Một phương pháp lặp giải một lớp bất đẳng thức biến phân hai cấp với ràng buộc điểm bất động tách

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ❍❖⑨◆● ❚❍➚ ❚❍❯ ❍×❒◆● ▼❐❚ P❍×❒◆● P❍⑩P ▲➄P ●■❷■ ▼❐❚ ▲❰P ❇❻❚ ✣➃◆● ❚❍Ù❈ ❇■➌◆ P❍❹◆ ❍❆■ ❈❻P ❱❰■ ❘⑨◆● ❇❯❐❈ ✣■➎▼ ❇❻❚ ✣❐◆● ❚⑩❈❍ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ö♥❣ ▼➣ sè✿ ✽ ✹✻ ✵✶ ✶✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❈⑩◆ ❇❐ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈ P●❙✳❚❙✳ ◆●❯❨➍◆ ❚❍➚ ❚❍❯ ❚❍Õ❨ ❚❍⑩■ ◆●❯❨➊◆ ✕ ✷✵✷✶ ✐✐ ▲í✐ ❝↔♠ ì♥ ▲✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❑❤♦❛ ❚♦→♥ ✲ ❚✐♥✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ữợ sỹ ữợ t t Põ s÷✱ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❚❤✉ ❚❤õ②✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❦➼♥❤ trå♥❣ ✈➔ ❜✐➳t ì♥ s➙✉ s➢❝ ✤è✐ ✈ỵ✐ ❝ỉ ◆❣✉②➵♥ ❚❤à ❚❤✉ ❚❤õ② ✭❱✐➺♥ ❚♦→♥ ù♥❣ ❞ư♥❣ ✈➔ ❚✐♥ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❇→❝❤ ❦❤♦❛ ❍➔ ữớ ổ t st ữợ t➟♥ t➻♥❤ ✈➔ ✤ë♥❣ ✈✐➯♥ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ tø ❦❤✐ ❧ü❛ ❝❤å♥ ✤➲ t➔✐ ❝❤♦ ✤➳♥ ❦❤✐ t❤ü❝ ❤✐➺♥ ✈➔ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❚→❝ ❣✐↔ ❝ô♥❣ ỷ ỡ t tợ qỵ ❚❤➛②✱ ❈ỉ ❣✐→♦ t❤✉ë❝ ❑❤♦❛ ❚♦→♥ ✲ ❚✐♥✱ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t➟♥ t➻♥❤ ❣✐↔♥❣ ❞↕② ❧ỵ♣ ❈❛♦ ❤å❝ ❚♦→♥ ❑✶✷❆✸✱ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐✱ ✤ë♥❣ ✈✐➯♥ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ t↕✐ ❚r÷í♥❣✳ ❈✉è✐ ❝ị♥❣✱ t→❝ ❣✐↔ ①✐♥ tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ❇❛♥ ❣✐→♠ ❤✐➺✉✱ t➟♣ t❤➸ ❝→❝ ❚❤➛②✱ ❈ỉ ❣✐→♦ ❝õ❛ tr÷í♥❣ ❚r✉♥❣ ❤å❝ ♣❤ê t❤ỉ♥❣ ▲÷ì♥❣ ❚❤➳ ❱✐♥❤ ♥ì✐ t→❝ ❣✐↔ ✤❛♥❣ ❝ỉ♥❣ t→❝✱ ❣✐❛ ✤➻♥❤ ✈➔ ♥❣÷í✐ t❤➙♥ ❧✉ỉ♥ ❦❤✉②➳♥ ❦❤➼❝❤ ✤ë♥❣ ✈✐➯♥ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ ❝❛♦ ❤å❝ ✈➔ ✈✐➳t ❧✉➟♥ ✈➠♥ ♥➔②✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❚→❝ ❣✐↔ ❍♦➔♥❣ ❚❤à ❚❤✉ ❍÷ì♥❣ ✐✐✐ ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ỵ s t tt ❉❛♥❤ s→❝❤ ❜↔♥❣ ✈✐ ▼ð ✤➛✉ ✶ ❈❤÷ì♥❣ ✶✳ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ✈➔ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ✸ ✶✳✶ ✶✳✷ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✶ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✷ ⑩♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳✶ ⑩♥❤ ①↕ ✤ì♥ ✤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳✷ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✷✳✸ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ❈❤÷ì♥❣ ✷✳ Pữỡ ởt ợ t tự ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✶✹ ✷✳✶ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ✈ỵ✐ t♦→♥ tû ❣✐↔ ✤ì♥ ✤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✶✳✶ ▼ỉ t↔ ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✶✳✷ ❙ü ❤ë✐ tö ✷✳✶✳✸ ⑩♣ ❞ö♥❣ ✈➔ ✈➼ ❞ö ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✐✈ ✷✳✷ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ✈ỵ✐ r➔♥❣ ❜✉ë❝ ✤✐➸♠ ❜➜t ✤ë♥❣ t→❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✷✳✶ ▼ỉ t↔ ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✷✳✷ ❙ü ❤ë✐ tö ✷✳✷✳✸ ❱➼ ❞ö ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✹✸ ✈ ❇↔♥❣ ỵ s t tt R Rn ❝→❝ sè t❤ü❝ H ❑❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ∅ ∀x ❚➟♣ ré♥❣ x∈D x∈ /D x t❤✉ë❝ t➟♣ D x ❦❤æ♥❣ t❤✉ë❝ t➟♣ D x, y x ❑❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ n ợ x ổ ữợ x ✈➔ y ❈❤✉➞♥ ❊✉❝❧✐❞❡ ❝õ❛ x A∗ I ❚♦→♥ tû ❧✐➯♥ ❤ñ♣ ❝õ❛ t♦→♥ tû A C ∩D C⊆D ●✐❛♦ ❝õ❛ ❤❛✐ t➟♣ C ✈➔ D C⊂D C ❧➔ t➟♣ ❝♦♥ t❤ü❝ sü ❝õ❛ D ❋✐①(T ) ❚➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T PC (x) P❤➨♣ ❝❤✐➳✉ trü❝ ❣✐❛♦ ✭♠➯tr✐❝✮ ❝õ❛ ♣❤➛♥ tû x ❧➯♥ t➟♣ C ✈✳✤✳❦ ❱✐➳t t➢t ❝õ❛ ❝ư♠ tø ✧✈ỵ✐ ✤✐➲✉ ❦✐➺♥✧ A ▼❛ tr➟♥ ❝❤✉②➸♥ ✈à ❝õ❛ ♠❛ tr➟♥ A det(A) ❱■P ✣à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ ✈✉æ♥❣ A ❙❋PP ❇➔✐ t♦→♥ ✤✐➸♠ t t tỷ ỗ t C t ❝♦♥ ❤♦➦❝ ❜➡♥❣ t➟♣ D ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈✐ ❉❛♥❤ s→❝❤ ❜↔♥❣ ✷✳✶ ❑➳t q✉↔ sè ❝❤♦ ❱➼ ❞ư ✷✳✶✳✽ ✈ỵ✐ x0 = (1, 1) ✱ αk = ✷✳✷ λk = 0.01✱ µ = 0.1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ k+1 ✱ ηk = 2(k+3) ✱ ❑➳t q✉↔ sè ❝❤♦ ❱➼ ❞ö ✷✳✶✳✽ ✈ỵ✐ x0 = (2, 3) ✱ αk = k+3 ✷✳✸ ✷✳✹ k+3 ✱ ηk = k+1 2(k+3) ✱ λk = 0.01✱ µ = 0.1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ k+1 ❑➳t q✉↔ ❝❤↕② ❝❤÷ì♥❣ tr➻♥❤ ✈ỵ✐ λk = ✱ αk = ✳ ✳ ✳ ✳ ✳ ✸✾ k+2 2(k + 3) k 0.01 + t q ữỡ tr ợ k = ✱ αk = ✳ ✳ ✹✵ 1.7k + 2(k 0.01 + 3) ✶ ▼ð ✤➛✉ ❈❤♦ H ❧➔ ♠ët ổ rt tỹ ợ t ổ ữợ , C ởt t ỗ ✤â♥❣✱ ❦❤→❝ ré♥❣ ❝õ❛ H✱ F ❧➔ →♥❤ ①↕ ✤✐ tø ♠ët t➟♣ tr♦♥❣ H ❝❤ù❛ C ✈➔♦ H✳ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ✮ ✐♥❡q✉❛❧✐t② ✈ỵ✐ →♥❤ ①↕ F ✈➔ t➟♣ r➔♥❣ ❜✉ë❝ C ✱ ỵ PC, F ữủ t ữ s❛✉✿ ❚➻♠ x∗ ∈ C s❛♦ ❝❤♦ F (x∗ ), x − x∗ ≥ ∀x ∈ C ✭❱■P✮ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✤÷đ❝ ♥❤➔ t♦→♥ ❤å❝ ♥❣÷í✐ ■t❛❧✐❛✱ ❙t❛♠♣❛❝❝❤✐❛ ✭①❡♠ ❬✼❪ ✈➔ ❬✶✵❪✮✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ✤÷❛ r❛ ❧➛♥ ✤➛✉ t✐➯♥ ✈➔♦ ❝✉è✐ ♥❤ú♥❣ ♥➠♠ ✻✵ ỳ t trữợ ứ ✤â ✤➳♥ ♥❛②✱ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❧✉æ♥ ❧➔ ♠ët ✤➲ t➔✐ t❤í✐ sü✱ t❤✉ ❤ót ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ❞♦ ✈❛✐ trá q✉❛♥ trồ t tr ỵ tt t ụ ♥❤÷ tr♦♥❣ ♥❤✐➲✉ ù♥❣ ❞ư♥❣ t❤ü❝ t➳✳ ❑❤✐ t➟♣ r➔♥❣ ❜✉ë❝ C ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ P ữủ ữợ t ❜➜t ✤ë♥❣ ❝õ❛ ♠ët →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ❤♦➦❝ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ✭❤ú✉ ❤↕♥ ❤♦➦❝ ✈æ ❤↕♥✮ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ t❤➻ ❜➔✐ t♦→♥ ✭❱■P✮ ❝á♥ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ❝→❝ ❜➔✐ t♦→♥ tỹ t ữ ỷ ỵ t ổ t tố ữ ố ợ ❧ỵ♣ ❜➔✐ t♦→♥ ♥➔②✱ ♥➠♠ ✷✵✵✶ ❨❛♠❛❞❛ ❬✶✶❪ ✤➣ ✤➲ ①✉➜t ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ✤➸ ❣✐↔✐✳ ❉ü❛ tr➯♥ ❝→❝❤ t✐➳♣ ❝➟♥ ❝õ❛ ❨❛♠❛❞❛✱ ✤➣ ❝â ♥❤✐➲✉ ♥❣❤✐➯♥ ❝ù✉ ♥❤➡♠ ♠ð rë♥❣ ✈➔ ❝↔✐ ❜✐➯♥ ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ❞↕♥❣ ✤÷í♥❣ ❞è❝ ♥❤➜t ❝❤♦ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr➯♥ t➟♣ r➔♥❣ ❜✉ë❝ C ❧➔ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥✱ ❤å ✈æ ❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳ ▼ư❝ t✐➯✉ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ t➻♠ ❤✐➸✉ ✈➔ tr➻♥❤ ❜➔② ♠ët số ữỡ ợ t t t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ tr♦♥❣ tr÷í♥❣ ❤đ♣ ❜➔✐ t♦→♥ ữợ t t tự ❣✐↔ ✤ì♥ ✤✐➺✉ ❤♦➦❝✴✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ t→❝❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ tr♦♥❣ ❜➔✐ ❜→♦ ❬✸❪ ✈➔ ❬✹❪ ❝æ♥❣ ❜è ♥➠♠ ✷✵✶✻ ✈➔ ✷✵✶✼✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ ✧❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ✈➔ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣✧ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠✱ ✈➼ ❞ö ✈➲ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣✱ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✱ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ✈➔ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣✳ ❈❤÷ì♥❣ ✷ Pữỡ ởt ợ t t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣✧ tr➻♥❤ ❜➔② ♠ët sè ữỡ ợ ữủ t tr ❬✹❪ ①➜♣ ①➾ ♥❣❤✐➺♠ ❝❤♦ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ tr♦♥❣ tr÷í♥❣ ❤đ♣ ❜➔✐ t♦→♥ ❝➜♣ ữợ t tự ỡ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ t→❝❤✱ tr➻♥❤ ❜➔② ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✈➔ ♠ët sè →♣ ❞ư♥❣ tr♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t✱ ỗ tớ t sü ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣✳ ❈❤÷ì♥❣ tr➻♥❤ t❤ü❝ ♥❣❤✐➺♠ ✤÷đ❝ ✈✐➳t ❜➡♥❣ ♥❣ỉ♥ ♥❣ú ▼❆❚▲❆❇✳ ✸ ❈❤÷ì♥❣ ✶ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ✈➔ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠✱ ✈➼ ❞ö ✈➲ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣✱ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✱ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ✈➔ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ♠ö❝✳ ▼ö❝ ✶✳✶ tr➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ♥ë✐ ❞✉♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❧✉➟♥ ✈➠♥✱ ❣✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ ♥➔②✳ ▼ư❝ ✶✳✷ ❣✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✱ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✱ ✈➼ ❞ư ✈➲ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔ tr➻♥❤ ❜➔② ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ t ũ sỹ tỗ t ❝õ❛ ❜➔✐ t♦→♥✳ ❑✐➳♥ t❤ù❝ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ tê♥❣ ❤đ♣ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶✱ ✷✱ ✹✕✻✱ ✽❪✳ ✶✳✶ ✶✳✶✳✶ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ▼➺♥❤ ✤➲ ✶✳✶✳✶ ✭①❡♠ ❬✶❪✮ ❈❤♦ H ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ x, y ∈ H✱ x, y = x + y − x−y = x+y − x − y ✹ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷ ✭①❡♠ ❬✶❪✮ ❈❤♦ ❤❛✐ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥ ❜➜t ❦ý X ✈➔ Y ✳ ▼ët →♥❤ ①↕ A : X −→ Y ❣å✐ ❧➔ t✉②➳♥ t➼♥❤ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❤❛② t♦→♥ tû ♥➳✉✿ ✭❛✮ A(x1 + x2 ) = Ax1 + Ax2 ✱ ∀x1 , x2 ∈ X ✳ ✭❜✮ A(αx) = αAx✱ ∀x ∈ X ✱ ∀α ∈ R✳ ◆➳✉ Y = X ✱ t❛ ❝ô♥❣ ♥â✐ A ❧➔ ♠ët t♦→♥ tû tr♦♥❣ X ✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝❤✉♥❣ ✈➲ →♥❤ ①↕ ❧✐➯♥ tö❝✱ →♥❤ ①↕ A : X −→ Y ❣å✐ ❧➔ tư❝ ❧✐➯♥ ♥➳✉ xn −→ x0 ❧✉ỉ♥ ❧✉ỉ♥ ❦➨♦ t❤❡♦ Axn −→ Ax0 ✳ ❚✉② ♥❤✐➯♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ❜➜t ❦ý✱ t♦→♥ tû t✉②➳♥ t➼♥❤ ❦❤æ♥❣ ♥❤➜t t❤✐➳t ❧✐➯♥ tư❝✳ ❑❤✐ ✤â ✤✐➲✉ ❦✐➺♥ ❧✐➯♥ tư❝ t÷ì♥❣ ữỡ ợ t ữủ ữ s ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸ ✭①❡♠ ❬✶❪✮ ▼ët t♦→♥ tû A : X −→ Y ❣å✐ ❧➔ ♥ë✐ ✮ ❜à ❝❤➦♥ ✭❣✐ỵ✐ ♥➳✉ ❝â ♠ët ❤➡♥❣ sè K > s❛♦ ❝❤♦✿ Ax ≤ K x ✭✶✳✶✮ ∀x ∈ X ❙è K ≥ ♥❤ä ♥❤➜t t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✶✳✶✮ ✤÷đ❝ t tỷ A ỵ A ✳ ◆❤÷ ✈➟②✿ ✶✳ Ax ≤ A x ∀x ∈ X ✳ ✷✳ ◆➳✉ ∀x ∈ X ✱ Ax ≤ K x t❤➻ A ≤ K ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹ ✭①❡♠ ❬✶❪✮ ❈❤♦ A : X −→ Y ❧➔ ♠ët t♦→♥ tû t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝✳ ⑩♥❤ ①↕ A∗ : Y −→ X ✤÷đ❝ ❣å✐ ❧➔ t♦→♥ tû ❧✐➯♥ ❤ñ♣ ❝õ❛ A ♥➳✉ Ax, y = x, A∗ y ✱ ∀x ∈ X, ∀y ∈ Y ✳ ❚♦→♥ tû ❧✐➯♥ ❤ñ♣ ❝õ❛ ♠ët t♦→♥ tû t✉②➳♥ t➼♥❤ ❝â ❝→❝ t➼♥❤ ❝❤➜t s❛✉✳ ▼➺♥❤ ✤➲ ✶✳✶✳✺ ✭①❡♠ ❬✶❪✮ ❈❤♦ A : X −→ Y ✱ B : X −→ Y ❧➔ ❝→❝ t♦→♥ tû t✉②➳♥ t➼♥❤✱ ✈ỵ✐ A∗ : Y −→ X ✱ B ∗ : Y −→ X ❧➛♥ ❧÷đt ❧➔ ❝→❝ t♦→♥ tû ❧✐➯♥ ❤đ♣ t÷ì♥❣ ù♥❣ ❝õ❛ A ✈➔ B ✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ α, β ∈ R✱ t❛ ❝â✿ ✶✳ A∗ = A ✳ ✸✵ ❙û ❞ư♥❣ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✈➔ t➼♥❤ ❦❤ỉ♥❣ ❣✐➣♥ ❝õ❛ ♣❤➨♣ ❝❤✐➳✉ PC ✱ t❛ t❤✉ ✤÷đ❝✿ y k − x∗ = PC (xk + δA∗ (Suk − Axk )) − PC (x∗ ) ≤ (xk − x∗ ) + δA∗ (Suk − Axk ) 2 = xk − x∗ + δA∗ (Suk − Axk ) ≤ xk − x∗ + δ A∗ ≤ xk − x∗ + δ2 A = xk − x∗ − δ(1 − δ A ) Suk − Axk ❑❤✐ ✤â✱ ✈ỵ✐ δ ∈ 0, 2 + 2δ xk − x∗ , A∗ (Suk − Axk ) Suk − Axk Suk − Axk A +1 2 + 2δ A(xk − x∗ ), Suk − Axk − δ uk − Axk 2 − δ Suk − Axk − δ uk − Axk ✭✷✳✸✵✮ ✱ t❛ ❝â y k − x∗ ≤ xk − x∗ ∀k ✭✷✳✸✶✮ ❚ø ❇ê ✤➲ ✶✳✷✳✹ ✈➔ t➼♥❤ ❦❤æ♥❣ ❣✐➣♥ ❝õ❛ ♣❤➨♣ ❝❤✐➳✉ PC ✱ t❛ ❝â✿ z k − x∗ = PC (y k − λk µF (y k )) − PC (x∗ ) ≤ y k − λk µF (y k ) − x∗ = (I − λk µF )(y k ) − (I − λk µF )(x∗ ) − λk µF (x∗ ) ≤ (I − λk µF )(y k ) − (I − λk µF )(x∗ ) + λk µ F (x∗ ) ≤ (1 − λk τ ) y k − x∗ + λk µ F (x∗ ) ✭✷✳✸✷✮ ❑➳t ❤ñ♣ T (x∗ ) = x∗ ✱ t➼♥❤ ❦❤æ♥❣ ❣✐➣♥ ❝õ❛ T ✱ ✭✷✳✸✷✮ ✈➔ ✭✷✳✸✶✮✱ t❛ t❤✉ ✤÷đ❝ xk+1 − x∗ = αk (xk − x∗ ) + (1 − αk )(T (z k ) − x∗ ) ≤ αk xk − x∗ + (1 − αk ) T (z k ) − x∗ = αk xk − x∗ + (1 − αk ) T (z k ) − T (x∗ ) ≤ αk xk − x∗ + (1 − αk ) z k − x∗ ≤ αk xk − x∗ + (1 − αk ) (1 − λk τ ) y k − x∗ + λk µ F (x∗ ) ≤ αk xk − x∗ + (1 − αk ) (1 − λk τ ) xk − x∗ + λk µ F (x∗ ) = [1 − λk (1 − αk )τ ] xk − x∗ + (1 − αk )λk µ F (x∗ ) k = [1 − λk (1 − αk )τ ] x − x ∗ µ F (x∗ ) + λk (1 − αk )τ τ ✸✶ ❉♦ ✤â✱ k+1 x −x ∗ ≤ max µ F (x∗ ) x −x , τ k ∗ , ✈➔ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ q✉② ♥↕♣ t♦→♥ ❤å❝✱ t❛ t❤✉ ✤÷đ❝ xk − x∗ ≤ max x0 − x∗ , µ F (x∗ ) τ ∀k ≥ ❉♦ ✤â✱ ❞➣② {xk } ❜à ❝❤➦♥✱ tø ✤â s✉② r❛ ❝→❝ ❞➣② {y k } ✈➔ {F (y k )} ❜à ❝❤➦♥✳ ❉ü❛ ✈➔♦ t➼♥❤ ❦❤æ♥❣ ❣✐➣♥ ❝õ❛ PC ✱ t❛ ❝â y k+1 − y k = PC (xk+1 + δA∗ (Suk+1 − Axk+1 )) − PC (xk + δA∗ (Suk − Axk )) ≤ xk+1 + δA∗ (Suk+1 − Axk+1 ) − xk − δA∗ (Suk − Axk ) = (xk+1 − xk ) + δA∗ (Suk+1 − Suk + Axk − Axk+1 ) = xk+1 − xk 2 2 + δ A∗ (Suk+1 − Suk + Axk − Axk+1 ) + 2δ xk+1 − xk , A∗ (Suk+1 − Suk + Axk − Axk+1 ) ✭✷✳✸✸✮ ❈❤ó þ r➡♥❣✿ δ A∗ (Suk+1 − Suk +Axk − Axk+1 ) ≤ δ A∗ = δ2 A 2 Suk+1 − Suk + Axk − Axk+1 Suk+1 − Suk + Axk − Axk+1 ✭✷✳✸✹✮ ỵ k := xk+1 xk , A (Suk+1 − Suk + Axk − Axk+1 ) ✈➔ sû ❞ư♥❣ t➼♥❤ ❦❤ỉ♥❣ ❣✐➣♥ ❝õ❛ S ✈➔ PQ ✱ t❛ ✤÷đ❝ Θk = 2δ A(xk+1 − xk ), Suk+1 − Suk + Axk − Axk+1 = 2δ Axk+1 − Axk , Suk+1 − Suk − 2δ Axk+1 − Axk =δ Suk+1 − Suk 2 − Axk+1 − Axk − (Suk+1 − Suk ) − Axk+1 − Axk ≤δ uk+1 − uk − Axk+1 − Axk − (Suk+1 − Suk ) 2 − Axk+1 − Axk ✸✷ =δ PQ (Axk+1 ) − PQ (Axk ) − Axk+1 − Axk − (Suk+1 − Suk ) − Axk+1 − Axk 2 ✭✷✳✸✺✮ ≤ −δ Axk+1 − Axk − (Suk+1 − Suk ) ⑩♣ ❞ö♥❣ ✭✷✳✸✹✮ ✈➔ ✭✷✳✸✺✮ ✈➔♦ ✭✷✳✸✸✮✱ ❦➳t ❤đ♣ ✈ỵ✐ < δ < y k+1 −y k A +1 ✱ t❛ ❝â ≤ xk+1 − xk − δ(1 − δ A ) Suk+1 − Suk + Axk − Axk+1 ≤ xk+1 − xk ❉♦ ✤â y k+1 − y k ≤ xk+1 − xk ∀k ≥ ✭✷✳✸✻✮ ✣➸ ✤ì♥ ❣✐↔♥ tr♦♥❣ ✈✐➺❝ ỵ t tk = T (z k ) ứ t➼♥❤ ❦❤æ♥❣ ❣✐➣♥ ❝õ❛ ❝→❝ →♥❤ ①↕ T ✈➔ PC ✱ ✭✷✳✸✻✮✱ ✈➔ ❇ê ✤➲ ✶✳✷✳✹✱ t❛ ❝â tk+1 −tk = T (z k+1 ) − T (z k ) ≤ z k+1 − z k = PC (y k+1 − λk+1 µF (y k+1 )) − PC (y k − λk µF (y k )) ≤ y k+1 − λk+1 µF (y k+1 ) − y k + λk µF (y k ) = (I − λk µF )(y k+1 ) − (I − λk µF )(y k ) + µ(λk − λk+1 )F (y k+1 ) ≤ (1 − λk τ ) y k+1 − y k + µ |λk − λk+1 | F (y k+1 ) ≤ (1 − λk τ ) xk+1 − xk + µ |λk − λk+1 | F (y k+1 ) ❇➜t ✤➥♥❣ t❤ù❝ ❝✉è✐ ❞➝♥ tỵ✐ tk+1 − tk − xk+1 − xk ≤ −λk τ xk+1 − xk + µ |λk − λk+1 | F (y k+1 ) ❱➻ {xk }✱ {F (y k )} ❜à ❝❤➦♥ ✈➔ lim λk = 0✱ ♥➯♥ t❛ ❝â k−→∞ lim sup( tk+1 − tk − xk+1 − xk ) ≤ k−→∞ ❉♦ ✈➟②✱ tø ❇ê ✤➲ ✶✳✷✳✺✱ t❛ t❤✉ ✤÷đ❝ lim tk − xk = k−→∞ ✸✸ ❚❛ t❤➜② xk+1 − xk = (1 − αk ) tk − xk ≤ tk − xk , ✈➔ xk − T (y k ) ≤ xk − tk + tk − T (y k ) = xk − tk + T (z k ) − T (y k ) ≤ x k − tk + z k − y k ≤ xk − tk + PC (y k − λk µF (y k )) − PC (y k ) ≤ xk − tk + y k − λk µF (y k ) − y k = xk − tk + λk µ F (y k ) , ❦➳t ❤đ♣ ✈ỵ✐ lim tk − xk = 0✱ lim λk = 0✱ ✈➔ t➼♥❤ ❜à ❝❤➦♥ ❝õ❛ ❞➣② {F (y k )}✱ k−→∞ s✉② r❛ k−→∞ lim xk+1 − xk = 0, ✭✷✳✸✼✮ lim xk − T (y k ) = k−→∞ k−→∞ ▲➛♥ ❧÷đt sû ❞ư♥❣ t➼♥❤ ❦❤ỉ♥❣ ❣✐➣♥ ❝õ❛ →♥❤ ①↕ T ✈➔ T (x∗ ) = x∗ ✱ t❛ ❝â✱ ✈ỵ✐ ♠å✐ k✱ xk+1 − x∗ = αk (xk − x∗ ) + (1 − αk )(T (z k ) − x∗ ) ≤ αk x k − x ∗ + (1 − αk ) T (z k ) − x∗ = αk x k − x ∗ + (1 − αk ) T (z k ) − T (x∗ ) ≤ αk x k − x ∗ + (1 − αk ) z k − x∗ ✭✷✳✸✽✮ ❚ø ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✸✵✮ ✈➔ ✭✷✳✸✷✮✱ t❛ t❤✉ ✤÷đ❝ z k − x∗ ≤ (1 − λk τ ) y k − x∗ + λk µ F (x∗ ) = (1 − λk τ )2 y k − x∗ xk − x∗ ≤ (1 − λk τ )2 + λk µ F (x∗ ) ≤ xk − x∗ 2 + λk µ F (x∗ ) 2 2(1 − λk τ ) y k − x∗ + λk µ F (x∗ ) − δ(1 − δ A ) Suk − Axk − δ uk − Axk + uk − Axk 2(1 − λk τ ) y k − x∗ + λk µ F (x∗ ) − δ(1 − λk τ )2 (1 − δ A ) Suk − Axk + λk µ F (x∗ ) 2(1 − λk τ ) y k − x∗ + λk µ F (x∗ ) ✭✷✳✸✾✮ ✸✹ ❚❤❛② ✭✷✳✸✾✮ ✈➔♦ ✭✷✳✸✽✮✱ t❛ ❝â xk+1 − x∗ ≤ xk − x∗ − δ(1 − αk )(1 − λk τ )2 × (1 − δ A ) Suk − Axk + (1 − αk )λk µ F (x∗ ) ❙û ❞ö♥❣ δ ∈ 0, A 2 + uk − Axk 2(1 − λk τ ) y k − x∗ + λk µ F (x∗ ) ✈➔ ✤à♥❤ ♥❣❤➽❛ +1 νk := δ(1 − αk )(1 − λk τ )2 , ψ := (1 − αk )λk µ F (x∗ ) 2(1 − λk τ ) y k − x∗ + λk µ F (x∗ ) , t❛ ✤÷đ❝ νk (1 − δ A ) Suk − Axk ≤ ( xk − x∗ 2 + uk − Axk − xk+1 − x∗ ) + ψk ≤ ( xk − x∗ + xk+1 − x∗ ) xk+1 − xk + ψk ✭✷✳✹✵✮ ❱➻ lim xk+1 − xk = 0, {xk }, {y k } ❜à ❝❤➦♥✱ lim λk = 0, lim αk = α ∈ (0, 1)✱ k−→∞ k−→∞ k−→∞ t❛ ❝â lim νk = δ(1 − α) > ✈➔ ✈➳ ♣❤↔✐ ❝õ❛ ✭✷✳✹✵✮ ❞➛♥ ✤➳♥ ❦❤✐ k −→ ∞✳ ❚ø k−→∞ ✤â s✉② r❛ lim Suk − Axk = 0, k−→∞ lim uk − Axk = k−→∞ ✭✷✳✹✶✮ ▲↕✐ sû ❞ư♥❣ t➼♥❤ ❦❤ỉ♥❣ ❣✐➣♥ ❝õ❛ t♦→♥ tû ❝❤✐➳✉ PC ✱ {xk } ⊂ C ✱ t❛ ❝â t❤➸ ✈✐➳t y k − xk = PC (xk + δA∗ (Suk − Axk )) − PC (xk ) ≤ xk + δA∗ (Suk − Axk ) − xk = δA∗ (Suk − Axk ) ≤ δ A∗ Suk − Axk = δ A Suk − Axk , t ủ ợ t ữủ lim y k xk = k−→∞ ✭✷✳✹✷✮ ✸✺ ❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝✱ t❛ ❝â y k − T (y k ) ≤ xk − y k + xk − T (y k ) uk − Suk ≤ uk − Axk + Suk − Axk , ❦➳t ❤đ♣ ✈ỵ✐ ✭✷✳✹✷✮✱ ✭✷✳✸✼✮ ✈➔ ✭✷✳✹✶✮ ❞➝♥ tỵ✐ lim y k − T (y k ) = 0, ✭✷✳✹✸✮ lim uk − Suk = k−→∞ k−→∞ ❇➙② ❣✐í t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ lim sup F (x∗ ), x∗ − y k + λk µF (y k ) ≤ k−→∞ ▲➜② ♠ët ❞➣② ❝♦♥ {y ki } ❝õ❛ {y k } s❛♦ ❝❤♦ lim sup F (x∗ ), x∗ − y k = lim F (x∗ ), x∗ − y ki i−→∞ k−→∞ ❱➻ {y ki } ❜à ❝❤➦♥✱ t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ y ki ❤ë✐ tö ②➳✉ ✤➳♥ y ✳ ❑❤✐ ✤â✱ lim sup F (x∗ ), x∗ − y k = lim F (x∗ ), x∗ − y ki i−→∞ k−→∞ = F (x∗ ), x∗ − y ❉➵ t❤➜② r➡♥❣ y ∈ C ✈➻ y ki ⊂ C ✱ y ki y ✈➔ C ✤â♥❣ ②➳✉✳ ●✐↔ sû r➡♥❣ y ∈ / ❋✐①(T )✱ ♥❣❤➽❛ ❧➔ y = T (y)✳ ❱➻ y ki y ✈➔ T ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✱ ♥➯♥ tø ✭✷✳✹✸✮ ✈➔ ❇ê ✤➲ ❖♣✐❛❧✱ t❛ ❝â✿ lim inf y ki − y < lim inf y ki − T (y) i−→∞ i−→∞ ≤ lim inf i−→∞ y ki − T (y ki ) + T (y ki ) − T (y) = lim inf T (y ki ) − T (y) i−→∞ ≤ lim inf y ki − y i−→∞ ▼➙✉ t❤✉➝♥ ♥➔② ❝❤ù♥❣ tä ❣✐↔ sû ♣❤↔♥ ❝❤ù♥❣ ❜❛♥ ✤➛✉ s❛✐✱ tù❝ ❧➔ y ∈ ❋✐①(T )✳ ❱➻ y ki y ✈➔ lim y k − xk = 0✱ ♥➯♥ xki k−→∞ y ✳ ❱➻ t❤➳ Axki Ay ✳ ❚ø ✤✐➲✉ ♥➔② ✈➔ ✭✷✳✹✶✮ t❛ ❝â uki Ay ✭✷✳✹✹✮ ✸✻ ❱➻ {uki } ⊂ Q ✈➔ Q ✤â♥❣ ②➳✉✱ ♥➯♥ tø ✭✷✳✹✹✮ t❛ s✉② r❛ Ay ∈ Q✳ ❚✐➳♣ t❤❡♦✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ Ay ∈ ❋✐①(S)✳ ❚❤➟t ✈➟②✱ ♥➳✉ S(Ay) = Ay ✱ tø ❇ê ✤➲ ❖♣✐❛❧ ✈➔ ✭✷✳✹✸✮✱ t❛ ❝â lim inf uki − Ay < lim inf uki − S(Ay) i−→∞ i−→∞ = lim inf uki − Suki + Suki − S(Ay) i−→∞ ≤ lim inf ( uki − Suki + Suki − S(Ay) ) i−→∞ = lim inf Suki − S(Ay) i−→∞ ≤ lim inf uki Ay , i ổ ỵ õ ự tọ Ay ∈ ❋✐①(S)✳ ❱➻ y ∈ ❋✐①(T ) ✈➔ Ay ∈ ❋✐①(S)✱ t❛ s✉② r❛ y ∈ Ω✳ ▼➔ x∗ ∈ ❙♦❧(Ω, F )✱ ♥➯♥ F (x∗ ), y − x∗ ≥ ❉♦ ✤â✱ tø t➼♥❤ ❜à ❝❤➦♥ ❝õ❛ {F (y k )} ✈➔ lim λk = t❛ s✉② r❛ k−→∞ lim sup F (x∗ ), x∗ − y k + λk µF (y k ) k−→∞ = lim sup F (x∗ ), x∗ − y k + λk µ F (x∗ ), F (y k ) k−→∞ = lim sup F (x∗ ), x∗ − y k k−→∞ = F (x∗ ), x∗ − y ≤ ❈✉è✐ ❝ị♥❣✱ t❛ ❝❤ù♥❣ ♠✐♥❤ xk ❤ë✐ tư ✤➳♥ ✤✐➸♠ x∗ ✳ ❚ø t➼♥❤ ❦❤æ♥❣ ❣✐➣♥ ❝õ❛ PC ✱ ❜➜t ✤➥♥❣ t❤ù❝ x−y ≤ x − y, x − y , ✤ó♥❣ ✈ỵ✐ ♠å✐ x, y ∈ H1 ✱ sû ❞ö♥❣ ❇ê ✤➲ ✶✳✷✳✹ ✈➔ ✭✷✳✸✶✮✱ t❛ ❧➛♥ ❧÷đt t❤✉ ✤÷đ❝ ✸✼ z k − x∗ = PC (y k − λk µF (y k )) − PC (x∗ ) ≤ y k − λk µF (y k ) − x∗ 2 = (I − λk µF )(y k ) − (I − λk µF )(x∗ ) − λk µF (x∗ ) ≤ (I − λk µF )(y k ) − (I − λk µF )(x∗ ) 2 − 2λk µ F (x∗ ), y k − λk µF (y k ) − x∗ ≤ (1 − λk τ )2 y k − x∗ ≤ (1 − λk τ ) xk − x∗ 2 − 2λk µ F (x∗ ), y k − λk µF (y k ) − x∗ − 2λk µ F (x∗ ), y k − λk µF (y k ) − x∗ ❚❤❛② ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✈➔♦ ✭✷✳✸✽✮✱ t❛ ❝â xk+1 − x∗ ≤ αk xk − x∗ + (1 − αk ) z k − x∗ ≤ αk xk − x∗ + (1 − αk )(1 − λk τ ) xk − x∗ 2 − 2λk µ(1 − αk ) F (x∗ ), y k − λk µF (y k ) − x∗ = [1 − λk (1 − αk )τ ] xk − x∗ tr♦♥❣ ✤â θk = + λk (1 − αk )τ θk , ✭✷✳✹✺✮ 2µ F (x∗ ), x∗ − y k + λk µF (y k ) τ ❱➻ lim sup F (x∗ ), x∗ − y k + λk µF (y k ) ≤ 0, k−→∞ ♥➯♥ lim sup θk ≤ k−→∞ ❈❤ó þ r➡♥❣ ∞ λk (1 − αk )τ = ∞ t❤❡♦ ✤✐➲✉ ❦✐➺♥ ✭❜✮✱ →♣ ❞ö♥❣ ❇ê ✤➲ ✶✳✷✳✼ ✈➔♦ k=0 ✭✷✳✹✺✮✱ t❛ s✉② r❛ xk −→ x∗ ✱ ✤➙② ❝❤➼♥❤ ❧➔ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ◆❤➟♥ ①➨t ✷✳✷✳✹ ❚❛ õ t k = tr ỵ ✷✳✷✳✸✳ k+1 ✱ αk = t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ k+2 2(k + 3) ✸✽ ✷✳✷✳✸ ❱➼ ❞ö ♠✐♥❤ ❤å❛ ❱➼ ❞ư ✷✳✷✳✺ ❳➨t ❜➔✐ t♦→♥ ✭❱■P✮✕✭❙❋PP✮ ✈ỵ✐ H1 = R2 ✱ H2 = R3 ✱ A : R2 −→ R3 ①→❝ ✤à♥❤ ❜ð✐ Ax = (x1 + 2x2 , 3x1 − x2 , x2 ), ✈ỵ✐ t♦→♥ tû ❧✐➯♥ ❤ñ♣ A∗ : R3 −→ R2 ✱ ①→❝ ✤à♥❤ ❜ð✐ A∗ y = (y1 + 3y2 , 2y1 − y2 + y3 ) ❈❤♦ C = x ∈ R2 | 2x1 + x2 ≤ ✱ Q = x ∈ R3 | 2x1 + x2 − x3 + = tữỡ ự t ỗ õ ré♥❣ ❝õ❛ R2 ✈➔ R3 ✳ ⑩♥❤ ①↕ ❣✐→ F = I ✱ ✈ỵ✐ I ❧➔ t♦→♥ tû ✤ì♥ ✈à tr♦♥❣ H1 ✭t❤ä❛ ♠➣♥ ✶✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ ✶✲❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③ tr➯♥ C ✮✳ ⑩♥❤ ①↕ T : C −→ C ✈➔ S : Q −→ Q ❧➛♥ ❧÷đt ①→❝ ✤à♥❤ ❜ð✐✿ T (x) = PC (x), S(x) = PQ (x) ❑❤✐ ✤â ❜➔✐ t♦→♥ ✭❱■P✮✕✭❙❋PP✮ trð t❤➔♥❤✿ ❚➻♠ x∗ ∈ Ω s❛♦ ❝❤♦ x∗ , x − x∗ ≥ ∀x ∈ Ω, ✭❱■P✶✮ tr♦♥❣ ✤â Ω ❧➔ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ t→❝❤ t➻♠ x∗ ∈ C s❛♦ ❝❤♦ x∗ = PC (x∗ ), Ax∗ ∈ Q ✈➔ Ax∗ = PQ (Ax∗ ) ✭❙❋P✶✮ ●✐↔✐ t ữợ P t x = T (x) ✈ỵ✐ ♠å✐ x ∈ C ✳ ❚❛ ❝â✿ Ax ∈ Q ⇔ 2(x1 + 2x2 ) + (3x1 − x2 ) − x2 + = ⇔ 5x1 + 2x2 + = ỗ tớ u = S(u), ∀u ∈ Q✳ ❉♦ ✤â✱ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭❙❋P✶✮ ❧➔✿ Ω = C ∩ x ∈ R2 | 5x1 + 2x2 + = = x ∈ R2 | 5x1 + 2x2 + = 0, x1 ≥ −1 ❉➵ t❤➜② Ω ❦❤→❝ ré♥❣✳ ❚✐➳♣ t❤❡♦✱ t❛ ❣✐↔✐ ❜➔✐ t♦→♥ ❝➜♣ tr➯♥ ✭❱■P✶✮✳ ❚❛ t❤➜②✿ x∗ , x − x∗ ≥ ∀x ∈ Ω ⇔ x∗ , x∗ ≤ x∗ , x ⇔ x∗ ≤ x∗ ⇔ x∗ ≤ x x ∀x ∈ Ω ∀x ∈ Ω ✭❈❛✉❝❤②✕❙❝❤✇❛r③✮ ∀x ∈ Ω, ♥❣❤➽❛ ❧➔ ❜➔✐ t♦→♥ ✭❱■P✶✮ trð t❤➔♥❤ ❜➔✐ t♦→♥✿ t➻♠ x∗ ∈ Ω s❛♦ ❝❤♦ x∗ ❧➔ ♣❤➛♥ tû ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t tr♦♥❣ Ω✳ ❚ù❝ ❧➔ ♥❣❤✐➺♠ ✤ó♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❧➔ x∗ = − , − 29 29 ✸✾ ✭❝❤➼♥❤ ❧➔ ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ O tr➯♥ Ω✮✳ ❚❤➟t ✈➟②✿ ▲➜② x∗ = (x∗1 , x∗2 ) ∈ C ✱ t❛ ❝â −1 − 5x∗1 2 ∗ ∗ ∗ ∗ ∗ x = x1 + x2 ✳ ❱➻ Ax ∈ Q ♥➯♥ x2 = ✱ ❦❤✐ ✤â t❛ ❝â✿ ∗ x = x∗1 + x∗2 −1 − 5x∗1 = x∗1 = 29x∗1 + 10x∗1 + + 2 + = 29 x∗1 + 29 29 √ 29 ≥ 29 −5 ∗ −2 ❉➜✉ “ = ” ①↔② r❛ ❦❤✐ x∗1 = ,x = 29 29 ❙û ❞ư♥❣ ❚❤✉➟t t♦→♥ ✷✳✷✳✷ ✈ỵ✐ ✤✐➸♠ ①✉➜t ♣❤→t x0 = (0, 0) ✱ ❝→❝ t❤❛♠ sè ❤➡♥❣ δ = ✱ µ = 2✳ ✣✐➲✉ ❦✐➺♥ ❞ø♥❣ ❧➔ s❛✐ sè ❣✐ú❛ ♥❣❤✐➺♠ ①➜♣ ①➾ ✈➔ ♥❣❤✐➺♠ ✤ó♥❣ ✤õ ♥❤ä✱ tù❝ ❧➔ xk − x∗ ≤ ε, ð ✤➙② ❝❤å♥ ε = 10−6 ✳ ❈❤å♥ ❝→❝ λk ✈➔ αk ❦❤→❝ ♥❤❛✉ ✤➸ ①❡♠ ①➨t sü t❤❛② ✤ê✐ t❤í✐ ữỡ tr ợ k = k+1 ✱ αk = ✱ t❛ t❤✉ ✤÷đ❝ ❦➳t q✉↔ tr♦♥❣ ❇↔♥❣ ✷✳✸✳ k+2 2(k + 3) k 0.01 + ❼ ❱ỵ✐ λk = ✱ αk = ✱ t❛ t❤✉ ✤÷đ❝ ❦➳t q✉↔ tr♦♥❣ ❇↔♥❣ ✷✳✹✳ 1.7k + 2(k 0.01 + 3) ữợ (k) ✶✺✸✻✼✽ xk1 xk2 xk − xk−1 xk − x∗ −0.17227111 −0.06890844 1.538323 × 10−7 ✵✳✵✵✵✶✺✸ −0.17239952 −0.06895981 1.536942 × 10−9 1.536788 × 10−5 −0.17241237 −0.06896495 1.536815 × 10−11 1.536789 × 10−6 −0.17241286 −0.06896515 6.507027 × 10−12 9.999993 × 10−7 ❇↔♥❣ t q ữỡ tr ợ k = k +1 ✱ αk = 2(kk ++13) ✳ ❚❤í✐ ❣✐❛♥ s ữợ (k) ✶✵✵✵ ✶✵✵✵✵ ✾✵✸✾✾ xk1 xk2 xk − xk−1 xk − x∗ −0.1722459 −0.06889836 3.624207 × 10−7 ✵✳✵✵✵✶✽✵ −0.17232985 −0.06893194 9.050217 × 10−8 9.040552 × 10−5 −0.1724054 −0.06896216 9.040966 × 10−10 9.039997 × 10−6 −0.17241286 −0.06896515 1.106215 × 10−11 9.999936 × 10−7 0.01 ❚❤í✐ ❣✐❛♥ ✭s✮ ✵✳✵✸✶✷✹ ✵✳✵✻✷✹✾ ✵✳✺✼✽✶✵ ✺✳✷✺✼✶✼ ❇↔♥❣ t q ữỡ tr ợ k = 1.7k1+ ✱ αk = 2(kk 0.01++13) ✳ ◆❣❤✐➺♠ ①➜♣ ①➾ ❤ë✐ tư ❞➛♥ ✈➲ ♥❣❤✐➺♠ ✤ó♥❣✿ x∗ = − ,− 29 29 ≈ (−0.17241379, −0.06896551) ✹✶ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➣ ✤↕t ✤÷đ❝ ♠ư❝ t✐➯✉ ✤➲ r❛ ự ởt ữỡ ởt ợ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝❀ ✤÷❛ r❛ ✈➔ t➼♥❤ t♦→♥ ✈➼ ❞ư ♠✐♥❤ ❤å❛✧✳ ❑➳t q✉↔ ❝õ❛ ❧✉➟♥ ✈➠♥ ▲✉➟♥ ✈➠♥ ✤➣ tr ởt số ữỡ ởt ợ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣✳ ❈ö t❤➸✿ ✶✳ ●✐ỵ✐ t❤✐➺✉ ✈➲ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣✱ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✱ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣✱ ♥➯✉ ✈➼ ❞ö ✈➔ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣✳ ✷✳ ▼ỉ t↔ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ①➜♣ ①➾ ♥❣❤✐➺♠ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ tr♦♥❣ ❤❛✐ tr÷í♥❣ ❤đ♣✿ ❜➔✐ t♦→♥ ❝➜♣ ữợ t tự ỡ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ t→❝❤✱ ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣✳ ✸✳ ✣÷❛ r❛ ❤❛✐ ✈➼ ❞ö sè ♠✐♥❤ ❤å❛ ❝❤♦ sü ❤ë✐ tö ♠↕♥❤ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣✱ ❝❤÷ì♥❣ tr➻♥❤ t❤ü❝ ♥❣❤✐➺♠ ✤÷đ❝ tỹ ổ ỳ ữợ t tr ❧✉➟♥ ✈➠♥ tr♦♥❣ t÷ì♥❣ ❧❛✐ ✶✳ ◆❣❤✐➯♥ ❝ù✉ ♠ët sè t tỹ t ữủ ổ t ữợ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣✳ ✹✷ ✷✳ ◆❣❤✐➯♥ ❝ù✉ ❝↔✐ t✐➳♥ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❤✐➺♥ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ✈➔ ♠ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥✳ ✹✸ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ❍♦➔♥❣ ❚ö② ✭✷✵✶✽✮✱ ❍➔♠ t❤ü❝ ✈➔ ●✐↔✐ t➼❝❤ ❤➔♠✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❚✐➳♥❣ ❆♥❤ ❬✷❪ ❘✳P✳ ❆❣❛r✇❛❧✱ ❉✳ ❖✬❘❡❣❛♥✱ ❉✳❘✳ ❙❛❤✉ ✭✷✵✵✾✮✱ ▲✐♣s❝❤✐t③✐❛♥✲t②♣❡ ▼❛♣♣✐♥❣s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❢♦r ❙♣r✐♥❣❡r✳ ❬✸❪ ❚✳❱✳ ❆♥❤ ✭✷✵✶✼✮✱ ✏❆ str♦♥❣❧② ❝♦♥✈❡r❣❡♥t s✉❜❣r❛❞✐❡♥t ❡①tr❛❣r❛❞✐❡♥t✲ ❍❛❧♣❡r♥ ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ ❛ ❝❧❛ss ♦❢ ❜✐❧❡✈❡❧ ♣s❡✉❞♦♠♦♥♦t♦♥❡ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✑✱ ❱✐❡t♥❛♠ ❏✳ ▼❛t❤✳✱ ✹✺✱ ♣♣✳ ✸✶✼✕✸✸✷✳ ❬✹❪ ❚✳❱✳ ❆♥❤✱ ▲✳❉✳ ▼✉✉ ✭✷✵✶✻✮✱ ✏❆ ♣r♦❥❡❝t✐♦♥✲❢✐①❡❞ ♣♦✐♥t ♠❡t❤♦❞ ❢♦r ❛ ❝❧❛ss ♦❢ ❜✐❧❡✈❡❧ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ✇✐t❤ s♣❧✐t ❢✐①❡❞ ♣♦✐♥t ❝♦♥str❛✐♥ts✑✱ ♠✐③❛t✐♦♥✱ ❖♣t✐✲ ✻✺✭✻✮✱ ♣♣✳ ✶✷✷✾✕✶✷✹✸✳ ❬✺❪ ❈✳❊✳ ❈❤✐❞✉♠❡ ✭✷✵✵✾✮✱ ❧✐♥❡❛r ✐t❡r❛t✐♦♥s✑ ✱ ✏●❡♦♠❡tr✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❇❛♥❛❝❤ s♣❛❝❡s ❛♥❞ ♥♦♥✲ ❙♣r✐♥❣❡r ❱❡r❧❛❣ ❙❡r✐❡s✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ■❙❇◆ ✾✼✽✲✶✲✽✹✽✽✷✲✶✽✾✲✼✳ ❬✻❪ ■✳❱✳ ❑♦♥♥♦✈✱ ❊✳ ▲❛✐t✐♥❡♥ ✭✷✵✵✷✮✱ ✏❚❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✑✱ ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❙❝✐❡♥❝❡s✱ ❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡✱ ❯♥✐✈❡rs✐t② ♦❢ ❖✉❧✉✱ ■❙❇◆ ✾✺✶✲✹✷✲✻✻✽✽✲✾✳ ❬✼❪ ❏✳▲✳ ▲✐♦♥s✱ ●✳ ❙t❛♠♣❛❝❝❤✐❛ ✭✶✾✻✼✮✱ ✏❱❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✑✱ ❆♣♣❧✳ ▼❛t❤✳✱ ✷✵✱ ♣♣✳ ✹✾✸✕✺✶✾✳ ❈♦♠♠✳ P✉r❡ ✹✹ ❬✽❪ ❉✳❙✳ ▼✐tr✐♥♦✈✐✁❝✱ ❏✳❊✳ P❡✞❝❛r✐✁❝✱ ❆✳▼✳ ❋✐♥❦ ✭✶✾✾✸✮✱ ✏❇❡ss❡❧✬s ■♥❡q✉❛❧✐t②✑✱ ❈❧❛s✲ s✐❝❛❧ ❛♥❞ ◆❡✇ ■♥❡q✉❛❧✐t✐❡s ✐♥ ❆♥❛❧②s✐s✳ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s ✭❊❛st ❊✉r♦♣❡❛♥ ❙❡r✐❡s✮✱ ✈♦❧ ✻✶✳ ❙♣r✐♥❣❡r✱ ❉♦r❞r❡❝❤t✳ ❬✾❪ ▲✳❉✳ ▼✉✉✱ ◆✳❱✳ ◗✉② ✭✷✵✶✺✮✱ ✏❖♥ ❡①✐st❡♥❝❡ ❛♥❞ s♦❧✉t✐♦♥ ♠❡t❤♦❞s ❢♦r str♦♥❣❧② ♣s❡✉❞♦♠♦♥♦t♦♥❡ ❡q✉✐❧✐❜r✐✉♠ ♣r♦❜❧❡♠s✑✱ ❱✐❡t♥❛♠ ❏✳ ▼❛t❤✳✱ ✹✸✱ ♣♣✳ ✷✷✾✕✷✸✽✳ ❬✶✵❪ ●✳ ❙t❛♠♣❛❝❝❤✐❛ ✭✶✾✻✹✮✱ ✏❋♦r♠❡s ❜✐❧✐♥➨❛✐r❡s ❝♦❡r❝✐t✐✈❡s s✉r ❧❡s ❡♥s❡♠❜❧❡s ❝♦♥✈❡①❡s✑✱ ❈✳ ❘✳ ❆❝❛❞✳ ❙❝✐✳ P❛r✐s✱ ✷✺✽✱ ♣♣✳ ✹✹✶✸✕✹✹✶✻✳ ❬✶✶❪ ■✳ ❨❛♠❛❞❛ ✭✷✵✵✶✮✱ ✏❚❤❡ ❤②❜r✐❞ st❡❡♣❡st ❞❡s❝❡♥t ♠❡t❤♦❞ ❢♦r t❤❡ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ♣r♦❜❧❡♠ ♦✈❡r t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ ❢✐①❡❞ ♣♦✐♥t s❡ts ♦❢ ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣s✑✱ ■♥✿ ❇✉t♥❛r✐✉✱ ❉✳✱ ❈❡♥s♦r✱ ❨✳✱ ❘❡✐❝❤✱ ❙✳ ✭❡❞s✳✮ ■♥❤❡r❡♥t❧② P❛r✲ ❛❧❧❡❧ ❆❧❣♦r✐t❤♠s ❢♦r ❋❡❛s✐❜✐❧✐t② ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ❚❤❡✐r ❆♣♣❧✐❝❛t✐♦♥s✱ ❊❧s❡✈✐❡r✱ ❆♠st❡r❞❛♠✱ ♣♣✳ ✹✼✸✲✺✵✹✳