Phương pháp lai ghép giải một lớp bất đẳng thức biến phân (Luận văn thạc sĩ)Phương pháp lai ghép giải một lớp bất đẳng thức biến phân (Luận văn thạc sĩ)Phương pháp lai ghép giải một lớp bất đẳng thức biến phân (Luận văn thạc sĩ)Phương pháp lai ghép giải một lớp bất đẳng thức biến phân (Luận văn thạc sĩ)Phương pháp lai ghép giải một lớp bất đẳng thức biến phân (Luận văn thạc sĩ)Phương pháp lai ghép giải một lớp bất đẳng thức biến phân (Luận văn thạc sĩ)Phương pháp lai ghép giải một lớp bất đẳng thức biến phân (Luận văn thạc sĩ)Phương pháp lai ghép giải một lớp bất đẳng thức biến phân (Luận văn thạc sĩ)Phương pháp lai ghép giải một lớp bất đẳng thức biến phân (Luận văn thạc sĩ)Phương pháp lai ghép giải một lớp bất đẳng thức biến phân (Luận văn thạc sĩ)Phương pháp lai ghép giải một lớp bất đẳng thức biến phân (Luận văn thạc sĩ)Phương pháp lai ghép giải một lớp bất đẳng thức biến phân (Luận văn thạc sĩ)Phương pháp lai ghép giải một lớp bất đẳng thức biến phân (Luận văn thạc sĩ)
✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ❍❖⑨◆● ❚❍➚ ❍❾❯ P❍×❒◆● P❍⑩P ▲❆■ ●❍➆P ●■❷■ ▼❐❚ ▲❰P ❇❻❚ ✣➃◆● ❚❍Ù❈ ❇■➌◆ P❍❹◆ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆✱ ✺✴✷✵✶✾ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ❍❖⑨◆● ❚❍➚ ❍❾❯ P❍×❒◆● P❍⑩P ▲❆■ ●❍➆P ●■❷■ ▼❐❚ ▲❰P ❇❻❚ ✣➃◆● ❚❍Ù❈ ❇■➌◆ P❍❹◆ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ö♥❣ ▼➣ sè✿ ✽✹✻✵✶✶✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ●■⑩❖ ❱■➊◆ ❍×❰◆● ❉❼◆ P●❙✳❚❙✳ ◆●❯❨➍◆ ❚❍➚ ế ử ỵ ữỡ ởt ợ t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✹ ✶✳✶ ✶✳✷ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ỗ ✳ ✳ ✳ ✺ ✶✳✶✳✷ P❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✼ ✶✳✶✳✸ ⑩♥❤ ①↕ ❧♦↕✐ ✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✽ ❇➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❧♦↕✐ ✤ì♥ ✤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✷✳✶ ❇➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✤ì♥ ✤✐➺✉ ✶✳✷✳✷ ❇➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ j ✲✤ì♥ ✤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ❈❤÷ì♥❣ ✷ P❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ❣✐↔✐ ♠ët ❧ỵ♣ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✶✺ ✷✳✶ ✷✳✷ P❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✶✳✶ ❇➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✶✳✷ P❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ❙ü ❤ë✐ tö ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷✳✶ ❙ü ❤ë✐ tư ❝õ❛ P❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✳✷✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷✳✷ ❙ü ❤ë✐ tư ❝õ❛ P❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✳✸✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✷✳✸ ❙ü ❤ë✐ tư ❝õ❛ P❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✳✹✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✷✳✹ ❱➼ ❞ö ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✐✈ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✸✻ ✸✼ ỵ H ổ rt tỹ E ổ ❣✐❛♥ ❇❛♥❛❝❤ E∗ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝õ❛ E SE ♠➦t ❝➛✉ ✤ì♥ ✈à ❝õ❛ E R t➟♣ ❝→❝ sè t❤ü❝ R+ t➟♣ ❝→❝ sè t❤ü❝ ❦❤æ♥❣ ➙♠ ∅ t➟♣ ré♥❣ ∀x ✈ỵ✐ ♠å✐ x D(F ) ♠✐➲♥ ①→❝ ✤à♥❤ ❝õ❛ →♥❤ ①↕ F R(F ) ♠✐➲♥ ❣✐→ trà ❝õ❛ →♥❤ ①↕ F F −1 →♥❤ ①↕ ♥❣÷đ❝ ❝õ❛ →♥❤ F I ỗ t lp , ≤ p < ∞ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② sè ❦❤↔ tê♥❣ ❜➟❝ p l∞ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② sè ❜à ❝❤➦♥ Lp [a, b], ≤ p < ∞ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ t➼❝❤ ❜➟❝ p tr➯♥ ✤♦↕♥ [a, b] d(x, C) ❦❤♦↔♥❣ ❝→❝❤ tø ♣❤➛♥ tû x ✤➳♥ t➟♣ ❤đ♣ C lim supn→∞ xn ❣✐ỵ✐ ❤↕♥ tr➯♥ ❝õ❛ ❞➣② sè {xn } lim inf n→∞ xn ❣✐ỵ✐ ❤↕♥ ữợ số {xn } xn x0 {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ x0 xn ❞➣② {xn } ❤ë✐ tö ②➳✉ ✈➲ x0 x0 J →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ j →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ✤ì♥ trà ❋✐①(T ) t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ✷ ▼ð ✤➛✉ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỉ ❤↕♥ ❝❤✐➲✉ ✤÷đ❝ ♥❤➔ t♦→♥ ❤å❝ ♥❣÷í✐ ■t❛❧✐❛ ❧➔ ❙t❛♠♣❛❝❝❤✐❛ ✭①❡♠ ❬✶✷❪✮ ✈➔ ỗ sỹ ữ r t ỳ ♥➠♠ ✤➛✉ ❝õ❛ t❤➟♣ ♥✐➯♥ ✻✵ t❤➳ ❦➾ ❳❳ tr♦♥❣ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❜➔✐ t♦→♥ ❜✐➯♥ tü ❞♦✳ ❇➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❝â ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ự t ỵ tt t tố ÷✉✱ ❜➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥✱ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣✱ ❜➔✐ t♦→♥ ❜ò✱ ❜➔✐ t♦→♥ ❣✐→ trà ❜✐➯♥ ✈✳✈ ✳ ✳ ✳ ❉♦ ✤â✱ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✤❛♥❣ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ✤➲ t➔✐ t❤✉ ❤ót ✤÷đ❝ sü q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ t tr ữợ ✤➣ ♥❤➟♥ ✤÷đ❝ ♥❤✐➲✉ ❦➳t q✉↔ ❤❛②✱ s➙✉ s➢❝✳ ❇➯♥ ❝↕♥❤ ✤â✱ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❝á♥ ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ ❝❤♦ ❝→❝ ❜➔✐ t♦→♥ t❤ü❝ t➳ ♥❤÷ ♠ỉ ❤➻♥❤ ❝➙♥ ❜➡♥❣ tr♦♥❣ ❦✐♥❤ t➳✱ ❣✐❛♦ t❤æ♥❣✱ ❜➔✐ t♦→♥ ❦❤ỉ✐ ♣❤ư❝ t➼♥ ❤✐➺✉✱ ❜➔✐ t♦→♥ ❝ỉ♥❣ ♥❣❤➺ ❧å❝ ❦❤ỉ♥❣ ❣✐❛♥✱ ❜➔✐ t♦→♥ ♣❤➙♥ ♣❤è✐ ❜➠♥❣ t❤æ♥❣ ✈✳✈ ✳ ✳ ✳ ✭①❡♠ ❬✽❪✱ ❬✶✵❪✱ ❬✶✶❪✮✳ ❈❤♦ ✤➳♥ ♥❛② ✈➝♥ ❝á♥ ♥❤✐➲✉ ✈➜♥ ✤➲ ♠ỵ✐ ✈➔ ❦❤â ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ữủ q t ự ợ ỳ ổ t ởt tr ỳ ữợ ♥❣❤✐➯♥ ❝ù✉ ✤❛♥❣ ✤÷đ❝ q✉❛♥ t➙♠ ❧➔ ①➙② ❞ü♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈ỵ✐ t➟♣ r➔♥❣ ❜✉ë❝ ❧➔ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✱ t➟♣ ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å →♥❤ ①↕ ❧♦↕✐ j ✲✤ì♥ ✤✐➺✉✱ t➟♣ ♥❣❤✐➺♠ ❝❤✉♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣✱ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✱ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ▼ư❝ t✐➯✉ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ❣✐↔✐ ♠ët ❧ỵ♣ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❜➔✐ ❜→♦ ❬✺❪✱ ❬✻❪ ✈➔ ✸ ❬✾❪ ❝õ❛ ◆❣✉②➵♥ ❇÷í♥❣✱ ◆❣✉②➵♥ ❙♦♥❣ ❍➔ ✈➔ ◆❣✉②➵♥ ❚❤à ❚❤✉ ❚❤õ② ❝ỉ♥❣ ❜è ♥➠♠ ✷✵✶✻✱ ✷✵✶✼ ✈➔ ✷✵✶✽✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ ✧▼ët ❧ỵ♣ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✧✳ ữỡ ợ t ởt số tự ỡ ổ ỗ ỡ j ỡ tr ổ ỗ t❤í✐✱ tr➻♥❤ ❜➔② ✈➲ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✤ì♥ ✤✐➺✉ ✈➔ j ✲✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ữỡ Pữỡ ởt ợ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✧✳ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❜❛ ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ j ✲✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❝ò♥❣ ỵ tử ữỡ P❤➛♥ ❝✉è✐ ❝õ❛ ❝❤÷ì♥❣ ❧➔ ♠ët ✈➼ ❞ư ♠✐♥❤ ❤å❛ tr ỵ tử ♠↕♥❤ ✤÷đ❝ t❤ä❛ ♠➣♥✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ❚r♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ✤➸ t→❝ ❣✐↔ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉✳ ❚→❝ ❣✐↔ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❝→❝ t❤➛②✱ ❝æ tr♦♥❣ ❑❤♦❛ ❚♦→♥ ✲ ❚✐♥✱ tr♦♥❣ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ✣➦❝ ❜✐➺t✱ t→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ P●❙✳❚❙✳ ◆❣✉②➵♥ ❚❤à ❚❤✉ ❚❤õ② ✲ ữớ t t ữợ t t ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ✤÷đ❝ ❣û✐ ❧í✐ ỡ tợ trữớ P ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ✤➸ t→❝ ❣✐↔ ✤÷đ❝ t❤❛♠ ❣✐❛ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✾ ❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥ ❍♦➔♥❣ ❚❤à ❍➟✉ ✹ ữỡ ởt ợ t tự tr ổ ữỡ ợ t ởt số t❤ù❝ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✤ì♥ ✤✐➺✉✱ j ✲✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❈ư t❤➸ ▼ư❝ ✶✳✶ tr➻♥❤ ❜➔② ✈➲ ❦❤ỉ♥❣ ỗ tr tr ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ →♥❤ ①↕ ✤ì♥ ✤✐➺✉✱ →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ▼ư❝ ✶✳✷ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ✈➔ ✈➼ ❞ö ✈➲ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✤ì♥ ✤✐➺✉ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ j ✲✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ ✈✐➳t tr➯♥ ❝ì sð tê♥❣ ❤đ♣ ❦✐➳♥ t❤ù❝ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪✱ ❬✸❪ ✈➔ ❬✼❪✳ ✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❈❤♦ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ỵ E ổ ố ❝õ❛ E ✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② t❛ sû ❞ö♥❣ ỵ ổ E ✈➔ E ∗ ✳ ❱ỵ✐ ♠é✐ x ∈ E ✈➔ x∗ ∈ E ∗ t❛ ✈✐➳t x∗ , x ❤♦➦❝ x, x∗ ✭t➼❝❤ ✤è✐ ♥❣➝✉✮ t❤❛② ❝❤♦ x∗ (x)✳ ◆➳✉ E = H ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤➻ t➼❝❤ ố t ổ ữợ , ❝↔♠ s✐♥❤ ❝❤✉➞♥ t÷ì♥❣ ù♥❣ ✳ ✺ ✶✳✶✳✶ ❑❤ỉ♥❣ ỗ ❬✶❪✮ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✤÷đ❝ ❣å✐ ❧➔ ♣❤↔♥ ①↕✱ ♥➳✉ ✈ỵ✐ ♠å✐ ♣❤➛♥ tû x∗∗ ∈ E ∗∗ ✱ ❦❤ỉ♥❣ ❣✐❛♥ ❧✐➯♥ ❤đ♣ t❤ù ❤❛✐ ❝õ❛ E ✱ ✤➲✉ tỗ t tỷ x E s x (x) = x∗∗ (x∗ ) ∀x∗ ∈ E ∗ ỵ E ổ ❇❛♥❛❝❤✳ ❑❤✐ ✤â✱ ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ (i) E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕✳ (ii) ▼å✐ ❞➣② ❜à ❝❤➦♥ tr♦♥❣ E ✤➲✉ ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tư ②➳✉✳ ❱➼ ❞ư ✶✳✶✳✸ ❈→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉✱ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✱ ❦❤æ♥❣ ❣✐❛♥ lp ✱ Lp [a, b], < p < ∞ ổ ỵ SE := {x ∈ E : x = 1} ❧➔ ♠➦t ❝➛✉ ✤ì♥ ✈à ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹ ✭①❡♠ ❬✷❪✮ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✤÷đ❝ ❣å✐ (i) ỗ t ợ x, y ∈ SE ✱ x = y ✱ s✉② r❛ (1 − λ)x + λy < ∀λ ∈ (0, 1); (ii) ỗ ợ (0, 2] ✈➔ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ x ≤ 1, y ≤ x y tọ t tỗ t↕✐ δ = δ(ε) > s❛♦ ❝❤♦ (x + y) ≤ − δ; (iii) trì♥ ♥➳✉ ❣✐ỵ✐ lim t0 tỗ t ợ x, y SE ✳ ❱➼ ❞ö ✶✳✶✳✺ x + ty − x t ✻ (i) ❑❤ỉ♥❣ ❣✐❛♥ E = Rn ✈ỵ✐ ❝❤✉➞♥ x n x ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ 1/2 x2i = , x = (x1 , x2 , , xn ) ∈ Rn i=1 ❧➔ ❦❤æ♥❣ ỗ t ổ E = Rn , n ≥ ✈ỵ✐ ❝❤✉➞♥ x ①→❝ ✤à♥❤ ❜ð✐ x = |x1 | + |x2 | + + |xn |, x = (x1 , x2 , , xn ) ∈ Rn ❦❤æ♥❣ ♣❤↔✐ ổ ỗ t (ii) ổ rt H ổ ỗ ổ ỗ ổ ỗ t (i) ❈❤✉➞♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ●➙t❡❛✉① ♥➳✉ ✈ỵ✐ ♠é✐ y ∈ SE ❣✐ỵ✐ ❤↕♥ lim t0 x + ty x t tỗ t ợ x SE ỵ y, x ❑❤✐ ✤â ✭✶✳✶✮ x ✤÷đ❝ ❣å✐ ❧➔ ✤↕♦ ❤➔♠ ●➙t❡❛✉① ❝õ❛ ❝❤✉➞♥✳ (ii) ❈❤✉➞♥ ❝õ❛ E ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉ ♥➳✉ ✈ỵ✐ ♠é✐ y ∈ SE ✱ ợ t ữủ ợ x SE ✳ (iii) ❈❤✉➞♥ ❝õ❛ E ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ♥➳✉ ✈ỵ✐ ♠é✐ x ∈ SE ✱ ❣✐ỵ✐ tỗ t ợ y SE ✳ (iv) ❈❤✉➞♥ ❝õ❛ E ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ rt ợ tỗ t ợ ♠å✐ x, y ∈ SE ✳ ❱➼ ❞ư ✶✳✶✳✽ ❑❤ỉ♥❣ ●➙t❡❛✉① ✈ỵ✐ x ❣✐❛♥ ❍✐❧❜❡rt H ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝â ❝❤✉➞♥ ❦❤↔ ✈✐ = x/ x , x = 0✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠é✐ x ∈ H ✈ỵ✐ ✷✹ ❱➻ ∞ k=0 λk = ∞✱ ∞ k=0 bk = ∞✳ ❚ø ❇ê ✤➲ ✷✳✷✳✶ ✈ỵ✐ ak = xk − p∗ ✱ ✭✷✳✶✶✮ ✈➔ t➼♥❤ ❧✐➯♥ tö❝ ②➳✉ s❛♦ ❝õ❛ j ✈ỵ✐ ✭✷✳✶✹✮✱ s✉② r❛ lim xk − p∗ k→∞ = ỵ ữủ ự ỹ ❤ë✐ tư ❝õ❛ P❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✳✸✮ ❇ê ✤➲ ✷✳✷✳✼ ✭①❡♠ ❬✻❪✮ ❈❤♦ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕ t❤ü❝✱ ỗ t õ t F : E → E ❧➔ →♥❤ ①↕ η ✲j ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ γ ✲❣✐↔ ❝♦ ❝❤➦t ✈ỵ✐ η + γ > 1✳ ❈❤♦ S˜ ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✽✮✳ ❑❤✐ ✤â✱ ♥➳✉ ❞➣② {xk }∞ k=1 ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✼✮ ❜à ❝❤➦♥ t❤➻ lim sup F (x∗ ), j(x∗ − xk ) ≤ ✭✷✳✶✺✮ k→∞ ð ✤➙② x∗ ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✮✳ ❙ü ❤ë✐ tö ♠↕♥❤ Pữỡ ữủ tr tr ỵ ữợ ỵ E ổ tỹ ỗ t õ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉✱ F : E → E ❧➔ →♥❤ ①↕ η ✲j ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ γ ✲❣✐↔ ❝♦ ❝❤➦t ✈ỵ✐ η + γ > 1✱ {Ti }∞ i=1 ❧➔ ♠ët ❤å ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr➯♥ E ✈ỵ✐ C := ∩∞ i=1 ❋✐①(Ti ) = ∅✳ ●✐↔ sû λk ∈ (0, 1) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭▲✶✮✱ ✭▲✷✮ ✈➔ {si}∞i=1 ❧➔ ❞➣② sè t❤ü❝ ❞÷ì♥❣ ❣✐↔♠ ♥❣➦t✱ ❤ë✐ tư ✈➲ 0✳ ❑❤✐ ✤â✱ ❞➣② {xk }∞ k=1 ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✼✮ ❤ë✐ tư ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t x∗ ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❱■∗ (F, C) ✭✷✳✶✮ k ự ữợ ự ❞➣② {xk }∞k=1 ❜à ❝❤➦♥✳ ❱➻ S˜k ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr➯♥ E ✈➔ ❋✐①(T i ) = ❋✐①(Ti ) s✉② r❛ S˜k p = p ∞ ✈ỵ✐ ❜➜t ❦ý ✤✐➸♠ p ∈ k ❋✐①(Ti ) ⊆ i=1 ❋✐①(Ti ) ✈ỵ✐ ♠é✐ k ≥ 1✳ ❉♦ ✤â✱ sû i=1 ✷✺ ❞ö♥❣ ▼➺♥❤ ✤➲ ✶✳✷✳✺✱ t❛ ❝â ✤→♥❤ ❣✐→ s❛✉✿ xk+1 − p = (I − λk F )S˜k (xk ) − p ≤ (1 − λk τ ) xk − p + (I − λk F )S˜k (p) − S˜k (p) ≤ (1 − λk τ ) xk − p + λk F (p) = (1 − λk τ ) xk − p + λk τ F (p) τ ≤ max { x1 − p , F (p) } τ ❚ø ✤➙② s✉② r❛ ❞➣② {xk }k≥1 ữợ ự xk+1 xk ❦❤✐ k → ∞✳ ❱➻ ❞➣② {xk } ❜à ❝❤➦♥ ♥➯♥ ❝→❝ ❞➣② {S˜k (xk )}k≥1 , {S˜k+1 (xk )}k≥1 , {F S˜k (xk )}k≥1 , {Ti (xk )}k≥1 , {T i (xk )}k≥1 ❝ơ♥❣ ❜à ❝❤➦♥ ✈ỵ✐ ♠å✐ i ≥ ✈➔ ❝→❝ ❞➣② {S˜k,−1 (xk )}k≥1 ✱ {S˜k+1,−1 (xk )}k≥1 ❝ô♥❣ ❜à ❝❤➦♥✱ ð ✤➙② S˜k,−1 (x) = s1 − sk k (si−1 − si )T i (x) i=2 ❑❤æ♥❣ ❧➔♠ ♠➜t t➼♥❤ tê♥❣ q✉→t t❛ ❣✐↔ sû ❝→❝ ❞➣② tr➯♥ ❜à ❝❤➦♥ ❜ð✐ sè t❤ü❝ ❞÷ì♥❣ M1 ✳ ❍ì♥ ♥ú❛ tø ✭✷✳✼✮ ✈➔ ✭✷✳✽✮✱ t❛ ❝â t❤➸ ✈✐➳t xk+1 = λk (I − F )S˜k (xk ) + (1 − λk )S˜k (xk ) (s0 − s1 )(1 − α1 ) = λk (I − F )S˜k (xk ) + (1 − λk ) xk s0 − sk (s0 − s1 )α1 s1 − sk ˜ + T1 (xk ) + Sk,−1 (xk ) s0 − sk s0 − sk = hk xk + (1 − hk )zk ✈ỵ✐ (1 − λk )(s0 − s1 )(1 − α1 ) , s0 − sk λk (I − F )S˜k (xk ) (1 − λk )(s0 − s1 )α1 zk = + T1 (xk ) − hk (1 − hk )(s0 − sk ) (1 − λk )(s1 − sk ) ˜ + Sk,−1 (xk ) (1 − hk )(s0 − sk ) hk = ✭✷✳✶✻✮ ✷✻ ❱➻ zk+1 − zk = C1 + (s0 − s1 )α1 C2 + C3 ✱ tr♦♥❣ ✤â λk+1 (I − F )S˜k+1 (xk+1 ) λk (I − F )S˜k (xk ) − − hk+1 − hk λk+1 = (I − F )S˜k+1 (xk+1 ) − (I − F )S˜k+1 (xk ) − hk+1 λk+1 + (I − F )S˜k+1 (xk ) − (I − F )S˜k (xk ) − hk+1 λk+1 λk + − (I − F )S˜k (xk ), − hk+1 − hk − λk+1 − λk C2 : = T1 (xk+1 ) − T1 (xk ) (1 − hk+1 )(s0 − sk+1 ) (s0 − sk )(1 − hk ) − λk+1 T1 (xk+1 ) − T1 (xk ) = (1 − hk+1 )(s0 − sk+1 ) (1 − λk+1 ) (1 − λk ) + − T1 (xk ), (1 − hk+1 )(s0 − sk+1 ) (1 − hk )(s0 − sk ) (1 − λk )(s1 − sk ) ˜ (1 − λk+1 )(s1 − sk+1 ) ˜ Sk+1,−1 (xk+1 ) − Sk,−1 (xk ) C3 : = (1 − hk+1 )(s0 − sk+1 ) (1 − hk )(s0 − sk ) (1 − λk+1 )(s1 − sk+1 ) ˜ = Sk+1,−1 (xk+1 ) − S˜k+1,−1 (xk ) (1 − hk+1 )(s0 − sk+1 ) (1 − λk+1 )(s1 − sk+1 ) ˜ + Sk+1,−1 (xk ) − S˜k,−1 (xk ) (1 − hk+1 )(s0 − sk+1 ) (1 − λk+1 )(s1 − sk+1 ) (1 − λk )(s1 − sk ) ˜ − + Sk,−1 (xk ), (1 − hk+1 )(s0 − sk+1 ) (1 − hk )(s0 − sk ) C1 : = S˜k+1,−1 (x) ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr➯♥ E ✱ ✈➔ s1 − sk+1 S˜k+1,−1 (xk ) − S˜k,−1 (xk ) = − s1 − sk = ≤ k+1 (si−1 − si )T i (xk ) k=2 k (si−1 − si )T i (xk ) k=2 1 − s1 − sk+1 s1 − sk k (si−1 − si )T i (xk ) + k=2 sk − sk+1 k+1 T (xk ) s1 − sk+1 −sk + sk+1 sk − sk+1 2M1 (sk − sk+1 ) (s1 − sk )M1 + , M1 = (s1 − sk+1 )(s1 − sk ) s1 − sk+1 (s1 − sk+1 ) ✷✼ t❛ ❝â zk+1 − zk ≤ C1 + (s0 − s1 )α1 C2 + C3 ≤ λk+1 4M1 τ1 xk+1 − xk + − hk+1 τ1 λk+1 λk + − 2M1 − hk+1 − hk (1 − λk+1 )(s0 − s1 )α1 xk+1 − xk + (1 − hk+1 )(s0 − sk+1 ) (1 − λk ) (1 − λk+1 ) + (s0 − s1 )α1 − M1 (1 − hk+1 )(s0 − sk+1 ) (1 − hk )(s0 − sk ) (1 − λk+1 )(s1 − sk+1 ) + xk+1 − xk (1 − hk+1 )(s0 − sk+1 ) (1 − λk+1 )(s1 − sk+1 )2M1 (sk − sk+1 ) + (1 − hk+1 )(s0 − sk+1 )(s1 − sk+1 ) (1 − λk+1 )(s1 − sk+1 ) (1 − λk )(s1 − sk ) − + M1 , (1 − hk+1 )(s0 − sk+1 ) (1 − hk )(s0 − sk ) tr♦♥❣ ✤â τ1 = (1 − η)/γ ✳ ❱➻ ✈➟②✱ λk+1 (1 − τ1 ) (1 − λk+1 )(s0 − s1 )α1 + − hk+1 (1 − hk+1 )(s0 − sk+1 ) (1 − λk+1 )(s1 − sk+1 ) − xk+1 − xk + ck + (1 − hk+1 )(s0 − sk+1 ) ✭✷✳✶✼✮ zk+1 − zk − xk+1 − xk ≤ ✈ỵ✐ ck = 4M1 λk+1 λk+1 λk + − 2M1 − hk+1 − hk+1 − hk (1 − λk ) (1 − λk+1 ) − M1 + (s0 − s1 )α1 (1 − hk+1 )(s0 − sk+1 ) (1 − hk )(s0 − sk ) (1 − λk+1 )(s1 − sk+1 )2M1 (sk − sk+1 ) + (1 − hk+1 )(s0 − sk+1 )(s1 − sk+1 ) (1 − λk+1 )(s1 − sk+1 ) (1 − λk )(s1 − sk ) + − M1 (1 − hk+1 )(s0 − sk+1 ) (1 − hk )(s0 − sk ) ❘ã r➔♥❣✱ sk → 0✱ − hk → [s0 − (s0 − s1 )(1 − α1 )]/s0 ❦❤✐ k → ∞✱ ✈➔ ❞♦ ✷✽ ✤â✱ ck → ❦❤✐ k → ∞✳ ❍ì♥ ♥ú❛✱ (1 − λk+1 )(s0 − s1 )α1 (1 − λk+1 )(s1 − sk+1 ) lim + −1 = k→∞ (1 − hk+1 )(s0 − sk+1 ) (1 − hk+1 )(s0 − sk+1 ) (s0 − s1 )α1 + s1 − = = s0 − (s0 − s1 )(1 − α1 ) ❉♦ ✤â tø ✭✷✳✶✼✮ s✉② r❛ lim sup zk+1 − zk − xk+1 − xk ≤ k→∞ ❚ø ❇ê ✤➲ ✷✳✷✳✷✱ lim xk − zk = 0, k→∞ ❦➳t ❤đ♣ ✈ỵ✐ ✭✷✳✶✻✮ s✉② r❛ lim xk+1 − xk = lim − k→∞ k→∞ (1 − λk )(s0 − s1 )(1 − α1 ) s0 − sk xk zk = ữợ ự limk xk = x∗✳ ⑩♣ ❞ö♥❣ ▼➺♥❤ ✤➲ ✶✳✷✳✺ t❛ ✤→♥❤ ❣✐→ xk+1 − x∗ xk+1 − x∗ = (I − λk F )S˜k (xk ) − x∗ ♥❤÷ s❛✉✿ = (I − λk F )S˜k (xk ) − (I − λk F )S˜k (x∗ ) − λk F (x∗ ) ≤ (1 − λk τ ) xk − x∗ − 2λk F (x∗ ), j(xk+1 − x∗ ) = (1 − λk τ ) xk − x∗ + 2λk [ F (x∗ ), j(x∗ − xk ) + F (x∗ ), j(x∗ − xk+1 ) − j(x∗ − xk ) ] = (1 − bk ) xk − x∗ + b k ck , ð ✤➙② bk = λk τ, ck = 2[ F (x∗ ), j(x∗ − xk ) + F (x∗ ), j(x∗ − xk+1 ) − j(x∗ − xk ) ]/τ ∞ ❱➻ ∞ λk = ∞ ✱ k=0 bk = ∞✱ ♥➯♥ tø ❇ê ✤➲ ✷✳✷✳✶ ✈ỵ✐ ak = xk − x∗ ✱ k=0 ✭✷✳✶✺✮ ✈➔ t➼♥❤ ❧✐➯♥ tư❝ s❛♦ ②➳✉ ❝õ❛ ❝❤✉➞♥ ❝õ❛ j ✈ỵ✐ ✭✷✳✶✽✮✱ s✉② r❛ lim xk − x∗ k→∞ = ỵ ữủ ự ỹ tử ❝õ❛ P❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✳✹✮ ❇ê ✤➲ ✷✳✷✳✾ ✭①❡♠ ❬✾❪✮ ❈❤♦ E ổ tỹ ỗ t ❝â ❝❤✉➞♥ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉✱ F : E → E ❧➔ →♥❤ ①↕ η ✲j ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ γ ✲❣✐↔ ❝♦ ❝❤➦t ✈ỵ✐ η + γ > 1✱ {Ti }∞ i=1 ❧➔ ♠ët ❤å ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr➯♥ E ✈ỵ✐ C := ∩∞ i=1 ❋✐①(Ti ) = ∅✳ ✣à♥❤ ♥❣❤➽❛ T := s˜k k k si Ti , k ≥ 1, i=1 k ð ✤➙② si ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ tr♦♥❣ ✭✷✳✻✮ ✈➔ s˜k = si ✳ ❑❤✐ ✤â✿ i=1 (i) ỗ t ởt ổ T : E → E ①→❝ ✤à♥❤ ❜ð✐ T (x) = lim T (x) = k→∞ s˜ ∞ k ∞ ✈ỵ✐ x∈E i=1 ∞ i Fix(T ) = i=1 si Ti (x), Fix(Ti ) = Fix(T ) i=1 (ii) ❱ỵ✐ ♠å✐ t➟♣ ❜à ❝❤➦♥ B tr♦♥❣ E ✱ t❛ ❝â lim sup T k (x) − T (x) = k→∞ x∈B (iii) ❱ỵ✐ α ∈ (0, 1)✱ ①→❝ ✤à♥❤ S := αI + (1 − α)T ◆➳✉ ❞➣② {xk }∞ k=1 tr♦♥❣ E ❜à ❝❤➦♥ ✈➔ lim xk − S(xk ) = 0✱ t❤➻ k→∞ lim sup F (p∗ ), j(p∗ − xk ) ≤ 0, ✭✷✳✶✾✮ k→∞ ð ✤➙② p∗ ∈ C ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✮✳ ❙ü ❤ë✐ tư ♠↕♥❤ ❝õ❛ P❤÷ì♥❣ ♣❤→♣ ✷✳✶✳✹ ✤÷đ❝ tr➻♥❤ ❜➔② tr ỵ ữợ ỵ E ổ tỹ ỗ ❝❤➦t ❝â ❝❤✉➞♥ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉✱ F : E → E ❧➔ →♥❤ ①↕ η ✲j ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ γ ✲❣✐↔ ❝♦ ❝❤➦t ✈ỵ✐ η + γ > 1✱ {Ti }∞ i=1 ❧➔ ♠ët ❤å ✈æ ❤↕♥ ✸✵ ữủ ổ tr E ợ C := ∩∞ i=1 ❋✐①(Ti ) = ∅✳ ●✐↔ sû λk ∈ (0, 1) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭▲✶✮✱ ✭▲✷✮ ✈➔ {si}∞i=1 t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✷✳✻✮✳ ❑❤✐ ✤â✱ ❞➣② {xk }∞ k=1 ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✾✮ ❤ë✐ tö ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t x∗ ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❱■∗ (F, C) ✭✷✳✶✮ ❦❤✐ k ự ữợ ự {xk }∞k=1 ❜à ❝❤➦♥✳ ❱ỵ✐ ❜➜t ❦ý p ∈ C ✱ tø ✭✷✳✾✮ ✈➔ ▼➺♥❤ ✤➲ ✶✳✷✳✺✱ t❛ ❝â ✤→♥❤ ❣✐→✿ xk+1 − p ≤ (I − λk F )S k (xk ) − S k (p) ≤ (1 − λk τ ) xk − p + (I − λk F )S k (p) − S k (p) = (1 − λk τ ) xk − p + λk F (p) = (1 − λk τ ) xk − p + λk τ F (p) /τ ≤ max { x1 − p , F (p) /τ }, ð ✤➙② S k = αI + (1 − α)T k ✈ỵ✐ T k ①→❝ ✤à♥❤ ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✷✳✷✳✾ ✈➔ τ =1− (1 − η)/γ ✳ ❉♦ ✤â ❞➣② {xk } ❜à ữợ ự xk+1 xk ❦❤✐ k → ∞✳ ❱➻ ❞➣② {xk }∞ k=1 ❜à ❝❤➦♥ ♥➯♥ ❝→❝ ❞➣② {S k (xk )}, {S k+1 (xk )}, {Ti (xk )} i ≥ 1, ❝ô♥❣ ❜à ❝❤➦♥ ✈➔ ❝→❝ ❞➣② {F S k (xk )}✱ {F S k+1 (xk )} ❝ơ♥❣ ❜à ❝❤➦♥✳ ❑❤ỉ♥❣ ❧➔♠ ♠➜t t➼♥❤ tê♥❣ q✉→t t❛ ❣✐↔ sû ❝→❝ ❞➣② ♥➔② ❜à ❝❤➦♥ ❜ð✐ ♠ët ❤➡♥❣ sè ❞÷ì♥❣ M1 ✳ ❚❤➯♠ ♥ú❛✱ tø ✭✷✳✾✮✱ t❛ ❝â t❤➸ ✈✐➳t xk+1 = λk (I − F )S k (xk ) + (1 − λk )S k (xk ) = λk (I − F )S k (xk ) + (1 − λk ) αxk + (1 − α)T k (xk ) = hk xk + (1 − hk )zk , ð ✤➙② hk = (1 − λk )α ✈➔ λk (I − F )S k (xk ) (1 − λk )(1 − α)T k (xk ) zk = + − hk − hk ✭✷✳✷✵✮ ✸✶ ❱➻ zk+1 − zk = C1 + (1 − α)C2 ✱ ð ✤➙② λk+1 (I − F )S k+1 (xk+1 ) λk (I − F )S k (xk ) C1 : = − − hk+1 − hk λk+1 = (I − F )S k+1 (xk+1 ) − (I − F )S k+1 (xk ) − hk+1 λk+1 + (I − F )S k+1 (xk ) − (I − F )S k (xk ) − hk+1 λk+1 λk + − (I − F )S k (xk ), − hk+1 − hk (1 − λk+1 )T k+1 (xk+1 ) (1 − λk )T k (xk ) − C2 : = − hk+1 − hk − λk+1 k+1 = T (xk+1 ) − T k+1 (xk ) − hk+1 − λk+1 k+1 + T (xk ) − T k (xk ) − hk+1 − λk+1 − λk k T (xk ), − + − hk+1 − hk ✈➔ T k+1 k (xk ) − T (xk ) = k+1 s˜k+1 i=1 si Ti (xk ) − s˜k k si Ti (xk ) i=1 k sk+1 si Ti (xk ) + Tk+1 (xk ) s˜k+1 s˜k i=1 s˜k+1 sk+1 sk+1 sk+1 ≤ M1 + M1 = M1 , s˜k+1 s˜k+1 s˜k+1 = − t❛ ❝â zk+1 − zk ≤ C1 + C2 (1 − α) λk+1 τ1 λk+1 λk xk+1 − xk + 2M1 + − 2M1 − hk+1 − hk+1 − hk (1 − λk+1 )(1 − α) sk+1 + xk+1 − xk + M1 − hk+1 s˜k+1 − λk+1 − λk + − M1 (1 − α) − hk+1 − hk λk+1 τ1 (1 − λk+1 )(1 − α) ≤ + xk+1 − xk + ck , − hk+1 − hk+1 ≤ ✸✷ ð ✤➙② τ1 = (1 − η)/γ ✈➔ ck = M1 ✣➦t c˜k := ✤÷đ❝ λk+1 τ1 λk λk+1 +2 − − hk+1 − hk+1 − hk sk+1 (1 − λk+1 )(1 − α) − λk+1 − λk + − +2 (1 − α) s˜k+1 − hk+1 − hk+1 − hk λk+1 τ1 (1 − λk+1 )(1 − α) + −1 − hk+1 − hk+1 xk+1 − xk + ck ✳ ❚❛ ♥❤➟♥ zk+1 − zk ≤ xk+1 − xk + c˜k ❘ã r➔♥❣ tø ✤✐➲✉ ❦✐➺♥ ✭▲✶✮ ✈➔ ✭✷✳✻✮✱ t❛ ❝â hk → α✱ sk → ✈➔ s˜k → s˜ > ❦❤✐ k → ∞✳ ❉♦ ✤â✱ lim k→∞ 1−α λk+1 τ1 (1 − λk+1 )(1 − α) + −1 = −1=0 − hk+1 − hk+1 1−α ✈➔ λk+1 τ1 λk λk+1 +2 − k→∞ − hk+1 − hk+1 − hk − λk+1 − λk sk+1 (1 − λk+1 )(1 − α) + − +2 (1 − α) s˜k+1 − hk+1 − hk+1 − hk 1 = M1 − (1 − α) = 1−α 1−α lim ck = M1 lim k→∞ ●✐ỵ✐ ❤↕♥ tr➯♥ ✈➔ t➼♥❤ ❜à ❝❤➦♥ ❝õ❛ ❞➣② {xk } ✤↔♠ ❜↔♦ r➡♥❣ c˜k → ❦❤✐ k → ∞✳ ❉♦ ✤â✱ lim sup zk+1 − zk − xk+1 − xk ≤ k→∞ ❚ø ❇ê ✤➲ ✷✳✷✳✷✱ t❛ ❝â lim xk − zk = k→∞ ✭✷✳✷✶✮ ❚ø ✭✷✳✾✮ ✈➔ ✭✷✳✷✵✮ ✈ỵ✐ ✭✷✳✷✶✮ s✉② r❛ S k (xk ) − xk+1 ≤ λk M1 → 0✱ ❦❤✐ k → ∞ ✈➔ lim xk+1 − xk = lim − (1 − λk ) xk zk = k k ữợ ❈❤ù♥❣ ♠✐♥❤ limk→∞ xk = x∗✳ ✸✸ ❚ø ❇ê ✤➲ tỗ t t p C ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✷✳✶✮✳ ❱➻ B := {xk } ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✾✮ ❧➔ ❜à ❝❤➦♥✱ ♥➯♥ tø ❇ê ✤➲ ✷✳✷✳✾(ii) s✉② r❛ lim sup T k (x) − T (x) = ✈➔ ❞♦ ✤â k→∞ x∈B lim sup S k (x) − S(x) = k→∞ xB ú ỵ r xk S(xk ) xk − S k (xk ) + S k (xk ) − S(xk ) ≤ xk − S k (xk ) + sup S k (x) − S(x) x∈B ❚ø ữợ t õ lim xk S k (xk ) = 0✳ ❉♦ ✤â xk − S(xk ) → 0✱ ❦❤✐ k→∞ k → ∞✳ ❚ø ❇ê ✤➲ ✷✳✷✳✾(iii)✱ t❛ ♥❤➟♥ ✤÷đ❝ lim sup F (p∗ ), j(p∗ − xk ) ≤ k→∞ ❱➻ j ❧✐➯♥ tö❝ ✤➲✉ tr➯♥ ♠å✐ t➟♣ ❝♦♥ ❜à ❝❤➦♥ ❝õ❛ E ✱ ♥➯♥ ✭✷✳✷✷✮ lim j(p∗ − xk+1 ) − j(p∗ − xk ) = k→∞ ❈✉è✐ ❝ò♥❣ tø ▼➺♥❤ ✤➲ ✶✳✶✳✶✾ ✈➔ ✶✳✷✳✺✱ t❛ ❝â xk+1 − p∗ = (I − λk F )S k (xk ) − p∗ = (I − λk F )S k (xk ) − (I − λk F )S k (p∗ ) − λk F (p∗ ) ≤ (I − λk F )S k (xk ) − (I − λk F )S k (p∗ ) 2 + −λk F (p∗ ), j(xk+1 − p∗ ) ≤ (1 − λk τ )2 S k (xk ) − S k (p∗ ) − 2λk F (p∗ ), j(xk+1 − p∗ ) ≤ (1 − λk τ ) xk − p∗ − 2λk F (p∗ ), j(xk+1 − p∗ ) ≤ (1 − λk τ ) xk − p∗ + 2λk τ F (p∗ ), j(p∗ − xk ) + F (p∗ ), j(p∗ − xk+1 ) − j(p∗ − xk ) /τ = (1 − bk ) xk − p∗ + bk ck , ✭✷✳✷✸✮ ð ✤➙② bk = λk τ, ✸✹ ck = F (p∗ ), j(p∗ − xk+1 ) + F (p∗ ), j(p∗ − xk+1 ) − j(p∗ − xk ) /τ, τ =1− (1 − η)/γ ∞ ❱➻ ∞ λk = ∞ ♥➯♥ k=1 bk = ∞✳ ❚ø ❇ê ✤➲ ✷✳✷✳✶ ✈ỵ✐ ak = xk − p∗ ✱ k=1 ✭✷✳✷✷✮ ✈➔ ✭✷✳✷✸✮ s✉② r❛ lim xk = p ỵ ữủ ự k→∞ ✷✳✷✳✹ ❱➼ ❞ö ♠✐♥❤ ❤å❛ ❚r♦♥❣ ♠ö❝ ♥➔② t❛ ❝❤➾ r❛ ♠ët ✈➼ ❞ö ♠➔ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ỵ tử tr ữủ tọ sû E := Rn ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈ỵ✐ n t ổ ữợ x, y = n x = xi yi i=1 1/2 x2i i=1 ✈ỵ✐ ♠å✐ x = (x1 , x2 , , xn ), y = (y1 , y2 , , yn ) ∈ Rn ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ❧➔ →♥❤ ①↕ ✤ì♥ ✈à✳ ❑❤✐ ✤â✱ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ j ✲✤ì♥ ✤✐➺✉ trð t❤➔♥❤ ❜➔✐ t♦→♥ t➻♠ ♣❤➛♥ tû p∗ tr♦♥❣ C s❛♦ ❝❤♦ F (p∗ ), p∗ − p ≤ ∀p ∈ C F rt ỗ t t t tự tữỡ ữỡ ợ ❜➔✐ t♦→♥ ❝ü❝ trà t➻♠ p∗ ∈ C s❛♦ ❝❤♦ ϕ(p∗ ) = ϕ(x) x∈C ❉♦ ✤â ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ♥➯✉ tr➯♥ ❝â t❤➸ →♣ ❞ö♥❣ ❝❤♦ ❜➔✐ t♦→♥ ❝ü❝ trà t➻♠ ♣❤➛♥ tû p∗ ∈ C s❛♦ ❝❤♦ ϕ(p∗ ) = ϕ(x), x∈C C= Ci = ∅, i1 ỗ ợ ❤➔♠ ❝õ❛ ♥â ϕ (x) ❧➔ →♥❤ ①↕ ✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ ❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ tr➯♥ E ✱ ✈➔ Ci t ỗ õ E ❚r÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❦❤✐ E ≡ R2 ✱ ϕ(x) = x Ci = {x = (x1 , x2 ) ∈ R2 : ✈➔ x1 − x2 ≤ 0} i ✈ỵ✐ ♠å✐ i ≥ 1✱ t❛ ❝â p∗ = (0, 0) ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ✭✷✳✷✹✮✳ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔②✱ F (x) = 2x ✈➔ Ti = PCi ✈ỵ✐ ♠å✐ i ≥ 1✳ ❘ã r➔♥❣✱ F ❧➔ 2✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ 2✲❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ tr➯♥ E ✈➔ PCi ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ✸✻ ❑➳t ❧✉➟♥ ✣➲ t➔✐ ❧✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ❜❛ ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ j ✲✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❈ư t❤➸✿ ✭✶✮ ❚r➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕✱ ỗ õ t ✈➲ →♥❤ ①↕ ✤ì♥ ✤✐➺✉✱ j ✲✤ì♥ ✤✐➺✉✱ →♥❤ ①↕ ❣✐↔ ❝♦ ❝❤➦t ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❧✐➯♥ q✉❛♥✳ ✭✷✮ ❚r➻♥❤ ❜➔② ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✤ì♥ ✤✐➺✉✱ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ j ✲✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ♠ët sè ✈➼ ❞ư ✈➲ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉✳ ✭✸✮ ❚r➻♥❤ ❜➔② ❜❛ ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ j ✲✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ❝ò♥❣ ✈➼ ❞ư ♠✐♥❤ ❤å❛ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ỵ tử ữủ tọ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ❍♦➔♥❣ ❚ö② ✭✷✵✵✸✮✱ ❍➔♠ t❤ü❝ ✈➔ ●✐↔✐ t➼❝❤ ❤➔♠✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❚✐➳♥❣ ❆♥❤ ❬✷❪ P✳❘✳ ❆❣❛r✇❛❧✱ ❉✳ ❖✬❘❡❣❛♥✱ ❉✳❘✳ ❙❛❤✉ ✭✷✵✵✵✮✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❢♦r ▲✐♣s❝❤✐t③✐❛♥✲❚②♣❡ ▼❛♣♣✐♥❣s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✱ ❙♣r✐♥❣❡r ❉♦r✲ ❞r❡❝❤t ❍❡✐❞❡❧❜❡r❣ ▲♦♥❞♦♥ ◆❡✇ ❨♦r❦✳ ❬✸❪ ❨✳ ❆❧❜❡r✱ ■✳P✳ ❘②❛③❛♥ts❡✈❛ ✭✷✵✵✻✮✱ ◆♦♥❧✐♥❡❛r ■❧❧✲♣♦s❡❞ Pr♦❜❧❡♠s ♦❢ ▼♦♥♦t♦♥❡ ❚②♣❡✱ ❙♣r✐♥❣❡r✕❱❡r❧❛❣✱ ❇❡r❧✐♥✳ ❬✹❪ ❑✳ ❆♦②❛♠❛✱ ❍✳ ■✐❞✉❦❛✱ ❛♥❞ ❲✳ ❚❛❦❛❤❛s❤✐ ✭✷✵✵✻✮✱ ✧❲❡❛❦ ❝♦♥✈❡r✲ ❣❡♥❝❡ ♦❢ ❛♥ ✐t❡r❛t✐✈❡ s❡q✉❡♥❝❡ ❢♦r ❛❝❝r❡t✐✈❡ ♦♣❡r❛t♦rs ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✧✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❛♥❞ ❆♣♣❧✳✱ ✷✵✵✻✿ ✸✺✸✾✵✱ ✶✕✶✸✳ ❬✺❪ ◆❣✳ ❇✉♦♥❣✱ ◆❣✳❙✳ ❍❛✱ ❛♥❞ ◆❣✳❚✳❚✳ ❚❤✉② ✭✷✵✶✻✮✱ ✧❆ ♥❡✇ ❡①♣❧✐❝✐t ✐t❡r❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ❛ ❝❧❛ss ♦❢ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✧✱ ◆✉♠❡r✐❝❛❧ ❆❧❣♦r✐t❤♠s✱ ✼✷✭✷✮✱ ✹✻✼✕✹✽✶✳ ❬✻❪ ◆❣✳ ❇✉♦♥❣✱ ◆❣✳❙✳ ❍❛✱ ❛♥❞ ◆❣✳❚✳❚✳ ❚❤✉② ✭✷✵✶✼✮✱ ✧❍②❜r✐❞ st❡♣❡st✲ ❞❡s❝❡♥t ♠❡t❤♦❞ ✇✐t❤ ❛ ❝♦✉♥t❛❜❧② ✐♥❢✐♥✐t❡ ❢❛♠✐❧② ♦❢ ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣s ♦♥ ❇❛♥❛❝❤ s♣❛❝❡s✧✱ ◆♦♥❧✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✷✶✭✷✮✱ ✷✼✸✕ ✷✽✼✳ ✸✽ ❬✼❪ ▲✳✲❈✳ ❈❡♥❣✱ ◗✳❍✳ ❆♥s❛r✐✱ ❏✳✲❈✳ ❨❛♦ ✭✷✵✵✽✮✱ ✧▼❛♥♥✲t②♣❡ st❡❡♣❡st✲ ❞❡s❝❡♥t ❛♥❞ ♠♦❞✐❢✐❡❞ ❤②❜r✐❞ st❡❡♣❡st ❞❡s❝❡♥t ♠❡t❤♦❞s ❢♦r ✈❛r✐❛✲ t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✧✱ ◆✉♠❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❖♣t✐♠✳✱ ✷✾✭✾✲✶✵✮✱ ✾✽✼✕✶✵✸✸✳ ❬✽❪ ❋✳ ●✐❛♥♥❡ss✐✱ ❆✳ ▼❛✉❣❡r✐ ✭✶✾✾✺✮✱ ❱❛r✐❛t✐♦♥❛❧ ■♥❡q✉❛❧✐t✐❡s ❛♥❞ ◆❡t✲ ✇♦r❦ ❊q✉✐❧✐❜r✐✉♠ Pr♦❜❧❡♠s✱ ❙♣r✐♥❣❡r ❙❝✐❡♥❝❡ + ❇✉s✐♥❡ss ▼❡❞✐❛✱ ▲▲❈✳ ❬✾❪ ◆❣✳❙✳ ❍❛✱ ◆❣✳ ❇✉♦♥❣ ❛♥❞ ◆❣✳❚✳❚✳ ❚❤✉② ✭✷✵✶✽✮✱ ✧❆ ♥❡✇ s✐♠♣❧❡ ♣❛r❛❧❧❡❧ ✐t❡r❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ❛ ❝❧❛ss ♦❢ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✧✱ ❆❝t❛ ▼❛t❤✳ ❱✐❡t♥❛♠✳✱ ✹✸✭✷✮✱ ✷✸✾✕✷✺✺✳ ❬✶✵❪ ■✳ ■✐❞✉❦❛ ✭✷✵✶✷✮✱ ✧■t❡r❛t✐✈❡ ❛❧❣♦r✐t❤♠ ❢♦r tr✐♣❧❡✲❤✐❡r❛r❝❤✐❝❛❧ ❝♦♥✲ str❛✐♥❡❞ ♥♦♥❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥ t♦ ♥❡t✲ ✇♦r❦ ❜❛♥❞✇✐❞t❤ ❛❧❧♦❝❛t✐♦♥✧✱ ❙■❆▼ ❏✳ ❖♣t✐♠✳✱ ✷✷✱ ✽✻✷✕✽✼✽✳ ❬✶✶❪ ■✳ ■✐❞✉❦❛ ✭✷✵✶✸✮✱ ✧❋✐①❡❞ ♣♦✐♥t ♦♣t✐♠✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ❞✐s✲ tr✐❜✉t❡❞ ♦♣t✐♠✐③❛t✐♦♥ ✐♥ ♥❡t✇♦r❦ s②st❡♠s✧✱ ❙■❆▼ ❏✳ ❖♣t✐♠✳✱ ✷✸✱ ✶✕✷✻✳ ❬✶✷❪ ❉✳ ❑✐♥❞❡r❧❡❤r❡r✱ ❛♥❞ ●✳ ❙t❛♠♣❛❝❝❤✐❛ ✭✶✾✽✵✮✱ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❱❛r✐❛t✐♦♥❛❧ ■♥❡q✉❛❧✐t✐❡s ❛♥❞ ❚❤❡✐r ❆♣♣❧✐❝❛t✐♦♥s✱ ❆❝❛❞❡♠✐❝ Pr❡ss✱ ◆❡✇ ❨♦r❦✱ ◆❨✳ ❬✶✸❪ ❍✳❑✳ ❳✉ ✭✷✵✵✸✮✱ ✧❆♥ ✐t❡r❛t✐✈❡ ❛♣♣r♦❛❝❤ t♦ q✉❛❞r❛t✐❝ ♦♣t✐♠✐③❛t✐♦♥✧✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳✱ ✶✶✻✱ ✻✺✾✕✻✼✽✳