✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ◆●❯❨➍◆ ❱❿◆ ❍❷■ ❚❍❯❾❚ ❚❖⑩◆ ✣■➎▼ ●❺◆ ❑➋ ✣×❮◆● ❉➮❈ ◆❍❻❚ ●■❷■ ▼❐❚ ▲❰P ❇❻❚ ✣➃◆● ❚❍Ù❈ ❇■➌◆ P❍❹◆ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ❇❆◆◆❆❈❍ ❚❍⑩■ ◆●❯❨➊◆✱ ✶✵✴✷✵✶✽ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ◆●❯❨➍◆ ❱❿◆ ❍❷■ ❚❍❯❾❚ ❚❖⑩◆ ✣■➎▼ ●❺◆ ❑➋ ✣×❮◆● ❉➮❈ ◆❍❻❚ ●■❷■ ▼❐❚ ▲❰P ❇❻❚ ✣➃◆● ❚❍Ù❈ ❇■➌◆ P❍❹◆ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ❇❆◆◆❆❈❍ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ö♥❣ ▼➣ sè✿ ✽✹✻✵✶✶✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❾P ❚❍➎ ●■⑩❖ ❱■➊◆ ❍×❰◆● ❉❼◆ ●❙✳❚❙✳ ◆●❯❨➍◆ ❇×❮◆● ❚❙✳ ◆●❯❨➍◆ ❚❍➚ ❚❍Ĩ❨ ❍❖❆ ❚❍⑩■ ◆●❯❨➊◆✱ ✶✵✴✷✵✶✽ ử ỵ ữỡ ợ t t t tự ♣❤➙♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✹ ✶✳✶ ✶✳✷ ⑩♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ổ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✶✳✷ ⑩♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✸ ⑩♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✶✳✹ ❚♦→♥ tû ❣✐↔✐ ✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✶✵ ✶✳✷✳✶ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ j ✲✤ì♥ ✤✐➺✉ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✷ ✶✵ P❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ❈❤÷ì♥❣ ✷✳ P❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ✤÷í♥❣ ❞è❝ ♥❤➜t ①➜♣ ①➾ ♥❣❤✐➺♠ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✶✼ ✷✳✶ ✷✳✷ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✶✳✶ ●✐ỵ✐ ❤↕♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✶✳✷ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ➞♥ ✈➔ sü ❤ë✐ tö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❤✐➺♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✷✳✶ ✷✹ ▼ỉ t↔ ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✈ ✷✳✷✳✷ ❙ü ❤ë✐ tö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✷✳✸ ❱➼ ❞ö ♠✐♥❤ ❤å❛ ✸✺ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✽ ✶ ❇↔♥❣ ỵ H E E SE R R+ x D(A) R(A) A−1 I d(x, C) lim supn→∞ xn lim inf n→∞ xn xn → x0 xn x0 J j ❋✐①(T ) ∂f ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❦❤ỉ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝õ❛ E ♠➦t ❝➛✉ ✤ì♥ ✈à ❝õ❛ E t➟♣ ❝→❝ sè t❤ü❝ t➟♣ ❝→❝ sè t❤ü❝ ❦❤ỉ♥❣ ➙♠ t➟♣ ré♥❣ ✈ỵ✐ ♠å✐ x ♠✐➲♥ ①→❝ ✤à♥❤ ❝õ❛ t♦→♥ tû A ♠✐➲♥ ↔♥❤ ❝õ❛ t♦→♥ tû A t tỷ ữủ t tỷ A t tỷ ỗ ♥❤➜t ❦❤♦↔♥❣ ❝→❝❤ tø ♣❤➛♥ tû x ✤➳♥ t➟♣ ❤ñ♣ C ❣✐ỵ✐ ❤↕♥ tr➯♥ ❝õ❛ ❞➣② sè {xn } ❣✐ỵ✐ ữợ số {xn } {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ x0 ❞➣② {xn } ❤ë✐ tö ②➳✉ ✈➲ x0 →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ✤ì♥ trà t➟♣ ✤✐➸♠ ❜➜t T ữợ ỗ f E ổ tỹ ỵ E ổ ❤ñ♣ ❝õ❛ E ✱ x∗ , x ❧➔ ❣✐→ trà ❝õ❛ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ x∗ ∈ X ∗ t↕✐ x ∈ E ✈➔ ❝❤✉➞♥ ❝õ❛ E ✈➔ ỵ à t t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✤÷đ❝ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉✿ ❈❤♦ C ❧➔ ♠ët t➟♣ ❝♦♥ ỗ õ rộ ổ tỹ E ✱ F : E → E ❧➔ ♠ët →♥❤ ①↕ ①→❝ ✤à♥❤ tr➯♥ E ✳ ❚➻♠ ♣❤➛♥ tû x∗ ∈ C s❛♦ ❝❤♦ F (x∗ ), j(x − x∗ ) ≥ ∀x ∈ C, ✭✶✮ ð ✤➙② j ❧➔ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ✤ì♥ trà ❝õ❛ E ✱ →♥❤ ①↕ F ❧➔ →♥❤ ①↕ ❣✐→✱ C ❧➔ t➟♣ r➔♥❣ ❜✉ë❝✳ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ữủ ợ t t P✳ ❍❛rt♠❛♥ ✈➔ ●✳ ❙t❛♠♣❛❝❝❤✐❛ ❝æ♥❣ ❜è ♥❤ú♥❣ ♥❣❤✐➯♥ ❝ù✉ ✤➛✉ t✐➯♥ ❝õ❛ ♠➻♥❤ ✈➲ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❧✐➯♥ q✉❛♥ tỵ✐ ✈✐➺❝ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ❜✐➳♥ ♣❤➙♥✱ ❜➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ tè✐ ÷✉ ✈➔ ❝→❝ ❜➔✐ t♦→♥ tr ỵ tt ữỡ tr r t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✈æ ự õ ữủ ợ t❤✐➺✉ tr♦♥❣ ❝✉è♥ s→❝❤ ✧❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❱❛r✐❛t✐♦♥❛❧ ■♥❡q✉❛❧✐t✐❡s ❛♥❞ ❚❤❡✐r ❆♣♣❧✐❝❛t✐♦♥s✧ ❝õ❛ ❉✳ ❑✐♥❞❡r❧❡❤r❡r ✈➔ ●✳ ❙t❛♠✲ ♣❛❝❝❤✐❛ ①✉➜t ❜↔♥ ♥➠♠ ✶✾✽✵ ✈➔ tr♦♥❣ ❝✉è♥ s→❝❤ ✧❱❛r✐❛t✐♦♥❛❧ ❛♥❞ ◗✉❛✲ s✐✈❛r✐❛t✐♦♥❛❧ ■♥❡q✉❛❧✐t✐❡s✿ ❆♣♣❧✐❝❛t✐♦♥s t♦ ❋r❡❡ ❇♦✉♥❞❛r② Pr♦❜❧❡♠s✧ ❝õ❛ ❈✳ ❇❛✐♦❝❝❤✐ ✈➔ ❆✳ ❈❛♣❡❧♦ ①✉➜t ❜↔♥ ♥➠♠ ✶✾✽✹✳ ▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❜❛ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ tr ổ ỗ õ ❝❤✉➞♥ ❦❤↔ ✈✐ ●➙t❡❛✉① ✸ ✤➲✉ ✈ỵ✐ t➟♣ r➔♥❣ ❜✉ë❝ C ❧➔ t➟♣ ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ m✲ j ✲✤ì♥ ✤✐➺✉✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr tr ữỡ ữỡ ợ t ởt sè ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ỗ õ t ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝✱ →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉✱ t♦→♥ tỷ tr ổ ỗ tớ tr ♣❤÷ì♥❣ ♣❤→♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr➯♥ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✱ ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣ ❞è❝ ♥❤➜t✱ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ →♥❤ ①↕ m✲j ✲✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ❜❛ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❦➳t ❤đ♣ ợ ữỡ ữớ ố t ởt ữỡ ➞♥ ✈➔ ❤❛✐ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❤✐➺♥✮ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈ỵ✐ →♥❤ ①↕ ❣✐→ ❧➔ →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ ❣✐↔ ❝♦ ❝❤➦t✱ t➟♣ r➔♥❣ ❜✉ë❝ ❧➔ t➟♣ ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ ❝õ❛ ❝→❝ mj ỡ tr ổ ỗ ✤➲✉ ❝â ❝❤✉➞♥ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✕ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ✣➛✉ t✐➯♥✱ tỉ✐ ①✐♥ ❦➼♥❤ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ ✤➳♥ t❤➛② ●❙✳❚❙✳ ◆❣✉②➵♥ ❇÷í♥❣✱ ♥❣÷í✐ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❧➔♠ ❧✉➟♥ ✈➠♥✳ ❚æ✐ ①✐♥ ❣û✐ ❝→♠ ỡ qỵ ổ tr ❚✐♥ ❝õ❛ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t t tr t tự qỵ ❜→✉ ❝❤♦ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ tæ✐ ❤å❝ t➟♣ t trữớ ổ ỷ ỡ qỵ ❚❤➛② ❈ỉ tr♦♥❣ P❤á♥❣ ✣➔♦ t↕♦ ❝õ❛ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ❝❤♦ tỉ✐ ❤♦➔♥ t❤➔♥❤ ❝❤÷ì♥❣ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❈✉è✐ ❝ị♥❣✱ tỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝→♠ ì♥ ✤➳♥ ❣✐❛ ✤➻♥❤ ✈➔ ❜↕♥ ❜➧ ✤➣ ✤ë♥❣ ✈✐➯♥✱ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ❝❤♦ tỉ✐ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✽ ❚→❝ ữỡ ợ t❤✐➺✉ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ♠ư❝✳ ▼ư❝ ✶✳✶ ❣✐ỵ✐ t❤✐➺✉ ❦❤→✐ ♥✐➺♠ ✈➔ tr➻♥❤ ❜➔② ♠ët sè t t ổ ỗ õ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉✱ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝✱ →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ ✈➔ t♦→♥ tû ❣✐↔✐ tr♦♥❣ ổ tự ữỡ ợ t ✈➲ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ j ✲✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr➯♥ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ tr♦♥❣ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ ✈✐➳t tr➯♥ ❝ì sð ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✕❬✸❪✱ ❬✶✶❪✕❬✶✹❪ ✈➔ ❝→❝ t➔✐ ❧✐➺✉ ✤÷đ❝ t❤❛♠ ❝❤✐➳✉ tr♦♥❣ ✤â✳ ✶✳✶ ⑩♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ ❈❤♦ E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈ỵ✐ ❦❤ỉ♥❣ ❣✐❛♥ ố ỵ E ũ ỵ tr E E ✈➔ ✈✐➳t t➼❝❤ ✤è✐ ♥❣➝✉ x, x∗ t❤❛② ❝❤♦ ❣✐→ trà ❝õ❛ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ x∗ ∈ E ∗ t↕✐ ✤✐➸♠ x ∈ E ✱ tù❝ ❧➔ x, x∗ = x∗ (x)✳ ❱ỵ✐ ♠ët →♥❤ ①↕ A : E → 2E ✱ t❛ s➩ ✤à♥❤ ♥❣❤➽❛ ♠✐➲♥ ①→❝ ✤à♥❤✱ tr ỗ t õ tữỡ ự ✺ ♥❤÷ s❛✉✿ D(A) = {x ∈ E : A(x) = ∅}, R(A) = ∪{Az : z ∈ D(A)}, ✈➔ G(A) = {(x, y) ∈ E × E : x ∈ D(A), y ∈ A(x)} ⑩♥❤ ①↕ ♥❣÷đ❝ A−1 ❝õ❛ →♥❤ ①↕ A ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐✿ x ∈ A−1 (y) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ y ∈ A(x) ✶✳✶✳✶ ❑❤æ♥❣ ỗ ổ E ữủ ợ tû x∗∗ ∈ E ∗∗ ✱ ❦❤ỉ♥❣ ❣✐❛♥ ❧✐➯♥ ❤đ♣ tự E tỗ t tỷ x ∈ E s❛♦ ❝❤♦ x∗ (x) = x∗∗ (x∗ ) ∀x∗ ∈ E ∗ ◆➳✉ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕ t❤➻ ♠å✐ ❞➣② ❜à ❝❤➦♥ tr♦♥❣ E ✤➲✉ ❝â ❞➣② ❝♦♥ ❤ë✐ tö ②➳✉✳ ✣â ❧➔ ỵ s ỵ ✭①❡♠ ❬✸❪✮ ❈❤♦ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❑❤✐ ✤â✱ ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ (i) E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕✳ (ii) ▼å✐ ❞➣② ❜à ❝❤➦♥ tr♦♥❣ E õ ởt tử ỵ ❤✐➺✉ SE := {x ∈ E : x = 1} ❧➔ ♠➦t ❝➛✉ ✤ì♥ ✈à ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ổ ỗ t ỗ (i) ổ E ữủ ỗ t ợ ✤✐➸♠ x, y ∈ SE ✱ x = y ✱ s✉② r❛ (1 − λ)x + λy < ∀λ ∈ (0, 1) ✻ (ii) ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ữủ ỗ ợ (0, 2] ✈➔ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ x ≤ 1, y ≤ 1✱ x − y ≥ ε t❤ä❛ ♠➣♥ t tỗ t = () > s (x + y)/2 ≤ − δ ✳ ▼è✐ ❧✐➯♥ ỳ ổ ỗ ỗ t ữủ ỵ ữợ ỵ ổ ỗ ỗ t ổ E ữủ trỡ ợ ♠é✐ ✤✐➸♠ x ♥➡♠ tr➯♥ ♠➦t ❝➛✉ ✤ì♥ ✈à SE tỗ t t ởt gx E ∗ s❛♦ ❝❤♦ x, gx = x ✈➔ gx = ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✻ (i) ❈❤✉➞♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ữủ t ợ ộ y ∈ SE ❣✐ỵ✐ ❤↕♥ x + ty − x lim t0 t tỗ t ợ x SE ỵ y, x õ x ữủ ❣å✐ ❧➔ ✤↕♦ ❤➔♠ ●➙t❡❛✉① ❝õ❛ ❝❤✉➞♥✳ (ii) ❈❤✉➞♥ ❝õ❛ E ữủ t ợ ộ y SE ợ t ữủ ✤➲✉ ✈ỵ✐ ♠å✐ x ∈ SE ✳ ▼è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ trì♥ ✈➔ t➼♥❤ ❦❤↔ ✈✐ ●➙t❡❛✉① ữủ ổ ố tr ỵ s ỵ ổ E trỡ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❝❤✉➞♥ ❝õ❛ E ❦❤↔ ✈✐ ●➙t❡❛✉① tr➯♥ E \ {0}✳ ✶✳✶✳✷ ⑩♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✽ ⑩♥❤ ①↕ Js : E → 2E , ∗ s > ✭♥â✐ ❝❤✉♥❣ ❧➔ ✤❛ trà✮ ①→❝ ✤à♥❤ ❜ð✐ Js x = {uq ∈ E ∗ : x, us = x us , us = x s−1 }, ✷✺ ð ✤➙② {eki } ❧➔ ❞➣② s❛✐ sè✱ i = 1, 2, , N ✈➔ x1 ∈ E, ❧➔ ♣❤➛♥ tỷ tũ ỵ, y0k = (I tk F )xk , k + eki , i = 1, 2, · · · , N − 1, yik = JrAi i yi−1 k k k xk+1 = J ANN yN −1 + eN r ✭✷✳✾✮ k ❈→❝ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ✤÷đ❝ ①➙② ❞ü♥❣ ❞ü❛ tr➯♥ t ủ ữỡ ữớ ố t ợ tt t♦→♥ ✤✐➸♠ ❣➛♥ ❦➲✱ ❜ð✐ ✈➟② t❛ ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ✤÷í♥❣ ❞è❝ ♥❤➜t✳ ❚❛ s➩ tr➻♥❤ ❜➔② sü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✽✮ ữợ s tk tọ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭❈✶✮❀ ✭❜✮ rki ≥ ε > ✈ỵ✐ ♠å✐ k ≥ ✱ i = 1, 2, , N ❀ ✈➔ ✭❝✮ limk→0 eki /tk = ✈ỵ✐ i = 1, 2, , N ✳ ✷✳✷✳✷ ❙ü ❤ë✐ tö ❙ü ❤ë✐ tư ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✽✮ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✾✮ ữủ tr ỵ ữợ ỵ ✷✳✷✳✶ ✭①❡♠ ❬✻❪✮ ❈❤♦ E, F ✈➔ Ai ♥❤÷ tr♦♥❣ ▼➺♥❤ ✤➲ ✷✳✶✳✻✳ ●✐↔ sû r➡♥❣ ❝→❝ t❤❛♠ sè tk ✱ rki ✈➔ ❞➣② s❛✐ sè {eki } t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❛✮✱ ✭❜✮ ✈➔ ✭❝✮✳ ❑❤✐ ✤â ❞➣② {xk }✱ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ✭✷✳✽✮ ❤ë✐ tư ♠↕♥❤ tỵ✐ ♣❤➛♥ tû p∗ ✱ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✶✳✻✮✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø JrA p = p ✈ỵ✐ ✤✐➸♠ ❜➜t ❦➻ p ∈ ∩Ni=1ZerAi✱ tø t➼♥❤ i i k ✷✻ ❝❤➜t ❦❤æ♥❣ ❣✐➣♥ ❝õ❛ JrAi i ✱ ✭✷✳✽✮ ✈➔ ❇ê ✤➲ ✶✳✶✳✶✼ t❛ ♥❤➟♥ ✤÷đ❝ k AN k k xk+1 − p = JrANN (1 − tk F )yN −1 + eN −JrN p k ≤ ≤ ≤ ≤ k k k (1 − tk F )yN −1 − (1 − tk F )p − tk F p + eN k k (1 − tk τ ) yN −1 − p + tk F p + eN k k (1 − tk τ ) yN −2 − p + 2tk F p + eN + N k (1 − tk τ ) x − p + N tk F p + eki i=1 ekN −1 ≤ max { x − p , N ( F p + c˜)/τ }, tr♦♥❣ ✤â c˜ ❧➔ ♠ët ❤➡♥❣ sè ❞÷ì♥❣ s❛♦ ❝❤♦ eki /tk ≤ c˜ ✈ỵ✐ ♠å✐ k ≥ k ✈➔ i = 1, 2, , N ✳ ❉♦ ✈➟②✱ ❞➣② {xk } ❜à ❝❤➦♥✳ ❱➻ t❤➳ ❝→❝ ❞➣② {F yi−1 − k k eki /tk } ✈➔ {zi−1 − p} ❝ơ♥❣ ❜à ❝❤➦♥ ✈ỵ✐ p ∈ ∩N i=1 ZerAi ✱ ð ✤➙② zi−1 = k (I − tk F )yi−1 + eki ✈ỵ✐ i = 1, 2, , N ✳ ❑❤ỉ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ ❝❤ó♥❣ t❛ ❣✐↔ sû r➡♥❣ ❝❤ó♥❣ ❜à ❝❤➦♥ ❜ð✐ ❤➡♥❣ sè ❞÷ì♥❣ M2 ✳ ❚❤❡♦ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ▼➺♥❤ ✤➲ ✷✳✶✳✻ t❛ s✉② r❛ k −p JrAi i zi−1 k k k − p)/2 ≤ (zi−1 − p)/2 + (JrAi i zi−1 k k k − p /2 ≤ zi−1 − p /2 + JrAi i zi−1 k −g k ≤ zi−1 −p ❚❛ ✤→♥❤ ❣✐→ ❣✐→ trà xk+1 − p xk+1 − p k − JrAi i zi−1 k k − g zi−1 k zi−1 k /4 − JrAi i zi−1 k ♥❤÷ s❛✉✿ k = JrANN zN −1 − p k /4 k ≤ zN −1 − p AN k k − g zN −1 − JrN zN −1 /4 k = (I − −g k k tk F )yN −1 + eN − p AN k k zN −1 − JrN zN −1 /4 k ✷✼ ❤❛② xk+1 − p k ≤ yN −1 − p k k k − tk F yN −1 − eN , j(zN −1 − p) AN k k − g zN −1 − JrN zN −1 /4 k k yN −1 ≤ −p AN k k + 2tk M22 − g zN −1 − JrN zN −1 /4 k ≤ ··· ≤ xk − p k k + 2N tk M22 − g zi−1 − JrAi i zi−1 /4, k ✈ỵ✐ ♠é✐ i ∈ {1, 2, , N }✳ ❉♦ ✤â✱ k k g zi−1 − JrAi i zi−1 /4 − 2N tk M22 ≤ xk − p − xk+1 − p k ❈❤ó♥❣ t❛ ❝❤➾ ❝➛♥ ♣❤➙♥ t➼❝❤ tr♦♥❣ ❤❛✐ tr÷í♥❣ ❤đ♣✳ k k ✭❛✮ ❑❤✐ g( zi−1 − JrAi i zi−1 )/4 ≤ 2tk N M22 ✈ỵ✐ ♠å✐ k ≥ 1✱ tø ✤✐➲✉ k ❦✐➺♥ ❝õ❛ tk ✱ t❛ ❝â k k lim g( zi−1 − JrAi i zi−1 ) = ✭❜✮ ❑❤✐ g( k→∞ k k zi−1 − JrAi i zi−1 k k )/4 > 2N tk M22 ✱ t❛ ✤÷đ❝ M k k g zi−1 − JrAi i zi−1 /4 − 2N tk M22 ≤ x1 − p − xM +1 − p k k=1 ≤ x1 − p ❱➻ ✈➟②✱ ∞ k k g zi−1 − JrAi i zi−1 /4 − 2N tk M22 < +∞ k k=1 ❉♦ ✤â✱ k k lim g zi−1 − JrAi i zi−1 /4 − 2N tk M22 = 0, k→∞ k ❦➳t ❤đ♣ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❝õ❛ tk s✉② r❛ k k lim g zi−1 − JrAi i zi−1 = k→∞ k ✈➻ t❤➳ tø ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ g ✱ ❝â ✤÷đ❝ k k lim zi−1 − JrAi i zi−1 = k k k k ữ ỵ r➡♥❣ zi−1 − yi−1 ≤ tk M2 ✱ k k lim zi−1 − yi−1 = 0, ∀i = 1, 2, · · · , N k→∞ ✭✷✳✶✶✮ k ❱➻ ❞➣② {zi−1 } ❜à ❝❤➦♥ ♥➯♥ ❞➣② {yik } ❝ô♥❣ ❜à ❝❤➦♥✳ ❙✉② r❛ k k k = JrAi i (I − tk µF )yi−1 + eki −JrAi i yi−1 yik − JrAi i yi−1 k k ≤ (I − k k tk µF )yi−1 + eki − k yi−1 ≤ tk M2 → ✭✷✳✶✷✮ ❦❤✐ k → ∞ ✈ỵ✐ i = 1, 2, , N ✳ ❑➳t ❤đ♣ ✈ỵ✐ ✭✷✳✶✵✮ ✈➔ ✭✷✳✶✶✮ s✉② r❛ k lim yik − yi−1 = ∀i = 1, 2, · · · , N k→∞ ✭✷✳✶✸✮ k ✱ t❛ ❝â ❙û ❞ư♥❣ ▼➺♥❤ ✤➲ ✶✳✶✳✷✶ ✈ỵ✐ r = ri , t = rki ✱ A = Ai ✈➔ x = zi−1 ✤÷đ❝ Ai k k k Ai Ai k Ai k Ai k z −J J z ≤ J z −J z z −J z ≤ i i i i−1 i−1 r i−1 i−1 i−1 r rk rki i−1 rk ri k rk ε ❚ø ❜➜t ✤➥♥❣ t❤ù❝ ♥➔②✱ ✭✷✳✶✵✮ ✈➔ ✭✷✳✶✶✮ t❛ s✉② r❛ k k lim JrAi i yi−1 − yi−1 = 0, ∀i = 1, 2, · · · , N k→∞ ✭✷✳✶✹✮ ❚✐➳♣ t❤❡♦✱ ❧➜② i = tr♦♥❣ ✭✷✳✶✹✮ ✈➔ ❧÷✉ þ r➡♥❣ y0k = xk , lim JrA11 xk − xk = k→∞ ❇➙② ❣✐í ❧➜② i = tr♦♥❣ ✭✷✳✶✹✮ t❛ ♥❤➟♥ ✤÷đ❝ lim JrA22 y1k − y1k = k→∞ ●✐ỵ✐ ❤↕♥ ♥➔② ❦➳t ❤đ♣ ✈ỵ✐ y0k = xk ✱ ✭✷✳✶✷✮✱ ✭✷✳✶✸✮ s✉② r❛ lim JrA22 JrA11 xk − xk = k→∞ ❇➡♥❣ ✤→♥❤ ❣✐→ t÷ì♥❣ tü t❛ ♥❤➟♥ ✤÷đ❝ ✭✷✳✻✮✳ ◆❣❤➽❛ ❧➔ ❞➣② {xk }✱ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ✭✷✳✶✹✮✱ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ tr♦♥❣ ❇ê ✤➲ ✷✳✶✳✼✳ ✷✾ ❇➙② ❣✐í✱ t❛ ✤→♥❤ ❣✐→ ❣✐→ trà xk+1 − p∗ xk+1 − p∗ 2 ♥❤÷ s❛✉✿ AN k k = JrANN (I − tk F )yN −1 + eN −JrN p∗ k k k k ≤ (I − tk F )yN −1 + eN − p∗ k k = (I − tk F )yN −1 − (I − tk F )p∗ − tk F p∗ + eN k ≤ (1 − tk τ ) yN −1 − p∗ 2 k k k + 2tk F p∗ − ekN /tk , j(p∗ − yN −1 + tk F yN −1 − eN ) k ≤ (1 − tk τ ) yN −2 − p∗ + 2tk F p∗ − ekN −1 /tk , k k k j(p∗ − yN −2 + tk F yN −2 − eN −1 ) k k k + 2tk F p∗ − ekN /tk , j(p∗ − yN −1 + tk F yN −1 − eN ) ≤ (1 − bk ) xk − p∗ + bk ck , ✭✷✳✶✺✮ tr♦♥❣ ✤â bk = tk τ ✈➔ N F p∗ , j(p∗ − xk ) ck = (2/τ ) i=1 N + F p∗ , j(p∗ − + F p∗ − k yi−1 ) eki /tk , j(p∗ k − j(p∗ − x ) + − k yi−1 + k −eki /tk , j(p∗ − yi−1 ) i=1 k tk F yi−1 − eki ) − j(p∗ − xk ) ❚ø ✭✷✳✽✮ ✈➔ ✭✷✳✶✶✮ s✉② r❛ yik − xk → ❦❤✐ k → ∞✳ = ∞✱ ∞ k=1 bk = ∞✱ ♥➯♥ tø ✭✷✳✶✸✮✱ ✭✷✳✶✺✮✱ ✤✐➲✉ ❦✐➺♥ ✱ t➼♥❤ ❝❤➜t ❝õ❛ j ✱ ✤✐➲✉ ❦✐➺♥ ❝õ❛ tk ✱ ❇ê ✤➲ ✷✳✶✳✺ ✈➔ ❇ê ✤➲ ✷✳✶✳✼ t❛ ♥❤➟♥ ✤÷đ❝ limk→∞ xk+1 − p∗ = 0✱ s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❱➻ ✭❝✮ ∞ k=1 tk ỵ E, F ✈➔ Ai ♥❤÷ tr♦♥❣ ▼➺♥❤ ✤➲ ✷✳✶✳✻✳ ●✐↔ sû ❞➣② {tk }, {rki } ✈➔ {eki } ♥❤÷ tr♦♥❣ ✣à♥❤ ỵ õ {xk } ✭✷✳✾✮✱ ❤ë✐ tư ♠↕♥❤ tỵ✐ ♣❤➛♥ tû p∗ ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✶✳✻✮✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø JrA p = p ✈ỵ✐ ✤✐➸♠ ❜➜t ❦ý p ∈ ∩Ni=1ZerAi✱ tø t➼♥❤ i i k ✸✵ ❝❤➜t ❦❤æ♥❣ ❣✐➣♥ ❝õ❛ JrAi i ✱ t❛ ❝â k AN k k xk+1 − p = JrANN yN −1 + eN −JrN p k k k yN −1 k yN −2 ≤ ≤ −p + −p + ekN ekN + ekN −1 ≤ · · · N ≤ y0k − p + eki i=1 N k eki = (I − tk F )x − p + i=1 N k eki ≤ (1 − tk τ ) x − p + tk F p + i=1 ≤ max { x − p , ( F p + N c)/τ } k ❉♦ ✤â✱ ❞➣② {xk } ❜à ❝❤➦♥✱ ✈➻ t❤➳ ❝→❝ ❞➣② {xk −p−tk F xk } ✈➔ {zi−1 −p} k k k ❝ơ♥❣ ❜à ❝❤➦♥ ✈ỵ✐ p ∈ ∩N i=1 ZerAi ✱ tr♦♥❣ ✤â zi−1 = yi−1 + ei ✱ i = 2, , N ✳ ❚❛ ❣✐↔ t❤✐➳t r➡♥❣ ❝❤ó♥❣ ❜à ❝❤➦♥ ❜ð✐ ❤➡♥❣ sè ❞÷ì♥❣ M3 ✳ ❚✐➳♣ t❤❡♦ ♥❤÷ tr♦♥❣ ❝❤ù♥❣ ỵ t ữủ xk+1 p k = JrANN zN −1 − p k k ≤ zN −1 − p AN k k − g zN −1 − JrN zN −1 /4 k = k yN −1 ≤ k yN −1 −g + ekN −p AN k k − g zN −1 − JrN zN −1 /4 k k − p + ekN , j(zN −1 AN k k zN −1 − JrN zN −1 /4 − p) k ≤ k yN −1 −p AN k k + 2˜ctk M3 − g zN −1 − JrN zN −1 /4 k ≤ ··· ≤ y0k − p k k + 2˜cN tk M3 − g zi−1 − JrAi i zi−1 /4 k k = (I − tk F )x − p + 2˜cN tk M3 k k − g zi−1 − JrAi i zi−1 /4 k ✸✶ ❤❛② xk+1 − p ≤ (1 − tk τ ) xk − p − 2tk F p, j(xk − p − tk F xk ) k k + 2˜cN tk M3 − g zi−1 − JrAi i zi−1 /4 k k ≤ x −p + 2tk F p + c˜N M3 k k − g zi−1 − JrAi i zi−1 /4 k ✈ỵ✐ ♠é✐ i ∈ {1, 2, , N }✳ ❉♦ ✤â✱ k k g zi−1 −JrAi i zi−1 /4−2tk F p +cN M3 ≤ xk −p − xk+1 −p k ❚÷ì♥❣ tỹ ữ ự ỵ t t ữủ k k lim g zi−1 − JrAi i zi−1 = k→∞ k ❙û ❞ö♥❣ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ g s✉② r❛ k k lim zi−1 − JrAi i zi−1 = k k k k ỵ r zi−1 − yi−1 ≤ c˜tk ✱ k k lim zi−1 − yi−1 = 0, ∀i = 1, 2, , N k→∞ ✭✷✳✶✼✮ k ❱➻ ❞➣② {zi−1 } ❜à ❝❤➦♥ ♥➯♥ ❞➣② {yik } ❝ô♥❣ ❜à ❝❤➦♥✳ ❑❤✐ ✤â✱ k k k yik − JrAi i yi−1 = JrAi i yi−1 + eki −JrAi i yi−1 ≤ c˜tk → k k k ❦❤✐ k → ∞ ✈ỵ✐ i = 1, 2, , N ✳ ❑➳t ❤đ♣ ✈ỵ✐ ✭✷✳✶✻✮✱ ✭✷✳✶✼✮ s✉② r❛ ✭✷✳✶✸✮ ✈➔ ✭✷✳✶✹✮✳ ❍ì♥ ♥ú❛✱ ❜✤→♥❤ ❣✐→ t÷ì♥❣ tü ♥❤÷ ❝❤ù♥❣ ♠✐♥❤ ✣à♥❤ ỵ t ữủ {xk } ✭✷✳✾✮✱ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ tr♦♥❣ ❇ê ✤➲ ✷✳✶✳✼✳ ✸✷ ❇➙② ❣✐í ❣✐→ trà xk+1 − p∗ xk+1 − p∗ 2 ✤÷đ❝ ✤→♥❤ ❣✐→ ♥❤÷ s❛✉✿ AN k k = JrANN yN −1 + eN −JrN p∗ k k k k ≤ yN −1 + eN − p∗ k k k ≤ yN − ekN , j(p∗ − yN −1 − p∗ −1 − eN ) k k k ≤ yN − ekN −1 , j(p∗ − yN −2 − p∗ −2 − eN −1 ) k k − ekN , j(p∗ − yN −1 − eN ) N k ≤ y0k − p∗ − eki , j(p∗ − yi−1 − eki ) i=1 ≤ (1 − tk τ ) xk − p∗ + 2tk F p∗ , j(p∗ − xk + tk F xk ) N k eki /tk , j(p∗ − yi−1 − eki ) − 2tk i=1 = (1 − bk ) xk − p∗ + bk c k , ✭✷✳✶✽✮ tr♦♥❣ ✤â bk = tk τ ✈➔ ck = (2/τ ) F p∗ , j(p∗ − xk ) + F p∗ , j(p∗ − xk + tk F xk ) N k eki /tk , j(p∗ − yi−1 − eki ) − j(p∗ − xk ) − i=1 ❚ø ✭✷✳✾✮ ✈➔ ✭✷✳✶✼✮ s✉② r❛ yik −xk → ❦❤✐ k → ∞✳ ❱➻ ∞ k=1 bk ∞ k=1 tk = ∞✱ = ∞✱ ♥➯♥ tø ✭✷✳✶✼✮✱ ✭✷✳✶✽✮✱ t➼♥❤ ❝❤➜t ❝õ❛ j ✈ỵ✐ tk ✈➔ ❇ê ✤➲ ✷✳✶✳✺✱ ❇ê ✤➲ ✷✳✶✳✼ t❛ ♥❤➟♥ ✤÷đ❝ limk→∞ xk+1 − p∗ = 0✳ ❙✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ◆❤➟♥ ①➨t ✷✳✷✳✸ ❑❤✐ N = t❤➻ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✽✮ ❝â ❞↕♥❣ ✤ì♥ ❣✐↔♥ ❤ì♥ xk+1 = JrAk (I − tk F )xk + ek , ✭✷✳✶✾✮ tr♦♥❣ ✤â A ❧➔ ♠ët →♥❤ ①↕ ❝â m✲j ✲✤ì♥ ✤✐➺✉ tr♦♥❣ E ✳ ✣➦t y k = (I − ✸✸ tk F )xk + ek ✱ tø ✭✷✳✶✾✮ t❛ ♥❤➟♥ ✤÷đ❝ y k+1 = (I − tk+1 F )JrAk y k + ek+1 ✭✷✳✷✵✮ ❍ì♥ ♥ú❛ ♥➳✉ tk → t❤➻ ❞➣② {xk } ❤ë✐ tö ❦❤✐ ✈➔ ❝❤➾ ❞➣② {y k } ❤ë✐ tư ✈➔ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❝❤ó♥❣ ❧➔ trị♥❣ ♥❤❛✉✳ ❚❤➟t ✈➟②✱ tø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ y k ✱ s✉② r❛ y k − xk ≤ tk ( F xk + c˜)✳ ❉♦ ✤â ❦❤✐ ❞➣② {xk } ❤ë✐ tö t❤➻ ❞➣② {F xk } ❜à ❝❤➦♥✳ ❱➻ tk → ❦❤✐ k → ∞✱ tø ❜➜t ✤➥♥❣ t❤ù❝ ❝✉è✐ ✈➔ sü ❤ë✐ tö ❝õ❛ ❞➣② {xk } ❦➨♦ t❤❡♦ sü ❤ë✐ tö ❝õ❛ ❞➣② {y k } ✈➔ ❝â ❣✐ỵ✐ ❤↕♥ trị♥❣ ♥❤❛✉✳ ❚r÷í♥❣ ❤đ♣ ❦❤✐ {y k } ❤ë✐ tư t❛ ❝â ❦➳t ❧✉➟♥ t÷ì♥❣ tü✳ ❇➙② ❣✐í t❛ ✈✐➳t ❧↕✐ xk := y k ✱ tk := tk+1 ✈➔ ek := ek+1 tr♦♥❣ ✭✷✳✷✵✮✱ t❛ ✤÷đ❝ ♣❤÷ì♥❣ ♣❤→♣ xk+1 = (I − tk F )JrAk xk + ek , ✭✷✳✷✶✮ ✤➙② ❧➔ ♠ët ❞↕♥❣ ✤ì♥ ❣✐↔♥ ❤ì♥ ❝õ❛ ✭✶✳✶✾✮✳ ❚✐➳♣ t❤❡♦✱ ❝❤♦ ♠ët ❝è ✤à♥❤ a ∈ (0, 1)✱ ❧➜② f x = ax ✈➔ F = I − f ✳ õ tỗ t số ữỡ s❛♦ ❝❤♦ η + γ > ✈➔ ✈➔ ❣✐↔ t❤✐➳t F ❧➔ →♥❤ ①↕ η ✲j ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ γ ✲❣✐↔ ❝♦ ❝❤➦t✳ ❚❤❛② F ❜ð✐ I − f tr♦♥❣ ✭✷✳✷✶✮ t❛ ❝â ♣❤÷ì♥❣ ♣❤→♣ s❛✉ ✤➙②✿ xk+1 = (1 − tk )JrAk xk + ek , tk := tk (1 − a), ✭✷✳✷✷✮ ✤➙② ❧➔ ♠ët ❝↔✐ ❜✐➯♥ ♠ỵ✐ ❝õ❛ ✭✶✳✶✸✮✳ ❚❛ ❝â ❦➳t q✉↔ s❛✉✳ ✣à♥❤ ỵ E ổ ỗ õ t A : E → E ❧➔ →♥❤ ①↕ ❝â m✲j ✲✤ì♥ ✤✐➺✉ tr♦♥❣ E s❛♦ ❝❤♦ ZerA = ∅✳ ●✐↔ sû ❝→❝ ❞➣② t❤❛♠ sè tk ✱ rk ✈ỵ✐ ❞➣② s❛✐ sè {ek } t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❛✮✱ ✭❜✮ ✈➔ ✭❝✮✳ ❑❤✐ ✤â ❞➣② {xk }✱ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ✭✷✳✷✷✮ ❤ë✐ tư ♠↕♥❤ tỵ✐ ♣❤➛♥ tû p∗ tr♦♥❣ ZerA✱ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ p∗, j(p∗ − p) ≤ ✈ỵ✐ ♠å✐ p ∈ ZerA✳ ◆➳✉ t❛ t❤❛② F ❜ð✐ I − f ✈ỵ✐ f = aI + (1 − a)u tr♦♥❣ ✭✷✳✶✾✮✱ tr♦♥❣ ✸✹ ✤â u ❧➔ ✤✐➸♠ ❜➜t ❦ý tr♦♥❣ E ✱ t❛ ♥❤➟♥ ✤÷đ❝ xk+1 = JrAk (1 − tk )xk + tk u + ek , tk := tk (1 − a) ✭✷✳✷✸✮ ❱➻ ✈➟②✱ t❛ ❝â ♠ët sü ♠ð rë♥❣ ❝õ❛ ✭✶✳✶✻✮ tø ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❧➯♥ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥♥❛❝❤ ỗ õ t sỹ tử t ữủ ữợ ỡ ❤ì♥ s♦ ✈ỵ✐ ✭✶✳✶✻✮ ✈➔ ✭✶✳✶✽✮✳ ◆❤➟♥ ①➨t ✷✳✷✳✺ ❑❤✐ N = 2✱ ✤➦t A := A1 ✈➔ B = A2 ✈ỵ✐ βk = rk1, γk = rk2 , ek = ek1 ✈➔ e˜k = ek2 ✱ tø ✭✷✳✽✮ t❛ ❝â ♣❤÷ì♥❣ ♣❤→♣ y k = J A (I − t F )xk + ek , k βk xk+1 = J B (I − tk F )y k + e˜k , ✭✷✳✷✹✮ γk ✣â ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ✤÷í♥❣ ❞è❝ ♥❤➜t✳ ❚❤❛② F ❜ð✐ I − f ✈ỵ✐ f = aI + (1 − a)u✱ tr♦♥❣ ✤â a ∈ (0, 1) ✈➔ u ∈ E ✱ u ❧➔ ❝è ✤à♥❤✱ ✈✐➳t ❧↕✐ k := k − tr♦♥❣ ✭✷✳✷✹✮✱ t❛ ✤÷đ❝ x2k+1 = J A t u + (1 − t )x2k + ek , k = 0, 1, · · · , k βk k x2k = J B tk u + (1 − tk )x2k−1 + e˜k , k = 1, 2, · · · , γk ✭✷✳✷✺✮ ✈ỵ✐ tû x0 ∈ E ✳ Pữỡ ữủ ự ợ ✤➦t ❧➯♥ ❝→❝ t❤❛♠ sè βk ✈➔ γk ❦❤✐ A ✈➔ B ❧➔ ❝→❝ →♥❤ ①↕ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H ✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ ❜ê s✉♥❣ ✤➣ ✤÷đ❝ ❣✐↔♠ ♥❤➭ ❝❤♦ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✷✺✮ ✈➔ ❞↕♥❣ t÷ì♥❣ ✤÷ì♥❣ ❝õ❛ ♥â ♥❤÷ s❛✉ z 2k+1 = t u + (1 − t )J A z 2k + ek , k = 0, 1, · · · , k k rk z 2k = tk u + (1 − tk )J B z 2k−1 + e˜k , k = 1, 2, , rk ✭✷✳✷✻✮ ❱➻ t❤➳✱ ✭✷✳✷✺✮ ✈➔ ✭✷✳✷✻✮ ❝â t❤➸ ✤÷đ❝ ❝♦✐ ❧➔ ♠ð rë♥❣ ❝õ❛ ❝→❝ ❦➳t q✉↔ tø ❦❤æ♥❣ rt ổ trỡ ỗ ✤➲✉✳ ❚÷ì♥❣ tü✱ t❛ ❝â ❝↔✐ t✐➳♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ tr➯♥ ♥❤÷ s❛✉✿ x2k+1 = (1 − t )J A x2k + ek , k = 0, 1, · · · , k βk x2k = (1 − tk )J B x2k−1 + e˜k , k = 1, 2, , γk ✭✷✳✷✼✮ ✸✺ ✈➔ ♥❤➟♥ ✤÷đ❝ t q s ỵ E ởt ổ ỗ õ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉✱ A : E → E ✈➔ B : E → E ❧➔ ❤❛✐ →♥❤ ①↕ m✲j ✲✤ì♥ ✤✐➺✉ tr♦♥❣ E s❛♦ ❝❤♦ t➟♣ ❦❤ỉ♥❣ ✤✐➸♠ ❝❤✉♥❣ C := ZerA ∩ ZerB = ∅✳ ●✐↔ sû r➡♥❣ ❝→❝ t❤❛♠ sè tk ✱ βk ✈➔ γk ✈ỵ✐ ❝→❝ s❛✐ sè {ek } ✈➔ {˜ek } t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❛✮✱ ✭❜✮ ✈➔ ✭❝✮✳ ❑❤✐ ✤â ❞➣② {xk } ữủ tử tợ p∗ ∈ C ✱ ✈➔ t❤ä❛ ♠➣♥ p∗ , j(p∗ − p) ≤ ✈ỵ✐ ♠å✐ p ∈ C ✳ ỵ E2 ổ õ t ổ ữợ tữỡ ự ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝ x, y = x1 y1 + x2 y2 ✈➔ x = (x21 + x22 )1/2 ✱ ✈ỵ✐ x = (x1 , x2 )✱ y = (y1 , y2 ) ∈ E2 ✳ ❱➼ ❞ö ✷✳✷✳✼ ❳➨t ❜➔✐ t♦→♥ ❝ü❝ trà ❦❤æ♥❣ r➔♥❣ ❜✉ë❝✿ t➻♠ ♠ët ✤✐➸♠ p∗ ∈ E2 s❛♦ ❝❤♦ f1 (p∗ ) = inf f1 (x), x∈E ✭✷✳✷✽✮ tr♦♥❣ ✤â f1 (x) = x − PC x /2 ✈ỵ✐ ♠å✐ x ∈ E2 ✈➔ C = {x = (x1 , x2 ) ∈ E2 : x ≤ 1}✳ ❚❛ t❤➜② t➟♣ ♥❣❤✐➺♠ C ữợ f t↕✐ ✤✐➸♠ x ∈ E2 ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿ (0; 0), x ∈ C, ∂f1 (x) = f1 (x) = [(λ − 1)/λ](x1 , x2 ), x ∈ / C, ð ✤➙② λ = (x21 + x22 )1/2 ✳ ❉♦ ✤â✱ (x , x ), x ∈ C, −1 (I + r∂f1 ) (x) = [λ/(λ + r(λ − 1))](x1 , x2 ), x ∈ / C ▲➜② u = (0; 2)✱ t❛ ♥❤➟♥ ✤÷đ❝ ❧í✐ ❣✐↔✐ p∗ = (0; 1) = PC (0, 2)✳ ❈→❝ ❦➳t q✉↔ t➼♥❤ t♦→♥ ❝õ❛ ❝❤♦ ❞➣② ❧➦♣ ✭✷✳✷✷✮ ✈ỵ✐ tk = 0.5/(k + 1); rk = 0.02 + 1/k ✱ ✸✻ ek = ✈➔ ♠ët ✤✐➸♠ ①✉➜t ♣❤→t x1 = (1.0; 2.0) ✤÷đ❝ ♠ỉ t↔ tr♦♥❣ ❇↔♥❣ ✷✳✶✳ ❇↔♥❣ ✷✳✶ k xk+1 xk+1 k xk+1 xk+1 ✶✵ ✵✳✵✼✻✽✶✼✽✷✽✾ ✶✳✵✹✹✼✺✹✵✾✵✽ ✶✵✵ ✵✳✵✵✽✸✻✽✽✾✹✹ ✶✳✵✵✹✾✹✵✶✼✼✼ ✷✵ ✵✳✵✹✵✷✹✻✸✺✺✶ ✶✳✵✷✸✺✾✵✵✻✶✸ ✷✵✵ ✵✳✵✵✹✷✵✺✷✽✶✹ ✶✳✵✵✷✷✹✽✹✾✸✷ ✸✵ ✵✳✵✷✼✷✻✺✶✹✻✺ ✶✳✵✶✻✵✷✹✺✹✶✺ ✸✵✵ ✵✳✵✵✷✽✵✽✶✼✽✵ ✶✳✵✵✶✻✺✾✾✺✸✷ ✹✵ ✵✳✵✷✵✻✶✺✺✹✻✻ ✶✳✵✶✷✶✸✹✸✹✵✺ ✹✵✵ ✵✳✵✵✷✶✵✼✽✽✻✷ ✶✳✵✵✶✷✹✻✷✶✾✵ ✺✵ ✵✳✵✶✻✺✼✸✹✺✸✷ ✶✳✵✵✾✼✻✹✷✹✵✹ ✺✵✵ ✵✳✵✵✶✻✽✼✶✺✵✼ ✶✳✵✵✵✾✾✼✺✼✽✸ ❱➼ ❞ö ✷✳✷✳✽ ❳➨t ❜➔✐ t♦→♥ ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✿ t➻♠ ♠ët ✤✐➸♠ p∗ ∈ E2 s❛♦ ❝❤♦ ✭✷✳✷✾✮ fi (p∗ ) = inf fi (x), i = 1, 2, x∈E tr♦♥❣ ✤â f1 (x) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ tr♦♥❣ ❱➼ ❞ư ✷✳✷✳✼ ✈➔ f2 (x) = ✈ỵ✐ x ∈ E2 ♥➳✉ x2 ≤ 0.5 ✈➔ f2 (x) = x2 − 0.5✱ ♥➳✉ x2 > 0.5✳ ❚ø ✤â✱ (x , x ), x2 ≤ 0.5, −1 (I + r∂f2 ) (x) = (x1 , x2 /(1 + r)), x2 > 0.5 ❱ỵ✐ u = (0; 2)✱ ♥❣❤✐➺♠ p∗ = (0; 0.5) ❝õ❛ ✭✷✳✷✾✮ t❤ä❛ ♠➣♥ p∗ = PC u✳ ❚❛ sû ❞ö♥❣ ữỡ ợ tr tk ữ tr ❱➼ ❞ö ✷✳✷✳✼✱ rk = 0.02 + 1/(k + 1)✱ ek = e˜k = t2k (1; 1) ✈➔ ❝ò♥❣ ✤✐➸♠ ❦❤ð✐ ✤➛✉ z = x1 = (1; 2)✳ ❈→❝ ❦➳t q✉↔ ❜➡♥❣ sè ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❇↔♥❣ ✷✳✷✳ ❇↔♥❣ ✷✳✷ k z1k z2k k z1k z2k ✶✵ ✵✳✶✶✾✼✺✺✽✽✾✽ ✵✳✾✽✷✹✶✾✸✼✽✹ ✶✵✵ ✵✳✵✷✹✵✻✶✹✷✺✶ ✵✳✺✼✹✽✼✹✼✽✸✾ ✷✵ ✵✳✵✼✽✶✷✺✺✻✾✼ ✵✳✾✶✺✶✷✸✾✺✻✻ ✷✵✵ ✵✳✵✶✸✽✵✾✶✸✶✾ ✵✳✹✾✾✼✺✽✶✾✷✼ ✸✵ ✵✳✵✺✾✷✸✼✵✺✻✾ ✵✳✽✺✷✼✶✹✵✹✽✼ ✸✵✵ ✵✳✵✵✾✽✾✹✵✸✼✻ ✵✳✺✵✷✸✽✽✸✻✹✹ ✹✵ ✵✳✵✹✽✷✶✽✼✵✵✹ ✵✳✼✾✽✼✼✻✼✸✽✷ ✹✵✵ ✵✳✵✵✼✽✺✶✻✻✵✺ ✵✳✹✾✺✺✹✵✵✶✼✹ ✺✵ ✵✳✵✹✵✾✶✻✺✶✸✹ ✵✳✼✺✶✷✺✼✹✼✽✻ ✺✵✵ ✵✳✵✵✻✹✺✸✽✺✼✺ ✵✳✹✾✷✸✺✸✵✶✼✼ ✸✼ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ❝→❝ ♥ë✐ ❞✉♥❣ s❛✉ ✤➙②✿ ✭✶✮ ❚r➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ổ ỗ õ ●➙t❡❛✉① ✤➲✉❀ ♥➯✉ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝✱ →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉❀ t♦→♥ tû ❣✐↔✐ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ ✭✷✮ ●✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ j ✲✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ tr➻♥❤ ❜➔② ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛✲ ♥❛❝❤❀ tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ j ✲✤ì♥ ✤✐➺✉ tr➯♥ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ✭✸✮ ❚r➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣ ❞è❝ ♥❤➜t✱ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ✭✹✮ ❚r➻♥❤ ❜➔② ❜❛ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❦➳t ❤đ♣ ợ ữỡ ữớ ố t ởt ữỡ ➞♥ ✈➔ ❤❛✐ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❤✐➺♥✮ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr➯♥ t➟♣ r➔♥❣ ❜✉ë❝ ❧➔ t➟♣ ❦❤ỉ♥❣ ✤✐➸♠ ❝❤✉♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ m✲j ✲✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ✭✺✮ ❚r➻♥❤ ❜➔② ❤❛✐ ✈➼ ❞ư sè ❣✐↔✐ ❜➔✐ t♦→♥ ❝ü❝ trà tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❤❛✐ ❝❤✐➲✉✳ ✸✽ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ❚r➛♥ ❱ô ❚❤✐➺✉✱ ◆❣✉②➵♥ ❚❤à ❚❤✉ ❚❤õ② ✭✷✵✶✶✮✱ ●✐→♦ tr➻♥❤ ❚è✐ ÷✉ ♣❤✐ t✉②➳♥✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❬✷❪ ❍♦➔♥❣ ❚ö② ✭✷✵✵✺✮✱ ●✐↔✐ t➼❝❤ ❤➔♠✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❚✐➳♥❣ ❆♥❤ ❬✸❪ ❘✳P✳ ❆❣❛r✇❛❧✱ ❉✳ ❖✬❘❡❣❛♥✱ ❉✳❘✳ ❙❛❤✉ ✭✷✵✵✾✮✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❢♦r ▲✐♣s❝❤✐t③✐❛♥✲t②♣❡ ▼❛♣♣✐♥❣s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✱ ❙♣r✐♥❣❡r✳ ❬✹❪ ❑✳ ❆♦②❛♠❛✱ ❍✳ ■✐❞✉❦❛✱ ❲✳ ❚❛❦❛❤❛s❤✐ ✭✷✵✵✻✮✱ ✧❲❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛♥ ✐t❡r❛t✐✈❡ s❡q✉❡♥❝❡ ❢♦r ❛❝❝r❡t✐✈❡ ♦♣❡r❛t♦rs ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✧✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❆♣♣❧✳✱ ✷✵✵✻✱ ❆rt✳ ♥♦✳ ✸✺✸✾✵✳ ❬✺❪ ❖✳❆✳ ❇♦✐❦❛♥②♦ ❛♥❞ ●✳ ▼♦r♦s❛♥✉ ✭✷✵✶✵✮✱ ✧❆ ♣r♦①✐♠❛❧ ♣♦✐♥t ❛❧✲ ❣♦r✐t❤♠ ❝♦♥✈❡r❣✐♥❣ str♦♥❣❧② ❢♦r ❣❡♥❡r❛❧ ❡rr♦rs✧✱ ❖♣t✐♠✳ ▲❡tt✳✱ ✹✱ ✻✸✺✕✻✹✶✳ ❬✻❪ ◆❣✳ ❇✉♦♥❣ ✭✷✵✶✽✮✱ ✧❙t❡❡♣❡st✲❞❡s❝❡♥t ♣r♦①✐♠❛❧ ♣♦✐♥t ❛❧❣♦r✐t❤♠s ❢♦r ❛ ❝❧❛ss ♦❢ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✧✱ ♠❛t✐s❝❤❡ ◆❛❝❤r✐❝❤t❡♥✱ ✷✾✶✭✽✲✾✮✱ ✶✶✾✶✕✶✷✼✵✳ ▼❛t❤❡✲ ❬✼❪ ▲✳❈✳ ❈❡♥❣✱ ◗✳❍✳ ❆♥s❛r✐✱ ❛♥❞ ❏✳❈❤✳ ❨❛♦ ✭✷✵✵✽✮✱ ✧▼❛♥♥✲t②♣❡ st❡❡♣❡st✲❞❡s❝❡♥t ❛♥❞ ♠♦❞✐❢✐❡❞ ❤②❜r✐❞ st❡❡♣❡st ❞❡s❝❡♥t ♠❡t❤♦❞s ❢♦r ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✧✱ ❖♣t✐♠✳✱ ✷✾✭✾✲✶✵✮✱ ✾✽✼✕✶✵✸✸✳ ◆✉♠❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ✸✾ ❬✽❪ ❋✳ ❉❡✉ts❝❤✱ ■✳ ❨❛♠❛❞❛ ✭✶✾✾✽✮✱ ✧▼✐♥✐♠✐③✐♥❣ ❝❡rt❛✐♥ ❝♦♥✈❡① ❢✉♥❝✲ t✐♦♥s ♦✈❡r t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ❢✐①❡❞ ♣♦✐♥t s❡ts ♦❢ ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣s✧✱ ◆✉♠❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❖♣t✐♠✳✱ ✶✾✱ ✸✸✕✺✻✳ ❬✾❪ ❙❤✳ ❑❛❦✐♠✉r❛ ❛♥❞ ❲✳ ❚❛❦❛❤❛s❤✐ ✭✷✵✵✵✮✱ ✧❲❡❛❦ ❛♥❞ str♦♥❣ ❝♦♥✲ ✈❡r❣❡♥❝❡ ♦❢ s♦❧✉t✐♦♥s t♦ ❛❝❝r❡t✐✈❡ ♦♣❡r❛t♦r ✐♥❝❧✉s✐♦♥s ❛♥❞ ❛♣♣❧✐❝❛✲ t✐♦♥s✧✱ ❙❡t✲❱❛❧✉❡❞ ❆♥❛❧✳✱ ✽✱ ✸✻✶✕✸✼✹✳ ❬✶✵❪ ❙✳ ❘❡✐❝❤ ✭✶✾✼✸✮✱ ✧❆s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ ❝♦♥tr❛❝t✐♦♥s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✧✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✹✹✭✶✮✱ ✺✼✕✼✵✳ ❬✶✶❪ ❉✳❘✳ ❙❛❤✉ ❛♥❞ ❏✳❈❤✳ ❨❛♦ ✭✷✵✶✶✮✱ ✧❚❤❡ ♣r♦①✲❚✐❦❤♦♥♦✈ r❡❣✉❧❛r✐③❛✲ t✐♦♥ ♠❡t❤♦❞ ❢♦r ♣r♦①✐♠❛❧ ♣♦✐♥t ❛❧❣♦r✐t❤♠ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✧✱ ●❧♦❜❛❧ ❖♣t✐♠✳✱ ✺✶✱ ✻✹✶✕✻✺✺✳ ❏✳ ❬✶✷❪ ●✳ ❙t❛♠♣❛❝❝❤✐❛ ✭✶✾✻✹✮✱ ✧❋♦r♠❡s ❜✐❧✐♥➨❛✐r❡s ❝♦❡r❝✐t✐✈❡s s✉r ❧❡s ❡♥✲ s❡♠❜❧❡s ❝♦♥✈❡①❡s✧✱ ❈✳ ❘✳ ❆❝❛❞✳ ❙❝✐✳ P❛r✐s✱ ✷✺✽✱ ✹✹✶✸✕✹✹✶✻✳ ❬✶✸❪ ❈❤✳❆✳ ❚✐❛♥ ❛♥❞ ❨✳ ❙♦♥❣ ✭✷✵✶✸✮✱ ✧❙tr♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛ r❡❣✉✲ ❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ❘♦❝❦❛❢❡❧❧❛r✬s ♣r♦①✐♠❛❧ ♣♦✐♥t ❛❧❣♦r✐t❤♠✧✱ ●❧♦❜❛❧ ❖♣t✐♠✳✱ ✺✺✱ ✽✸✶✕✽✸✼✳ ❏✳ ❬✶✹❪ ❍✳❑✳ ❳✉ ✭✷✵✵✸✮✱ ✧❆♥ ✐t❡r❛t✐✈❡ ❛♣♣r♦❛❝❤ t♦ q✉❛❞r❛t✐❝ ♦♣t✐♠✐③❛✲ t✐♦♥✧✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳✱ ✶✶✻✱ ✻✺✾✕✻✼✽✳ ❬✶✺❪ ❍✳❑✳ ❳✉ ✭✷✵✵✻✮✱ ✧❙tr♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛♥ ✐t❡r❛t✐✈❡ ♠❡t❤♦❞ ❢♦r ♥♦♥❡①♣❛♥s✐✈❡ ❛♥❞ ❛❝❝r❡t✐✈❡ ♦♣❡r❛t♦rs✧✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✸✶✹✱ ✻✸✶✕✻✹✸✳ ❬✶✻❪ ❍✳❑✳ ❳✉ ✭✷✵✵✻✮✱ ✧❆ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞ ❢♦r t❤❡ ♣r♦①✐♠❛❧ ♣♦✐♥t ❛❧❣♦r✐t❤♠✧✱ ❏✳ ●❧♦❜❛❧ ❖♣t✐♠✳✱ ✸✻✱ ✶✶✺✕✶✷✺✳ ❬✶✼❪ ■✳ ❨❛♠❛❞❛ ✭✷✵✵✶✮✱ ✧❚❤❡ ❤②❜r✐❞ st❡❡♣❡st✲❞❡s❝❡♥t ♠❡t❤♦❞ ❢♦r ✈❛r✐❛✲ t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ♣r♦❜❧❡♠s ♦✈❡r t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ❢✐①❡❞ ♣♦✐♥t ■♥❤❡r❡♥t❧② P❛r❛❧❧❡❧ ❆❧❣♦r✐t❤♠s ✐♥ ❋❡❛s✐❜✐❧✐t② ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ❚❤❡✐r ❆♣♣❧✐❝❛t✐♦♥s✱ ❈❤❛♣✲ s❡ts ♦❢ ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣s✧✱ t❡r ✽✱ ✹✼✸✕✺✵✹✳