈➔ ❆✷ ✤➲✉ t❤ä❛ ♠➣♥✳ ❑❤✐ ✤â✱ ❧➔ ❞➣② ❤ë✐ tư ✈ỵ✐ ♠å✐ x∗ ∈ S(F ; C)❀ k ✭✐✐✮... ✭✷✳✽✮ t❛ ❝â +∞ δk < +∞ ✭✷✳✶✽✮ k=0 ❉♦ ✤â✱ tø ✭✷✳✶✼✮✱ ✭✷✳✶✽✮ ✈➔ ❇ê ✤➲ ✷✳✸ s✉② r❛ { xk − x∗ } ❧➔ ♠ët tử r tứ ỵ ✷✳✷✳ ●✐↔ sû F ❧✐➯♥ tö❝ ✈➔ ❝→❝ ❣✐↔ t❤✐➳t ❆✶✱ ❆✷ ✤➲✉ t❤ä❛ ♠➣♥✳ ❑❤✐ ✤â✱ t❛ ❝â lim sup