Xấp xỉ nghiệm cho bất đẳng thức biến phân với họ vô hạn các ánh xạ không giãn (tt)

Xấp xỉ nghiệm cho bất đẳng thức biến phân với họ vô hạn các ánh xạ không giãn (tt)Xấp xỉ nghiệm cho bất đẳng thức biến phân với họ vô hạn các ánh xạ không giãn (tt)Xấp xỉ nghiệm cho bất đẳng thức biến phân với họ vô hạn các ánh xạ không giãn (tt)Xấp xỉ nghiệm cho bất đẳng thức biến phân với họ vô hạn các ánh xạ không giãn (tt)Xấp xỉ nghiệm cho bất đẳng thức biến phân với họ vô hạn các ánh xạ không giãn (tt)Xấp xỉ nghiệm cho bất đẳng thức biến phân với họ vô hạn các ánh xạ không giãn (tt)Xấp xỉ nghiệm cho bất đẳng thức biến phân với họ vô hạn các ánh xạ không giãn (tt)Xấp xỉ nghiệm cho bất đẳng thức biến phân với họ vô hạn các ánh xạ không giãn (tt)Xấp xỉ nghiệm cho bất đẳng thức biến phân với họ vô hạn các ánh xạ không giãn (tt)Xấp xỉ nghiệm cho bất đẳng thức biến phân với họ vô hạn các ánh xạ không giãn (tt)

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ◆●❯❨➍◆ ❙❖◆● ❍⑨ ❳❻P ❳➓ ◆●❍■➏▼ ❈❍❖ ❇❻❚ ✣➃◆● ❚❍Ù❈ ❇■➌◆ P❍❹◆ ❱❰■ ❍➴ ❱➷ ❍❸◆ ❈⑩❈ ⑩◆❍ ❳❸ ❑❍➷◆● ●■❶◆ ◆❣➔♥❤✿ ❚♦→♥ ●✐↔✐ t➼❝❤ ▼➣ sè✿ ✾✹✻✵✶✵✷ ❚➶▼ ❚➁❚ ▲❯❾◆ ⑩◆ ❚■➌◆ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✽ ❈æ♥❣ tr➻♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐✿ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ữớ ữợ ●❙✳❚❙✳ ◆❣✉②➵♥ ❇÷í♥❣ P❤↔♥ ❜✐➺♥ ✶✿ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ P❤↔♥ ❜✐➺♥ ✷✿ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ P❤↔♥ ❜✐➺♥ ✸✿ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ▲✉➟♥ →♥ s➩ ✤÷đ❝ ❜↔♦ trữợ ỗ rữớ t↕✐✿ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ỗ t ♥➠♠ ✷✵✶✽ ❈â t❤➸ t➻♠ ❤✐➸✉ ❧✉➟♥ →♥ t↕✐ t❤÷ ✈✐➺♥✿ ✲ ❚❤÷ ✈✐➺♥ ◗✉è❝ ❣✐❛ ❱✐➺t ◆❛♠ ✲ ❚r✉♥❣ t➙♠ ❤å❝ ❧✐➺✉ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✲ ❚❤÷ ✈✐➺♥ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ▼ð ✤➛✉ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✤➣ ✤÷đ❝ ✤➲ ①✉➜t ✈➔♦ ♥❤ú♥❣ ♥➠♠ ✤➛✉ ❝õ❛ t❤➟♣ ♥✐➯♥ ✻✵ t❤➳ ❦➾ ❳❳✱ ❣➢♥ ❧✐➲♥ ✈ỵ✐ ♥❤ú♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ▲✐♦♥s✱ ❙t❛♠♣❛❝❝❤✐❛ ✈➔ ❝ë♥❣ sü ✭▲✐♦♥s ✈➔ ❙t❛♠♣❛❝❝❤✐❛✱ ✶✾✻✺✱ ✶✾✻✼❀ ❍❛rt♠❛♥ ✈➔ ❙t❛♠♣❛❝❝❤✐❛✱ ✶✾✻✻✮✳ ❚ø ✤â ✤➳♥ ♥❛②✱ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❧✉æ♥ ❧➔ ♠ët ❝❤õ ✤➲ ♥❣❤✐➯♥ ❝ù✉ ♠❛♥❣ t➼♥❤ t❤í✐ sü ✈➔ t❤✉ ❤ót ✤÷đ❝ sü q✉❛♥ t➙♠ ❝õ❛ ♥❤✐➲✉ ♥❤➔ ❦❤♦❛ ❤å❝ tr♦♥❣ ✈➔ ữợ t ữ t ỹ tr ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣❀ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣❀ ❜➔✐ t ũ ữỡ tr ợ t tỷ ỡ t♦→♥ ❜✐➯♥ ❝â ❞↕♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝â t❤➸ q✉② ✈➲ ♠æ ❤➻♥❤ ❜➔✐ t t tự ữợ tt t❤➼❝❤ ❤đ♣✳ ❱➻ t❤➳ ❜➔✐ t♦→♥ ♥➔② ❧➔ ♠ët ❝ỉ♥❣ ❝ư ♠↕♥❤ ✈➔ t❤è♥❣ ♥❤➜t tr♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ♥❤✐➲✉ ♠ỉ ❤➻♥❤ ❜➔✐ t♦→♥ ❧➼ t❤✉②➳t ✈➔ ù♥❣ ❞ö♥❣ t❤ü❝ t➳✳ Ð ❱✐➺t ◆❛♠✱ t❤❡♦ ♥❤✐➲✉ ❝♦♥ ✤÷í♥❣ t✐➳♣ ❝➟♥ ❦❤→❝ ♥❤❛✉✱ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ❝â ♥❤ú♥❣ ✤â♥❣ ❣â♣ q✉❛♥ trå♥❣ ❝❤♦ ❜➔✐ t♦→♥ ♥➔② ❝â t❤➸ ❦➸ ✤➳♥ ♥❤÷ ❝→❝ ♥❤â♠ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ●❙✳❚❙❑❍✳ P❤↕♠ ❑ý ❆♥❤ ✭P✳❑✳ ❆♥❤ ✈➔ ✤t❣✱ ✷✵✶✺✱ ✷✵✶✼✮❀ ●❙✳❚❙❑❍✳ P❤❛♥ ◗✉è❝ ❑❤→♥❤ ✭P✳◗✳ ❑❤→♥❤ ✈➔ ✤t❣✱ ✷✵✵✺✱ ✷✵✵✻✮❀ ●❙✳❚❙❑❍✳ ✣✐♥❤ ❚❤➳ ▲ö❝ ✭✣✳❚✳ ▲ư❝ ✈➔ ✤t❣✱ ✷✵✵✽✱ ✷✵✶✹✮❀ ●❙✳❚❙❑❍✳ ▲➯ ❉ơ♥❣ ▼÷✉ ✭▲✳❉✳ ▼÷✉ ✈➔ ✤t❣✱ ✷✵✵✺✱ ✷✵✶✷✮❀ ●❙✳❚❙❑❍✳ P❤↕♠ ❍ú✉ ❙→❝❤ ✭P✳❍✳ ❙→❝❤ ✈➔ ✤t❣✱ ✷✵✵✹✱ ✷✵✵✽✮❀ ●❙✳❚❙❑❍✳ ◆❣✉②➵♥ ❳✉➙♥ ❚➜♥ ✭◆✳❳✳ ❚➜♥ ✈➔ ✤t❣✱ ✷✵✶✷✱ ✷✵✶✸✮❀ ●❙✳❚❙❑❍✳ ◆❣✉②➵♥ ✣æ♥❣ ❨➯♥ ✭◆✳✣✳ ❨➯♥ ✈➔ ✤t❣✱ ✷✵✵✺✱ ✷✵✵✽✮❀ ●❙✳❚❙✳ ◆❣✉②➵♥ ❇÷í♥❣ ✭◆✳ ❇÷í♥❣ ✈➔ ✤t❣✱ ✷✵✶✶✱ ✷✵✶✸✱ ✷✵✶✺✱ ✷✵✶✻✮❀ P●❙✳❚❙✳ P❤↕♠ ◆❣å❝ ❆♥❤ ✭P✳◆✳ ❆♥❤ ✈➔ ✤t❣✱ ✷✵✵✹✱ ✷✵✵✺✱ ✷✵✶✵✮❀ P●❙✳❚❙✳ ◆❣✉②➵♥ ◗✉❛♥❣ ❍✉② ✭◆✳◗✳ ❍✉② ✈➔ ✤t❣✱ ✷✵✶✶✮ ✈➔ P●❙✳❚❙✳ ◆❣✉②➵♥ ❚❤à ❚❤✉ ❚❤õ② ✭◆✳❚✳❚✳ ❚❤õ② ✈➔ ✤t❣✱ ✷✵✶✸✱ ✷✵✶✻✮ ❇➯♥ ❝↕♥❤ ✤â✱ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔ ♠ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ❝ô♥❣ ✤➣ ✈➔ ✤❛♥❣ ❧➔ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ♥❤✐➲✉ t→❝ ❣✐↔ ❧➔ t s ự s tr ữợ ổ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❝ê ✤✐➸♥ ❝â ❞↕♥❣✿ ❚➻♠ x∗ ∈ C s❛♦ ❝❤♦✿ F (x∗), x − x∗ ≥ 0, ∀x ∈ C, ✭✵✳✶✮ tr♦♥❣ ✤â C t ỗ õ rộ ổ ❣✐❛♥ ❍✐❧❜❡rt H ✈➔ F : H → H ❧➔ →♥❤ ①↕ ①→❝ ✤à♥❤ tr➯♥ H ✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ t C t ữủ ữợ ➞♥ ❧➔ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ ❤❛② ✈æ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ t❤➻ ❜➔✐ t♦→♥ ✭✵✳✶✮ ❝â ❧✐➯♥ ❤➺ ✈ỵ✐ ♥❤✐➲✉ ❜➔✐ t♦→♥ t❤ü❝ t✐➵♥ ♥❤÷ ❜➔✐ t♦→♥ ❦❤ỉ✐ ♣❤ư❝ t➼♥ ❤✐➺✉✱ ❜➔✐ t♦→♥ ♣❤➙♥ ♣❤è✐ ❜➠♥❣ t❤ỉ♥❣✱ ❦✐➸♠ s♦→t ♥➠♥❣ ❧÷đ♥❣ ❝❤♦ ❤➺ t❤è♥❣ ♠↕♥❣ ✈✐➵♥ t❤æ♥❣ ❈❉▼❆ ✈➔ ❦➽ t❤✉➟t ①û ❧➼ t➼♥ ❤✐➺✉ ❜➠♥❣ t➛♥✳ ✣➸ ❝â t❤➸ ù♥❣ ❞ö♥❣ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔♦ t❤ü❝ t✐➵♥✱ ✤á✐ ❤ä✐ ♣❤↔✐ ❝â ♥❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ sè ❤✐➺✉ q✉↔ ❝❤♦ ❜➔✐ t♦→♥ ♥➔②✳ ❱➻ ❧➩ ✤â✱ ♠ët tr ỳ ữợ ự q trồ ❞➔♥❤ ✤÷đ❝ sü q✉❛♥ t➙♠ ❝õ❛ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ tr ữợ õ t ữỡ ợ t t ❝↔✐ t✐➳♥ ❤✐➺✉ q✉↔ ❝õ❛ ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ✤➣ ❝â✳ ❈❤♦ ✤➳♥ ♥❛② ♥❣÷í✐ t❛ ✤➣ t❤✐➳t ❧➟♣ ✤÷đ❝ ♥❤✐➲✉ ❦➽ t❤✉➟t ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❞ü❛ tr➯♥ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❝õ❛ ●♦❧❞st❡✐♥ ✭✶✾✻✹✮✱ P♦❧②❛❦ ✭✶✾✻✻✱ ✶✾✻✼✱ ✶✾✻✾✮✱ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❝õ❛ ▼❛rt✐♥❡t ✭✶✾✼✵✮✱ ❘♦❦❛❢❢❡❧❧❛r ✭✶✾✼✻✮✱ ỵ t ữỡ ❤✐➺✉ ❝❤➾♥❤ ❞↕♥❣ ❇r♦✇❞❡r✲❚✐❦❤♦♥♦✈ ✭❇r♦✇❞❡r✱ ✶✾✻✻❀ ❚✐❦❤♦♥♦✈✱ ✶✾✻✸✮✱ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❤✐➺✉ ❝❤➾♥❤ ❝õ❛ ▲❡❤❞✐❧✐ ✈➔ ▼♦✉❞❛❢✐ ✭✶✾✾✻✮✱ ❘②❛③❛♥ts❡✈❛ ✭✷✵✵✷✮ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ q✉→♥ t➼♥❤ ❞♦ ❆❧✈❛r❡③ ✈➔ ❆tt♦✉❝❤ ✭✷✵✵✶✮ ✤➲ ①✉➜t ❤♦➦❝ ❞ü❛ tr➯♥ ♠ët sè ❦➽ t❤✉➟t t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛s♥♦s❡❧✬s❦✐✐✲▼❛♥♥ ✭▼❛♥♥✱ ✶✾✺✸❀ ❑r❛s♥♦s❡❧✬s❦✐✐✱ ✶✾✺✺✮✱ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❍❛❧♣❡r♥ ✭✶✾✻✼✮ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ♠➲♠ ✭▼♦✉❞❛❢✐✱ ✷✵✵✵✮✳ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✤✐➸♥ ❤➻♥❤ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ✭✵✳✶✮ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣r❛❞✐❡♥t ✭●♦❧❞st❡✐♥✱ ✶✾✻✹❀ ❩❡✐❞❧❡r✱ ✶✾✾✵✮ ✤÷đ❝ ♠ỉ t↔ ♥❤÷ s❛✉✿  x ∈ C, xk+1 = PC (I − ρF )(xk ), k = 0, 1, 2, ✭✵✳✷✮ tr♦♥❣ ✤â PC ❧➔ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tø H ❧➯♥ C ✱ I ❧➔ →♥❤ ①↕ ✤ì♥ ✈à tr➯♥ H ✈➔ ρ ❧➔ ♠ët ❤➡♥❣ sè ❞÷ì♥❣ ❝è ✤à♥❤✳ P❤÷ì♥❣ ♣❤→♣ ✭✵✳✷✮ ❝â ❝➜✉ tró❝ ✤ì♥ ❣✐↔♥ ♥➯♥ ✈✐➺❝ ✈➟♥ ❞ư♥❣ tr♦♥❣ ♥❤ú♥❣ t➻♥❤ ❤✉è♥❣ ❝ư t❤➸ ❦❤→ t❤✉➟♥ t✐➺♥✳ P❤÷ì♥❣ ♣❤→♣ ♥➔② ❧➔ sü ❦➳t ❤đ♣ ❣✐ú❛ ✈✐➺❝ sû ❞ư♥❣ trü❝ t✐➳♣ ❞↕♥❣ ✤â♥❣ ❝õ❛ ♣❤➨♣ ❝❤✐➳✉ PC ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❦✐➸✉ ✤÷í♥❣ ❞è❝ ♥❤➜t✳ ◆❤í ❝â ♥❤ú♥❣ t✐➳♥ ❜ë ✤→♥❣ ❦➸ tr♦♥❣ ❧➼ t❤✉②➳t ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ð t❤➳ ❦➾ ❳❳✱ ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ✤÷đ❝ ❨❛♠❛❞❛ ✈➔ ❝ë♥❣ sü ✭❨❛♠❛❞❛ ✈➔ ✤t❣✱ ✶✾✾✽✱ ✶✾✾✾✮ ✤➲ ①✉➜t ♥❤÷ ❧➔ ♠ët ❜✐➳♥ t❤➸ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ✤➸ t➻♠ ❝ü❝ t✐➸✉ ❝õ❛ ♠ët ỗ tr t t →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳ ✣➦❝ ✤✐➸♠ ❝❤➼♥❤ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ❧➔ ❞ò♥❣ ❞↕♥❣ ✤â♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ❜➜t ❦➻ ♠➔ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♥â ❧➔ t➟♣ r➔♥❣ ❜✉ë❝ ❝õ❛ ❜➔✐ t♦→♥✳ ▼➦t ❦❤→❝✱ tr♦♥❣ ♥❤✐➲✉ ❜➔✐ t♦→♥ t❤ü❝ t➳✱ ❝❤➥♥❣ ❤↕♥ ❜➔✐ t♦→♥ ①û ❧➼ t➼♥ ❤✐➺✉ ✭■✐❞✉❦❛✱ ✷✵✶✵✮✱ ❦✐➸♠ s♦→t ♥➠♥❣ ❧÷đ♥❣ ❝❤♦ ❤➺ t❤è♥❣ ♠↕♥❣ ✈✐➵♥ t❤æ♥❣ ❈❉▼❆ ✭■✐❞✉❦❛✱ ✷✵✶✷✮ ❤♦➦❝ ♣❤➙♥ ♣❤è✐ ❜➠♥❣ t❤æ♥❣ ✭■✐❞✉❦❛ ✈➔ ❯❝❤✐❞❛✱ ✷✵✶✶✮ ❝â t❤➸ ✤÷❛ ✈➲ ❜➔✐ t♦→♥ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr➯♥ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ♠ët ❤♦➦❝ ♠ët ❤å ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ỡ ỳ ú t t r t ỗ õ õ t ữợ ✤÷đ❝ ❝õ❛ ❝→❝ ♥û❛ ❦❤ỉ♥❣ ❣✐❛♥✱ ❞♦ ✤â ❧➔ ❣✐❛♦ ✤➳♠ ✤÷đ❝ ❝õ❛ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ❧➔ ❝→❝ t♦→♥ tû ❝❤✐➳✉ ❧➯♥ ♥❤ú♥❣ ♥û❛ ❦❤æ♥❣ ❣✐❛♥ ♥➔②✳ ❱➻ t❤➳ ❜➔✐ t♦→♥ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✵✳✶✮ tr➯♥ ♠ët t➟♣ ❝♦♥ ỗ õ õ t q t t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr➯♥ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ❑❤✐ ✤â✱ ♠ët ✈➜♥ ✤➲ ✤➦t r❛ ❧➔ ①→❝ ✤à♥❤ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ①➜♣ ①➾ ♥❣❤✐➺♠ ❝❤♦ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✵✳✶✮ ♥❤÷ t❤➳ ♥➔♦ ♥➳✉ ❝❤ó♥❣ t❛ ❝â ❞↕♥❣ ❤✐➺♥ ❝õ❛ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ✸ ❣✐➣♥ Ti❄ ✭i ∈ I ✈ỵ✐ I ❧➔ t➟♣ ❝❤➾ sè ♥➔♦ õ t t tứ ỵ tữ ✤➣ ①➙② ❞ü♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ♠➔ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ❤ë✐ tư ♠↕♥❤ ✈➲ ♠ët t❤➔♥❤ ♣❤➛♥ ♥➡♠ tr♦♥❣ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ❤å ỳ ổ ỗ tớ tọ ♠➣♥ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✵✳✶✮✳ ❈ö t❤➸✱ ❦❤✐ C := ❋✐①(T ) ❧➔ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ♠ët →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✱ ❨❛♠❛❞❛ ✤➣ t❤✐➳t ❧➟♣ ✤÷đ❝ ✤à♥❤ ❧➼ ❤ë✐ tư ♠↕♥❤ s❛✉✳ ✣à♥❤ ❧➼ ✵✳✷✳ ❈❤♦ F : H → H ❧➔ →♥❤ ①↕ ❧✐➯♥ tư❝ L✲▲✐♣s❝❤✐t③ ✈➔ η ✲✤ì♥ T : H → H ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr➯♥ H ✈ỵ✐ ❋✐①(T ) = ∅✳ ●✐↔ λk ∈ (0, 1] t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥✿ ✭▲✶✮ k→∞ lim λk = 0, ✭▲✷✮ ∞ λk = ∞, k=1 ✭▲✸✮ ✤✐➺✉ ♠↕♥❤ tr➯♥ H ✳ ❈❤♦ sû ρ ∈ (0, 2η/L2) ✈➔ ❞➣② lim (λk − λk+1 )λ−2 k+1 = 0✳ k õ ợ tũ ỵ x0 ∈ H ✱ ❞➣② ❧➦♣ ①→❝ ✤à♥❤ ❜ð✐ xk+1 = T (xk ) − λk+1 ρF (T (xk )), k = 0, 1, 2, ✭✵✳✸✮ ❤ë✐ tö ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t x∗ ❝õ❛ ❜➔✐ t♦→♥ ✭✵✳✶✮✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ C ❧➔ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ Ti : H → H (i = 1, 2, 3, , N )✱ ❞➣② ❧➦♣ ①♦❛② ✈á♥❣ ①➜♣ ①➾ ♥❣❤✐➺♠ ❝❤♦ ❜➔✐ t♦→♥ ✭✵✳✶✮ ✤÷đ❝ ❨❛♠❛❞❛ ①➙② ❞ü♥❣ ❝â ❞↕♥❣ xk+1 = T[k+1] (xk ) − λk+1 ρF (T[k+1] (xk )), k = 0, 1, 2, ✭✵✳✹✮ ð ✤➙②✱ [k] := k ♠♦❞ N ❧➔ ❤➔♠ ♠♦❞✉❧♦ ❧➜② ❣✐→ trà tr♦♥❣ t➟♣ {1, 2, 3, , N }✳ ✣à♥❤ ❧➼ ✵✳✸✳ ❈❤♦ ❧➔ →♥❤ ①↕ ❧✐➯♥ tư❝ L✲▲✐♣s❝❤✐t③ ✈➔ η✲✤ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥ H ✳ ❈❤♦ N Ti : H → H ❧➔ ❤å ❤ú✉ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr➯♥ H ✈ỵ✐ C := ❋✐①(Ti) = ∅ ✈➔ F : H → H i=1 C = ❋✐①(T1 T2 TN ) = ❋✐①(T2 T3 TN T1 ) = · · · = ❋✐①(TN T1 TN −1 ) ●✐↔ sû ρ ∈ (0, 2η/L2) ✈➔ ❞➣② λk ∈ (0, 1] t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥✿ ∞ ∞ ∗ ✭▲✶✮ k→∞ lim λk = 0, ✭▲✷✮ λk = ∞, ✭▲✸✮ |λk − λk+N | < ∞ k=1 k=1 õ ợ tũ ỵ x0 ∈ H ✱ ❞➣② ❧➦♣ ✭✵✳✹✮ ❤ë✐ tö ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t x∗ ❝õ❛ ❜➔✐ t♦→♥ ✭✵✳✶✮✳ ❚ø ✤â ✤➳♥ ♥❛②✱ ✤➣ ❝â ♥❤✐➲✉ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ♥❤➡♠ ♠ð rë♥❣ ❤♦➦❝ ❝↔✐ t✐➳♥ ♣❤÷ì♥❣ ♣❤→♣ ❝õ❛ ❨❛♠❛❞❛ t ữợ t ữợ ❣✐↔♠ ♥❤➭ ✤✐➲✉ ❦✐➺♥ ✤➦t ❧➯♥ ❝→❝ ❞➣② t❤❛♠ sè ❧➦♣ ✭❳✉ ✈➔ ✤t❣✱ ✷✵✵✸❀ ❩❡♥❣ ✈➔ ✤t❣✱ ✷✵✵✼✮ ❤❛② ❧♦↕✐ ❜ä ❣✐↔ t❤✐➳t ✈➲ t➼♥❤ ❣✐❛♦ ❤♦→♥ tr➯♥ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ Ti ✭◆❣✉②➵♥ ❇÷í♥❣ ✈➔ ✤t❣✱ ✷✵✶✶✮✳ ❍♦➦❝✱ ①➨t ❜➔✐ t♦→♥ ✭✵✳✶✮ tr trữớ ủ tờ qt ỡ ợ C t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ✈æ ❤↕♥ ữủ ổ ữợ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✤÷đ❝ t❤✐➳t ❧➟♣ ①➜♣ ①➾ ♥❣❤✐➺♠ ❝❤♦ ❜➔✐ t♦→♥ ✭✵✳✶✮ t❤ỉ♥❣ q✉❛ ✈✐➺❝ ❞ò♥❣ →♥❤ ①↕ Wk ✭■❡♠♦t♦ ✈➔ ✤t❣ ✷✵✵✽❀ ❨❛♦ ✈➔ ✤t❣✱ ✷✵✶✵❀ ❲❛♥❣✱ ✷✵✶✶✮✳ ❚✉② ✈➟②✱ →♥❤ ①↕ Wk ❝â ❝➜✉ tró❝ ♣❤ù❝ t↕♣✳ ◆❣♦➔✐ r❛✱ ❝→❝ ❦➳t q✉↔ ♥â✐ tr➯♥ ✤➲✉ ✤÷đ❝ t❤✐➳t ❧➟♣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✈➔ ð ộ ữợ ữủ tỹ ✈á♥❣ ♥➯♥ ✤â ❧➔ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ t✉➛♥ tü✳ ▼ët ữợ ự rở tứ ổ ❍✐❧❜❡rt H tỵ✐ ❝→❝ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✭❈❡♥❣ ✈➔ ✤t❣✱ ✷✵✵✽❀ ❈❤✐❞✉♠❡ ✈➔ ✤t❣✱ ✷✵✶✶❀ ◆❣✉②➵♥ ❇÷í♥❣ ✈➔ ✤t❣✱ ✷✵✶✸✱ ✷✵✶✺✮✳ ◆ê✐ ❜➟t tr♦♥❣ ✤â ❧➔ ❤❛✐ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❞↕♥❣ ❤✐➺♥ ①➜♣ ①➾ ♥❣❤✐➺♠ ❝❤♦ ♠ët ❧ỵ♣ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❝õ❛ ◆❣✉②➵♥ ❇÷í♥❣ ✈➔ ❝ë♥❣ sü ✭✷✵✶✺✮✳ ❈→❝ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② sû ❞ư♥❣ →♥❤ ①↕ Sk ❝â ❝➜✉ tró❝ ✤ì♥ ❣✐↔♥ ✈➔ ❝â t❤➸ t➼♥❤ t♦→♥ s♦♥❣ s♦♥❣ ✤÷đ❝✳ ❈â t❤➸ ❦❤➥♥❣ ✤à♥❤ r➡♥❣✱ ✈✐➺❝ ①➙② ❞ü♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❧➔ ♠ët ✈➜♥ ✤➲ ✤÷đ❝ ♥↔② s✐♥❤ ♠ët ❝→❝❤ tü ♥❤✐➯♥ ✈➔ ❝➛♥ t❤✐➳t ✤➸ ❧➔♠ ♣❤♦♥❣ ♣❤ó ✈➔ t t ỵ tt t q trå♥❣ ♥➔②✳ ❱➻ ♥❤ú♥❣ ❧➼ ❞♦ ✤➣ ♣❤➙♥ t➼❝❤ ð tr➯♥✱ ❝❤ó♥❣ tỉ✐ ❧ü❛ ❝❤å♥ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ❝❤♦ ❧✉➟♥ →♥ ❧➔ ✧❳➜♣ ①➾ ♥❣❤✐➺♠ ❝❤♦ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈ỵ✐ ❤å ✈ỉ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✧✳ ▼ö❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ →♥ ♥➔② ❧➔ ♥❣❤✐➯♥ ❝ù✉ ✤➲ ①✉➜t ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❞↕♥❣ ❤✐➺♥ ①➜♣ ①➾ ♥❣❤✐➺♠ ❝❤♦ ♠ët ❧ỵ♣ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✳ ❈ư t❤➸✱ ❧ỵ♣ ❜➔✐ t♦→♥ ✤â ❧➔ ✧❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr➯♥ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕ tỹ ỗ t õ t ▲✉➟♥ →♥ ❣✐↔✐ q✉②➳t ❝→❝ ✈➜♥ ✤➲ s❛✉✿ ✶✳ ❳➙② ❞ü♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❞↕♥❣ ❤✐➺♥ ①➜♣ ①➾ ♥❣❤✐➺♠ ❝❤♦ ❧ỵ♣ ❜➔✐ t♦→♥ ♥❣❤✐➯♥ ❝ù✉ t❤ỉ♥❣ q✉❛ ✈✐➺❝ ✤➲ ①✉➜t ✈➔ sû ❞ư♥❣ ❝→❝ →♥❤ ①↕ ♠ỵ✐ S˜k , Sk S k ỗ tớ tt t tữỡ q ợ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ✤➣ ❝â✳ ✷✳ ⑩♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♠ỵ✐ ❝❤♦ ♠ët ❧ỵ♣ ❜➔✐ t♦→♥ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ ổ ữỡ ợ ❝❤♦ ♠ët ❧ỵ♣ ❜➔✐ t♦→♥ ①→❝ ✤à♥❤ ❦❤ỉ♥❣ ✤✐➸♠ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ →♥❤ ①↕ j ỡ ỹ ỗ ✤➛✉✱ ❜❛ ❝❤÷ì♥❣✱ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ữỡ ợ t sỡ ữủ ởt số ✤➲ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝➜✉ tró❝ ❤➻♥❤ ❤å❝ ❝õ❛ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ❧ỵ♣ ❜➔✐ t♦→♥ ♥❣❤✐➯♥ ❝ù✉✱ ♠ët sè ♠➺♥❤ ✤➲ ✈➔ ❜ê ✤➲ ❝➛♥ sû ❞ö♥❣ ❝❤♦ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ✤↕t ✤÷đ❝ ð ❝→❝ ❝❤÷ì♥❣ s❛✉ ❝õ❛ ❧✉➟♥ →♥✳ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ❜❛ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ♠ỵ✐ ❝õ❛ ❝❤ó♥❣ tỉ✐ ✈➲ ❝→❝ ✈➜♥ ✤➲ ♥➯✉ tr➯♥✳ ❈❤÷ì♥❣ ✸ ✤➲ ❝➟♣ ✤➳♥ ♠ët ❜➔✐ t♦→♥ t❤ü❝ t➳ ❧✐➯♥ q✉❛♥ ❝ò♥❣ ❝→❝ ✈➼ ❞ư ❝ư t❤➸ ♠✐♥❤ ❤å❛✳ ❈❤÷ì♥❣ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✸✳ ▼ët ❧ỵ♣ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✶✳✸✳✶ ▼æ ❤➻♥❤ ❜➔✐ t♦→♥ ❈❤♦ E ổ tỹ ỗ t ✈➔ ❝â ❝❤✉➞♥ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉✳ ❈❤♦ F : E → E ❧➔ →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈ỵ✐ ❤➺ sè η ✈➔ γ ✲❣✐↔ ❝♦ ❝❤➦t ✈ỵ✐∞η + γ > 1✳ ●✐↔ sû {Ti} ❧➔ ❤å ✈æ ❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr➯♥ E ✈ỵ✐ C := ❋✐①(Ti) = ∅✳ i=1 ∗ ▲ỵ♣ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✱ ❦➼ ❤✐➺✉ ❧➔ ❱■P (F, C)✱ ✤÷đ❝ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉✿ ❚➻♠ x∗ ∈ C s❛♦ ❝❤♦✿ F (x∗), j(x − x∗) ≥ 0, ∀x ∈ C, ✭✶✳✷✮ tr♦♥❣ ✤â j ❧➔ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ❝õ❛ E ✳ ✣✐➸♠ x∗ ∈ C t❤ä❛ ♠➣♥ ✭✶✳✷✮ ✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❱■P∗(F, C)✳ ✶✳✸✳✷ P❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ❚r♦♥❣ ♣❤➛♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ s➩ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t ♠ët sè ♥❣❤✐➯♥ ❝ù✉ ♠ð rë♥❣ ❤♦➦❝ ❝↔✐ ❜✐➯♥ ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ①➜♣ ①➾ ♥❣❤✐➺♠ ❝❤♦ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❝â ❞↕♥❣ ✭✵✳✶✮ ❤♦➦❝ ✭✶✳✷✮✳ ❑❤✐ C ❧➔ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✱ ♥➠♠ ✷✵✵✸✱ ❳✉ ✈➔ ❑✐♠ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❦➳t q✉↔ t÷ì♥❣ tü ✣à♥❤ ❧➼ ✵✳✷ ✈➔ ✣à♥❤ ❧➼ ✵✳✸ ❦❤✐ t❤❛② t❤➳ ✭▲✸✮ ✈➔ ✭▲✸✮∗ t÷ì♥❣ ù♥❣ ❜ð✐ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭▲✹✮ k→∞ lim λk /λk+1 = ✈➔ ✭▲✹✮∗ lim λk /λk+N = k→∞ ❈â t❤➸ t❤➜② r➡♥❣✱ ✤✐➲✉ ❦✐➺♥ ✭▲✹✮ ②➳✉ ❤ì♥ t❤ü❝ sü ✭▲✸✮✱ ❤ì♥ ♥ú❛ ✤✐➲✉ ❦✐➺♥ ✭▲✹✮ ❝❤♦ ♣❤➨♣ t❛ ❝â t❤➸ ❧ü❛ ❝❤å♥ ✈ỵ✐ ❞➣② t❤❛♠ sè ❝❤➼♥❤ t➢❝ {1/k} tr♦♥❣ ❦❤✐ ✤â ✭▲✸✮ ❦❤æ♥❣ t❤ä❛ ♠➣♥✳ ▼➦t ❦❤→❝✱ ❦❤æ♥❣ ❦❤â ❦❤➠♥ ✤➸ ❝❤➾ r❛ r➡♥❣ ✤✐➲✉ ❦✐➺♥ ✭▲✸✮∗ s✉② r❛ ✤✐➲✉ ❦✐➺♥ ✭▲✹✮∗ ♥➳✉ ❣✐ỵ✐ lim k /k+N tỗ t ❝ë♥❣ sü ✤➣ ✤➲ ①✉➜t ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ①♦❛② ✈á♥❣ k→∞ ✭✶✳✸✮ ✈ỵ✐ t❤❛♠ sè ρk+1 ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ❤➡♥❣ sè ❝è ✤à♥❤ ♥❤÷ tr♦♥❣ ✭✵✳✹✮ ✈➔ ✤✐➲✉ ❦✐➺♥ ✤➦t ❧➯♥ ❝→❝ ❞➣② t❤❛♠ sè ❧➦♣ ❝ơ♥❣ ✤÷đ❝ ❝↔✐ ❜✐➯♥ ✤➸ ✤↔♠ ❜↔♦ sü ❤ë✐ tö✳ xk+1 = T[k+1] (xk ) − λk+1 ρk+1 F (T[k+1] (xk )), k = 0, 1, 2, ✣à♥❤ ❧➼ ✶✳✸✳ ❈❤♦ ❧➔ →♥❤ ①↕ ❧✐➯♥ tư❝ L✲▲✐♣s❝❤✐t③ ✈➔ η✲✤ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥ H ✳ ❈❤♦ N Ti : H → H ❧➔ ❤å ❤ú✉ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr➯♥ H ✈ỵ✐ C := ❋✐①(Ti) = ∅ ✈➔ F : H → H i=1 C = ❋✐①(T1 T2 TN ) = ❋✐①(T2 T3 TN T1 ) = · · · = ❋✐①(TN T1 TN −1 ) ✻ ●✐↔ sû ρk ∈ (0, 2η/L2) ✈ỵ✐ ♠å✐ k ∈ N ✈➔ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❜↔♦ ✤↔♠✿ ✐✮ λk ∈ (0, 1) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭▲✷✮✱ ✐✐✮ |ρk − η/L2| ≤ η2 − aL2/L2 ✈ỵ✐ ➼t ♥❤➜t ♠ët a ∈ (0, η2/L2), ✐✐✐✮ k→∞ lim (ρk+N − (λk /λk+N )ρk ) = õ ợ tũ ỵ x0 H ✱ ♥➳✉ lim sup T[k+N ] T[k+1] (xk ) − xk+N , T[k+N ] T[k+1] (xk ) − xk ≤ k→∞ t❤➻ ❞➣② ❧➦♣ ✭✶✳✸✮ ❤ë✐ tư ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t x∗ ❝õ❛ ❜➔✐ t♦→♥ ✭✵✳✶✮✳ ❘ã r➔♥❣✱ ♥➳✉ ρk = ρ ✈ỵ✐ ♠å✐ k ≥ ✈➔ ρ ∈ (0, 2η/L2) t❤➻ t❛ ❝â ✐✐✮✳ ◆➳✉ t❤➯♠ ❣✐↔ t❤✐➳t ✭▲✹✮∗ t❤ä❛ ♠➣♥ t❤➻ ✤✐➲✉ ❦✐➺♥ ✐✐✐✮ tr♦♥❣ ✤à♥❤ ❧➼ tr➯♥ ✤÷đ❝ ❜↔♦ ✤↔♠✳ ❍ì♥ ♥ú❛✱ ❩❡♥❣ ✈➔ ❝ë♥❣ sü ❝ơ♥❣ ✤➣ ❝❤➾ r❛ r➡♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭▲✶✮✱ ✭▲✷✮ ✈➔ ✭▲✹✮∗ ❧➔ ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ {xk } ❜à ❝❤➦♥✳ ỗ tớ ữợ ữủ tọ lim sup T[k+N ] T[k+1] (xk ) − xk+N , T[k+N ] T[k+1] (xk ) − xk ≤ k→∞ ❱➻ t❤➳✱ ✣à♥❤ ❧➼ ✶✳✸ ❧➔ sü ❝↔✐ ❜✐➯♥ ✈➔ ❤ñ♣ ♥❤➜t ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤➦t ❧➯♥ ❝→❝ ❞➣② t❤❛♠ sè ❧➦♣ s♦ ✈ỵ✐ ❦➳t q✉↔ ♠➔ ❨❛♠❛❞❛✱ ❳✉ ✈➔ ❑✐♠ ✤➣ ♥❤➟♥ ✤÷đ❝✳ ◆➠♠ ✷✵✶✵✱ ▲✐✉ ✈➔ ❈✉✐ ✭▲✐✉ ✈➔ ✤t❣✱ ✷✵✶✵✮ ✤➣ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ C = ∅ t❤➻ N C := ❋✐①(Ti) = ❋✐①(T1T2 TN ) ✭✶✳✹✮ i=1 ❧➔ ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t ❣✐❛♦ ❤♦→♥ tr➯♥ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ Ti✳ ◆❤➡♠ ❧♦↕✐ ❜ä ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭▲✸✮✱ ✭▲✸✮∗ ✈➔ ❣✐↔ t❤✐➳t ✭✶✳✹✮✱ ♥➠♠ ✷✵✶✶✱ ◆❣✉②➵♥ ❇÷í♥❣ ✈➔ ▲➙♠ ❚❤ò② ❉÷ì♥❣ ①➙② ❞ü♥❣ ❞➣② ❧➦♣ ✭✶✳✺✮ xk+1 = (1 − βk0 )xk + βk0 (I − λk ρF )V˜k (xk ), k = 0, 1, 2, tr♦♥❣ ✤â✱ V˜k = TNk TNk −1 · · · T1k ✈➔ Tik = (1 − βki )I + βki Ti ✈ỵ✐ i = 1, 2, , N ✳ ❈→❝ t→❝ ❣✐↔ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❦➳t q✉↔ s❛✉✳ ✣à♥❤ ❧➼ ✶✳✹✳ ❈❤♦ ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝ L✲▲✐♣s❝❤✐t③ ✈➔ η✲✤ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥ H ✳ ❈❤♦ N Ti : H → H ❧➔ ❤å ❤ú✉ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr➯♥ H ✈ỵ✐ C := ❋✐①(Ti) = ∅✳ ●✐↔ sû i=1 ρ ∈ (0, 2η/L2 ) ❧➔ ❤➡♥❣ sè ❞÷ì♥❣ ❝è ✤à♥❤ ✈➔ ❞➣② λk ∈ (0, 1) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭▲✶✮ ✈➔ ✭▲✷✮✳ ỗ tớ tt r ki (, ) ợ i = 0, 1, 2, , N ✱ tr♦♥❣ ✤â α, β ∈ (0, 1) ✈➔ i lim |βk+1 − βki | = ✈ỵ✐ i = 1, 2, 3, , N ❑❤✐ ✤â✱ ợ tũ ỵ x0 H ❞➣② k→∞ ❧➦♣ ✭✶✳✺✮ ❤ë✐ tư ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t x∗ ❝õ❛ ❜➔✐ t♦→♥ ✭✵✳✶✮✳ F : H → H ✼ ❈â t❤➸ t❤➜② ♠ët tr♦♥❣ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ t÷ì♥❣ tü ✤↔♠ ❜↔♦ sü ❤ë✐ tư ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✭✵✳✸✮✱ ✭✵✳✹✮✱ ✭✶✳✸✮ ✈➔ ✭✶✳✺✮ ❧➔ ❣✐↔ t❤✐➳t t❤❛♠ sè ρ ♣❤ư t❤✉ë❝ ✈➔♦ ❤➺ sè ✤ì♥ ✤✐➺✉ ♠↕♥❤ η ✈➔ ❤➡♥❣ sè ▲✐♣s❝❤✐t③ L✳ ❚r➯♥ t❤ü❝ t➳✱ t❛ ❜✐➳t r➡♥❣ ✈✐➺❝ ①→❝ ✤à♥❤ η ❤♦➦❝ L ❦❤æ♥❣ ♣❤↔✐ ởt ổ ỗ tớ t t❤➜② r➡♥❣ ✭✵✳✹✮✱ ✭✶✳✸✮ ✈➔ ✭✶✳✺✮ ✤÷đ❝ t❤ü❝ ❤✐➺♥ ❧✉➙♥ ♣❤✐➯♥ ①♦❛② ✈á♥❣ ♥➯♥ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ❧➔ t✉➛♥ tü✳ ◆❣❤✐➯♥ ❝ù✉ ♠ð rë♥❣ ❝❤♦ tr÷í♥❣ ❤đ♣ C ❧➔ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ✈æ ❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ Ti : H → H ✱ ❜➡♥❣ ✈✐➺❝ sû ❞ö♥❣ →♥❤ ①↕ Wk ✱ ♥➠♠ ✷✵✵✽✱ ■❡♠♦t♦ ✈➔ ❚❛❦❛❤❛s❤✐ ✤➣ ①➙② ❞ü♥❣ ❞➣② ❧➦♣ ❤✐➺♥ ❝â ❞↕♥❣ xk+1 = (I − λk ρF )Wk (xk ), k = 1, 2, 3, x1 tũ ỵ tở H ✱ λk ∈ (0, 1] ✈➔ ρ > ❧➔ ❝→❝ t❤❛♠ sè ❧➦♣✳ ✣à♥❤ ❧➼ ✶✳✺✳ ❈❤♦ F : H → H ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝ L✲▲✐♣s❝❤✐t③ ✈➔ η∞✲✤ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥ H ✳ ❈❤♦ {Ti} ❧➔ ❤å ✈æ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr➯♥ H ✈ỵ✐ C := Fix(Ti) = ∅✳ ●✐↔ sû {αk } ❧➔ ❞➣② i=1 ❝→❝ sè t❤ü❝ t❤ä❛ ♠➣♥ < a ≤ αk ≤ b < 1✱ k = 1, 2, 3, ✈ỵ✐ a, b ∈ (0, 1)✳ ❑❤✐ ✤â✱ ♥➳✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❜↔♦ ✤↔♠ ✐✮ ρ ∈ (0, 2η/L2)✱ ✐✐✮ λk t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭▲✶✮ ✈➔ ✭▲✷✮✱ t❤➻ ❞➣② ❧➦♣ ✭✶✳✼✮ ❤ë✐ tư ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t x∗ ❝õ❛ ❜➔✐ t♦→♥ ✭✵✳✶✮✳ P❤÷ì♥❣ ♣❤→♣ ✭✶✳✼✮ sû ❞ư♥❣ →♥❤ ①↕ Wk ❦➳t ủ ợ ữỡ ữớ ố t rë♥❣ ❦➳t q✉↔ ❝õ❛ ❨❛♠❛❞❛ ❝❤♦ ❤å ✈æ ❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✳ ❈→❝ t→❝ ❣✐↔ ✤➣ ❧♦↕✐ ❜ä ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭▲✸✮ ❤♦➦❝ ✭▲✸✮∗✳ ❚✉② ✈➟②✱ ♥❣♦➔✐ ❦❤â ❦❤➠♥ ✤➸ ①→❝ ✤à♥❤ ρ✱ →♥❤ ①↕ Wk ❝â ❝➜✉ tró❝ ♣❤ù❝ t↕♣ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ✭✶✳✼✮ ❝ơ♥❣ ❧➔ t✉➛♥ tü✳ ◆➠♠ ✷✵✶✵✱ ❦➳t ❤đ♣ ♣❤÷ì♥❣ ♣❤→♣ ❦✐➸✉ ✤÷í♥❣ ❞è❝ ♥❤➜t✱ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥ ✈➔ sû ❞ư♥❣ →♥❤ ①↕ Wk ✱ ✈ỵ✐ x1 tũ ỵ tở H sỹ tt ởt ữủ ỗ ữ s  y = (I − λ F )(x ), k k k xk+1 = (1 − γk )yk + γk Wk (yk ), k = 1, 2, 3, ✭✶✳✽✮ tr♦♥❣ ✤â γk ∈ [0, 1] ✈➔ λk ≥ ❧➔ ❝→❝ t❤❛♠ sè ❧➦♣✳ ✣à♥❤ ❧➼ ✶✳✻✳ ❈❤♦ F : H → H ❧➔ →♥❤ ①↕ ❧✐➯♥ tư❝ L✲▲✐♣s❝❤✐t③ ✈➔∞η✲✤ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥ H ✳ ❈❤♦ {Ti} ❧➔ ❤å ✈æ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr➯♥ H ✈ỵ✐ C := Fix(Ti) = ∅✳ ●✐↔ sû {αk } ❧➔ ❞➣② ❝→❝ i=1 sè t❤ü❝ t❤ä❛ ♠➣♥ < αk ≤ b < 1✱ k = 1, 2, 3, ❑❤✐ ✤â✱ ♥➳✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❜↔♦ ✤↔♠ ✽ ✐✮ γk ∈ [γ, 1/2] ✈ỵ✐ γ > 0✱ ✐✐✮ λk t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭▲✶✮ ✈➔ ✭▲✷✮✱ t❤➻ ❞➣② ❧➦♣ ✭✶✳✽✮ ❤ë✐ tö ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t x∗ ❝õ❛ ❜➔✐ t♦→♥ ✭✵✳✶✮✳ ●✐è♥❣ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ✭✶✳✼✮✱ t❛ t❤➜② r➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✶✳✽✮ ❝ơ♥❣ ❝â ❝➜✉ tró❝ ♣❤ù❝ t↕♣ ✈➔ ✤â ❧➔ ♣❤÷ì♥❣ ♣❤→♣ t✉➛♥ tü✳ ▼ët ♥➠♠ s❛✉✱ ❲❛♥❣ ✭✷✵✶✶✮ ❝ô♥❣ ✤➣ ♥❤➟♥ ✤÷đ❝ ❦➳t q✉↔ t÷ì♥❣ tü ♥❤÷ ❝õ❛ ❨❛♦ ✈➔ sỹ ữợ tt ợ t ❞➣② t❤❛♠ sè ❧➦♣✳ ❑➳t q✉↔ ❝õ❛ ❲❛♥❣ t❤❛② t❤➳ ✭▲✶✮ ❜ð✐ ✤✐➲✉ ❦✐➺♥ < λk ≤ η/L2 − ε✱ ∀k ≥ k0✱ ✈ỵ✐ ➼t ♥❤➜t ♠ët sè ♥❣✉②➯♥ k0 > ❧➔ ♥❤➭ ❤ì♥ t❤ü❝ sü ✤✐➲✉ ❦✐➺♥ ✭▲✶✮✳ ◆❣♦➔✐ r❛✱ ✤✐➲✉ ❦✐➺♥ ✤➦t ❧➯♥ ❝❤♦ γk ❝❤➾ ✤á✐ ❤ä✐ t➼♥❤ ❣✐ỵ✐ ♥ë✐ ❝õ❛ ❞➣② t❤❛♠ sè ♥➔② tr♦♥❣ ✭✵✱✶✮✳ ❚✉② ♥❤✐➯♥ ✤✐➲✉ ❦✐➺♥ λk ✈➝♥ ②➯✉ ❝➛✉ ♣❤ư t❤✉ë❝ ✈➔♦ ❤➺ sè ✤ì♥ ✤✐➺✉ ♠↕♥❤ η ✈➔ ❤➡♥❣ sè ▲✐♣s❝❤✐t③ L✳ ▼➦t ❦❤→❝✱ ✤✐➲✉ ❦✐➺♥ ❜ê s✉♥❣ λk F (xk ) → ❦❤✐ k → ∞ ✤↔♠ ❜↔♦ sü ❤ë✐ tö ♣❤ö t❤✉ë❝ ✈➔♦ ❣✐→ trà F (xk ) t ộ ữợ t ❝❤å♥ t✐➯♥ ♥❣❤✐➺♠ {λk } t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ♥➔② s➩ ❦❤â ❦❤➠♥✳ ◆➠♠ ✷✵✵✽✱ ❈❡♥❣ ✈➔ ❝ë♥❣ sü ✤➣ ♥❣❤✐➯♥ ❝ù✉ ❝❤♦ tr÷í♥❣ ❤đ♣ C ❧➔ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ♠ët →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕ t❤ü❝✳ ▼ët ✤✐➲✉ ❦✐➺♥ q✉❛♥ trå♥❣ ✤↔♠ sỹ tử ố ợ ữỡ ợ ❝→❝ t→❝ ❣✐↔ ❧➔ ❣✐↔ t❤✐➳t ✈➲ t➼♥❤ ❧✐➯♥ tö❝ ②➳✉ t❤❡♦ ❞➣② ❝õ❛ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝✳ ❚✉② ♥❤✐➯♥✱ ✤✐➲✉ ✤â ✤➣ ❧➔♠ ❣✐ỵ✐ ❤↕♥ ♣❤↕♠ ✈✐ ự ữỡ ố ợ t ✤÷đ❝ t❤✐➳t ❧➟♣ tr♦♥❣ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ q✉❛♥ trå♥❣ ♠➔ ❦❤æ♥❣ ❝â t➼♥❤ ❝❤➜t ♥➔②✱ ❝❤➥♥❣ ❤↕♥ ❦❤æ♥❣ ❣✐❛♥ Lp[a, b] (1 < p < ∞)✳ ◆➠♠ ✷✵✶✶✱ ❈❤✐❞✉♠❡ ✈➔ ❝ë♥❣ sü ✤➣ ♠ð rë♥❣ ❦➳t q✉↔ ❝õ❛ ❳✉ ✈➔ ❑✐♠ tỵ✐ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ q✲trì♥ ✤➲✉ ✈ỵ✐ ❤➡♥❣ sè dq , q > 1✳ P❤÷ì♥❣ ♣❤→♣ ♥➔② ❝â t❤➸ →♣ ❞ư♥❣ tr➯♥ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ Lp[a, b], (1 < p < ∞)✳ ❚✉② ♥❤✐➯♥✱ ❣✐↔ t❤✐➳t ✤➦t k tữỡ tỹ ỗ t❤í✐ t❤❛♠ sè ρ ✈➝♥ ✤á✐ ❤ä✐ ♣❤ư t❤✉ë❝ ✈➔♦ ❤➺ sè η✱ L ✈➔ ❤➡♥❣ sè dq ✳ ❚❤➯♠ ✈➔♦ ✤â✱ ❝→❝ t→❝ ❣✐↔ ✈➝♥ ❝➛♥ sû ❞ö♥❣ ❣✐↔ t❤✐➳t ✈➲ t➼♥❤ ❣✐❛♦ ❤♦→♥ tr➯♥ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ Ti✳ ❑❤✐ C ❧➔ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ✈æ ❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ E ✱ t❤❛② ❝❤♦ ✈✐➺❝ sû ❞ö♥❣ →♥❤ ①↕ ♣❤ù❝ t↕♣ Wk ✱ t❛ ❝â t❤➸ sû ❞ö♥❣ →♥❤ ①↕ Vk ✤ì♥ ❣✐↔♥ ❤ì♥ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✭✶✳✶✶✮ Vk = Vk1 , Vki = T i T i+1 T k , T i = (1 − αi )I + αi Ti , i = 1, , k ∞ tr♦♥❣ ✤â αi ∈ (0, 1) ✈➔ αi < ∞ ◆❣✉②➵♥ ❇÷í♥❣ ✈➔ ❝→❝ ❝ë♥❣ sü ✭✷✵✶✸✮ ✤➣ ✤➲ ①✉➜t ✈➔ i=1 ❝❤ù♥❣ ữỡ ợ tử tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t x∗ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✷✮ ❧➔ xk = Vk (I − λk F )(xk ), ✭✶✳✶✸✮ ✶✶ ✷✳✶✳✸ ▼ët sè ❤➺ q✉↔ ◆➠♠ ✷✵✵✽✱ ❈❡♥❣ ✈➔ ❝ë♥❣ sü ❝↔✐ ❜✐➯♥ ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ❝õ❛ ❨❛♠❛❞❛ ✤➸ t❤✐➳t ❧➟♣ ❞➣② ❧➦♣ ❤✐➺♥ xk+1 = (I − λk F )(αk xk + (1 − αk )JrA (xk )), k ≥ 0, ✭✷✳✶✺✮ ①→❝ ✤à♥❤ ổ x A ỗ tớ x∗ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✷✮ ✈ỵ✐ x0 ∈ E tũ ỵ t t❤❛♠ sè ❧➦♣ λk , αk ∈ (0, 1) ✈➔ rk > ❧➔ ∞ ∞ ✐✮ k→∞ lim λk = 0, λk = ∞ ✈➔ | λk+1 − λk |< ∞, k k=1 ✐✐✮ rk ≥ ε ✈ỵ✐ ♠å✐ k ∈ N ✈➔ ∞ k=0 | rk+1 − rk |< ∞, k=0 ✐✐✐✮ < a ≤ αk ≤ b < ✈ỵ✐ ♠å✐ k ∈ N ✈➔ ∞ | αk+1 − αk |< ∞ k=0 ❇➡♥❣ ❝→❝❤ t❤❛② t❤➳ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ Ti ❜ð✐ ❝→❝ t♦→♥ tû ❣✐↔✐ J A := (I + Ai)−1 tr♦♥❣ ✭✷✳✶✮ t ú tổ ữủ t q t ợ ❜➔✐ t♦→♥ tê♥❣ q✉→t ❤ì♥ s❛✉ ✤➙②✳ ▼➺♥❤ ✤➲ ✷✳✶✳ ❈❤♦ E ✱ F ✱ αi✱ λk ✈➔ si ✤÷đ❝ ❣✐↔ t❤✐➳t t÷ì♥❣ ∞tü ✣à♥❤ ❧➼ ✷✳✶✳ ❈❤♦ {Ai} ❧➔ ❤å ✈ỉ ❤↕♥ ❝→❝ →♥❤ ①↕ ❥✲✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr➯♥ E ✈ỵ✐ C := Zer(Ai) = ∅✳ ❑❤✐ ➜②✱ ợ i=1 tũ ỵ x1 E ✱ ❞➣② {xk } ①→❝ ✤à♥❤ ❜ð✐ i k xk+1 = (I − λk F ) i=1 si (1 − αi )I + αi J Ai (xk ), s˜k k ≥ 1, ✭✷✳✶✻✮ ❤ë✐ tư ♠↕♥❤ tỵ✐ ❦❤ỉ♥❣ ✤✐➸♠ ❝❤✉♥❣ x∗ ∈ C ✈➔ x∗ ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✷✮ ❦❤✐ k → ∞✳ ◆❤➟♥ ①➨t ✷✳✹✳ P❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✺✮ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✻✮ ✤➲✉ sû ❞ư♥❣ ❜❛ t❤❛♠ sè ❧➦♣✳ ❘ã r➔♥❣✱ ✤✐➲✉ ❦✐➺♥ ✤➦t ❧➯♥ ❝→❝ ❞➣② t❤❛♠ sè {λk } ✈➔ {αk } ✤↔♠ ❜↔♦ sü ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✻✮ ❧➔ ♥❤➭ ❤ì♥ s♦ ✈ỵ✐ ❝→❝ ❣✐↔ t❤✐➳t ✐✮ ✈➔ ✐✐✐✮✳ ❚✉② ♥❤✐➯♥✱ ❝→❝ t❤❛♠ sè rk ✈➔ si t÷ì♥❣ ù♥❣ tr♦♥❣ ✭✷✳✶✺✮ ✈➔ ✭✷✳✶✻✮ ❧➔ ❦❤→❝ ❜✐➺t✱ ✤â♥❣ ✈❛✐ trá ❦❤→❝ ♥❤❛✉ ✈➔ ❦❤ỉ♥❣ s♦ s→♥❤ ✤÷đ❝✳ ❱➻ t❤➳✱ ✭✷✳✶✺✮ ✈➔ ✭✷✳✶✻✮ ❝❤♦ t❛ ❝→❝ q✉② t➢❝ t➻♠ ❦❤æ♥❣ ✤✐➸♠ ❦❤→❝ ♥❤❛✉✳ ◆❤➟♥ ①➨t ✷✳✺✳ ❚❛ ✤➦t f := aI ✈ỵ✐ a ∈ (0, 1) ❧➔ sè t❤ü❝ ❝è ✤à♥❤✳ ❑❤✐ ✤â✱ F := I − f ❧➔ →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈ỵ✐ ❤➺ sè η ✈➔ γ ✲❣✐↔ ❝♦ ❝❤➦t tr➯♥ E t❤ä❛ ♠➣♥ η + γ > t ợ x1 tũ ỵ tở E ♥➳✉ t❤❛② F ❜ð✐ I − f tr♦♥❣ ❝æ♥❣ t❤ù❝ t t õ ữủ ỗ k xk+1 = − λk k (1 − i=1 αi )ξik I αi ξik Ti (xk ), + k ≥ 1, ✭✷✳✶✽✮ i=1 ✤➸ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr➯♥ E ✱ tr♦♥❣ ✤â λk := (1 − a)λk ✈➔ ξik := si/˜sk ✳ ▼➺♥❤ ✤➲ s❛✉ ❧➔ ❤➺ q✉↔ trü❝ t✐➳♣ ❝õ❛ ✣à♥❤ ❧➼ ✷✳✶✳ ✶✷ ▼➺♥❤ ✤➲ ✷✳✷✳ ❈❤♦ E ✱ {Ti}✱ αi✱ λk ✈➔ si ✤÷đ❝ ❣✐↔ t❤✐➳t t÷ì♥❣ tü ✣à♥❤ ❧➼ ✷✳✶✳ ●✐↔ sû a ❧➔ sè t❤ü❝ ❝è ✤à♥❤ t❤✉ë❝ (0, 1)✳ ❑❤✐ ➜②✱ ✈ỵ✐ tũ ỵ x1 E {xk } ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✶✽✮ ❤ë✐ tư ♠↕♥❤ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ p∗ ∈ C ❦❤✐ k → ∞ ✈➔ t❤ä❛ ♠➣♥ p∗ , j(p∗ − p) ≤ p C ứ ú ỵ t ữủ t q ữợ ▼➺♥❤ ✤➲ ✷✳✸✳ ❈❤♦ E ✱ {Ai}✱ αi✱ λk ✈➔ si ✤÷đ❝ ❣✐↔ t❤✐➳t t÷ì♥❣ tü ▼➺♥❤ ✤➲ ✷✳✶✳ ●✐↔ sû a ❧➔ sè t❤ü❝ ❝è ✤à♥❤ t❤✉ë❝ (0, 1)✳ ợ tũ ỵ x1 E ✱ ❞➣② {xk } ①→❝ ✤à♥❤ ❜ð✐ k k (1 − xk+1 = − λk αi )ξik I αi ξik J Ai (xk ), + i=1 k ≥ 1, ✭✷✳✷✵✮ i=1 ❤ë✐ tư ♠↕♥❤ tỵ✐ ❦❤ỉ♥❣ ✤✐➸♠ ❝❤✉♥❣ p∗ ∈ C ❦❤✐ k → ∞ ✈➔ p∗ t❤ä❛ ♠➣♥ ✭✷✳✶✾✮✳ ◆➠♠ ✷✵✵✼✱ ◗✐♥ ✈➔ ❙✉ ✤➣ ①➙② ❞ü♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❦✐➸✉ ❍❛❧♣❡r♥ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ ♠ët →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ A tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ trì♥ ✤➲✉ E ❝â ❞↕♥❣ ✭✷✳✷✶✮ xk+1 = λk u + (1 − λk )(αk xk + (1 − αk )JrA (xk )), k ≥ 1, tr♦♥❣ ✤â x1 E tũ ỵ u E tû ❝è ✤à♥❤✱ αk , λk ✈➔ rk ❧➔ ❝→❝ t❤❛♠ sè ❧➦♣✳ ✣✐➲✉ ❦✐➺♥ ✤↔♠ ❜↔♦ sü ❤ë✐ tö ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✷✶✮ ❧➔ t÷ì♥❣ tü ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✺✮ ✈➔ ❝❤➾ ①➨t ❝❤♦ ❜➔✐ t♦→♥ ①→❝ ✤à♥❤ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ ♠ët →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ♠➔ ❦❤ỉ♥❣ ❣✐↔✐ ✤÷đ❝ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ♥➔♦✳ ❉♦ ✤â✱ ♥➳✉ ❝❤➾ ①➨t r✐➯♥❣ t❤❡♦ ❦❤➼❛ ❝↕♥❤ ♥➔② t❤➻ ❣✐↔ t❤✐➳t ✤➦t ❧➯♥ t❤❛♠ sè αk ✈➔ λk ❧➔ ♥❤ú♥❣ ②➯✉ ❝➛✉ ❝❤➦t ❝❤➩ ❤ì♥ s♦ ✈ỵ✐ ❦➳t q✉↔ ❝õ❛ ❝❤ó♥❣ tỉ✐ ♥➯✉ tr♦♥❣ ▼➺♥❤ ✤➲ ✷✳✶✳ ụ ữ ỵ r trú ữỡ s ợ t❤ü❝ sü ❦❤→❝ ❜✐➺t✳ ◆❤➟♥ ①➨t ✷✳✻✳ ❚❛ ✤➦t f := aI + (1 − a)u ✈ỵ✐ a ∈ (0, 1) ❧➔ sè t❤ü❝ ❝è ✤à♥❤ ✈➔ u ❧➔ ♣❤➛♥ tû ❝è ✤à♥❤ t❤✉ë❝ E ✳ ❑❤✐ ✤â✱ ❞➵ t❤➜② r➡♥❣ F := I − f ❝ô♥❣ ❧➔ →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈ỵ✐ ❤➺ sè η ✈➔ γ ✲❣✐↔ ❝♦ ❝❤➦t t❤ä❛ ♠➣♥ η + γ > 1✳ õ ợ x1 tũ ỵ tở E t F ❜ð✐ I − f tr♦♥❣ ✭✷✳✶✮✱ t❛ ❝â ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❦✐➸✉ ❍❛❧♣❡r♥ k k k xk+1 = λk u + − λk (1 − αi )ξik I + i=1 αi ξik Ti (xk ), k ≥ 1, ✭✷✳✷✷✮ αi ξik J Ai (xk ), k ≥ ✭✷✳✷✸✮ i=1 ✈➔ ♥➳✉ t❤❛② Ti ❜ð✐ J A t❤➻ ♥❤➟♥ ✤÷đ❝ i k xk+1 = λk u + − λk k (1 − i=1 αi )ξik I + i=1 tr♦♥❣ ✤â λk := (1 − a)λk ✈➔ ξik := si/˜sk ✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ t❛ ❝ơ♥❣ ♥❤➟♥ ữủ q trỹ t ữợ ❧➼ ✷✳✶✳ ✶✸ ▼➺♥❤ ✤➲ ✷✳✹✳ ❈❤♦ E ✱ {Ti}✱ αi✱ a✱ λk ✈➔ si ✤÷đ❝ ❣✐↔ t❤✐➳t t÷ì♥❣ tü ▼➺♥❤ ✤➲ ✷✳✷✳ ❑❤✐ ➜②✱ ✈ỵ✐ ♠å✐ u ∈ E ❝è ✤à♥❤✱ ❞➣② {xk } ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✷✷✮ ❤ë✐ tư ♠↕♥❤ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ p∗ ∈ C ❦❤✐ k → ∞ ✈➔ t❤ä❛ ♠➣♥ p∗ − u, j(p∗ − p) ≤ ∀p ∈ C ✭✷✳✷✹✮ ▼➺♥❤ ✤➲ ✷✳✺✳ ❈❤♦ E ✱ {Ai}✱ αi✱ a✱ λk ✈➔ si ✤÷đ❝ ❣✐↔ t❤✐➳t t÷ì♥❣ tü ▼➺♥❤ ✤➲ ✷✳✸✳ ❑❤✐ ➜②✱ ✈ỵ✐ ♠å✐ u ∈ E ❝è ✤à♥❤✱ ❞➣② {xk } ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✷✸✮ ❤ë✐ tư ♠↕♥❤ tỵ✐ ❦❤ỉ♥❣ ✤✐➸♠ ❝❤✉♥❣ p∗ ∈ C ❦❤✐ k → ∞ ✈➔ p∗ t❤ä❛ ♠➣♥ ✭✷✳✷✹✮✳ ✷✳✷✳ P❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ❞ò♥❣ →♥❤ ①↕ Sˆk ✷✳✷✳✶ ◆ë✐ ❞✉♥❣ ♣❤÷ì♥❣ ♣❤→♣ P❤÷ì♥❣ ♣❤→♣ t❤ù ❤❛✐ ✤÷đ❝ t❤✐➳t ❧➟♣ ❞ü❛ tr➯♥ ✈✐➺❝ sû ❞ö♥❣ →♥❤ ①↕ Sˆk ✳ ❳✉➜t ♣❤→t tø x1 tũ ỵ tở E ú tổ ❞ü♥❣ ❞➣② ❧➦♣ ❤✐➺♥ {xk } ♥❤÷ s❛✉✿ xk+1 = (I − λk F )Sˆk (xk ), k = 1, 2, 3, ✭✷✳✷✺✮ ð ✤➙② →♥❤ ①↕ Sˆk ①→❝ ✤à♥❤ ❜ð✐ Sˆk = s0 − sk k ✭✷✳✷✻✮ (si−1 − si )T i i=1 tr♦♥❣ ✤â T i ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✸✮✱ λk ∈ (0, 1) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭▲✶✮✱ ✭▲✷✮ ✈➔ {si } ❧➔ ❞➣② ❝→❝ sè t❤ü❝ ❣✐↔♠ ♥❣➦t✱ ❤ë✐ tö ✈➲ ❦❤✐ i → ∞✳ ✷✳✷✳✷ ❙ü ❤ë✐ tö ♠↕♥❤ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✣à♥❤ ❧➼ ✷✳✹✳ ✭✷✮ ✷ ❈❤♦ E ổ tỹ ỗ t ❝â ❝❤✉➞♥ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉✳ ❈❤♦ F : E → E ❧➔ →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈ỵ✐ ❤➺ sè η ✈➔ γ ✲❣✐↔ ❝♦ ❝❤➦t ✈ỵ✐ η + γ > 1✳ ∞ ❈❤♦ {Ti} ❧➔ ❤å ✈æ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr➯♥ E ✈ỵ✐ C := ❋✐①(Ti) = ∅✳ ●✐↔ sû λk ∈ (0, 1) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❤ë✐ tö ✈➲ 0✳ ❑❤✐ ➜②✱ ❞➣② {xk } ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✷✮ ❦❤✐ k → ∞✳ i=1 ❦✐➺♥ ✭▲✶✮✱ ✭▲✷✮ ✈➔ {si} ❧➔ ❞➣② sè t❤ü❝ ❞÷ì♥❣ ❣✐↔♠ ♥❣➦t✱ ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✷✺✮ ❤ë✐ tư tợ t x t Pữỡ ợ ữỡ ỡ ð ❞➣② t❤❛♠ sè {si} ✈➔ ✈✐➺❝ sû ❞ö♥❣ ♥â ✤➸ t❤✐➳t ❦➳ ❝→❝ →♥❤ ①↕ S˜k ✈➔ Sˆk ✳ ◆➳✉ ❝❤å♥ si = 1/(i + 1) (i = 0, 1, 2, ) t❤➻ ♥â t❤ä❛ ♠➣♥ ❣✐↔ t❤✐➳t ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✷✺✮ ♥❤÷♥❣ ❣✐↔ t❤✐➳t ❝õ❛ ♣❤÷ì♥❣ ∞ ♣❤→♣ ✭✷✳✶✮ t❤➻ ❦❤ỉ♥❣ ✤÷đ❝ ❜↔♦ ✤↔♠ ✈➻ ❝❤✉é✐ si ♣❤➙♥ ❦➻✳ ❚✉② ♥❤✐➯♥✱ ❝❤å♥ si = 1/(i+1)3 i=0 ✷ ❇✉♦♥❣✱ ◆❣✳✱ ❍❛✱ ◆❣✳ ❙✳✱ ❚❤✉② ◆❣✳ ❚✳ ❚✳ ✭✷✵✶✻✮✱ ✧❍②❜r✐❞ st❡❡♣❡st✲❞❡s❝❡♥t ♠❡t❤♦❞ ✇✐t❤ ❛ ❝♦✉♥t❛❜❧② ✐♥❢✐♥✐t❡ ❢❛♠✐❧② ♦❢ ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣s ♦♥ ❇❛♥❛❝❤ s♣❛❝❡s✧✱ ◆♦♥❧✐♥❡❛r ❋✉♥❝t✳ ❆♥❛❧✳ ❆♣♣❧✳ ✱ ✷✶✱ ♣♣✳ ✷✼✸✲✷✽✼✳ ✶✹ ♥➳✉ i ❝❤➤♥ ✈➔ si = 1/(i + 1)2 ♥➳✉ i ❧➫ (i = 0, 1, 2, ) t❤➻ ❣✐↔ t❤✐➳t ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✮ ✤÷đ❝ ❜↔♦ ✤↔♠ ♥❤÷♥❣ ❣✐↔ t❤✐➳t ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✷✺✮ ❧↕✐ ❦❤ỉ♥❣ ✈➻ ♥â ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ❞➣② sè ❣✐↔♠ ♥❣➦t✳ ❱➻ t❤➳✱ ♥❣♦➔✐ ✈✐➺❝ ✤↕t ✤÷đ❝ ♥❤ú♥❣ ♠ư❝ t✐➯✉ ✈➔ ❦➳t ❧✉➟♥ t÷ì♥❣ tü ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✮ ✤➣ ♥➯✉ tr♦♥❣ ▼ö❝ ✷✳✶✳✷ ✈➔ ▼ö❝ ✷✳✶✳✸ t❤➻ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✷✺✮ ❣â♣ ♣❤➛♥ ✤❛ ❞↕♥❣ ✈➔ ❤♦➔♥ t❤✐➺♥ t ữỡ ợ t♦→♥ ♥❣❤✐➯♥ ❝ù✉✳ ✷✳✷✳✸ ▼ët sè ❤➺ q✉↔ ▼➺♥❤ ✤➲ ✷✳✻✳ ❈❤♦ E ✱ F ✱ αi✱ λk ✈➔ si ✤÷đ❝ ❣✐↔ t❤✐➳t t÷ì♥❣ ∞tü ✣à♥❤ ❧➼ ✷✳✹✳ ❈❤♦ {Ai} ❧➔ ❤å ✈ỉ ❤↕♥ ❝→❝ →♥❤ ①↕ ❥✲✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr➯♥ E ✈ỵ✐ C si−1 − si (1 − αi )I + αi J Ai (xk ), s0 − sk xk+1 = (I − λk F ) i=1 ❑❤✐ ợ i=1 tũ ỵ x1 E ✱ ❞➣② {xk } ①→❝ ✤à♥❤ ❜ð✐ k Zer(Ai ) = ∅✳ := k ≥ 1, ❤ë✐ tö ♠↕♥❤ tỵ✐ ❦❤ỉ♥❣ ✤✐➸♠ ❝❤✉♥❣ x∗ ∈ C ✈➔ x∗ ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✷✮ ❦❤✐ k → ∞✳ ❚✐➳♣ t❤❡♦✱ ✤➦t βik := (si−1 − si)/(s0 − sk )✱ sû ❞ö♥❣ ❧↕✐ ❝→❝ ❦➼ ❤✐➺✉ ✈➔ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tü ◆❤➟♥ ①➨t ✷✳✺ ✈➔ ◆❤➟♥ ①➨t ✷✳✻✱ t ữủ q trỹ t ữợ ❝õ❛ ✣à♥❤ ❧➼ ✷✳✹✳ ▼➺♥❤ ✤➲ ✷✳✼✳ ❈❤♦ E ✱ {Ti}✱ αi✱ λk ✈➔ si ✤÷đ❝ ❣✐↔ t❤✐➳t t÷ì♥❣ tü ✣à♥❤ ❧➼ ✷✳✹✳ ●✐↔ sû a ❧➔ sè t❤ü❝ ❝è ✤à♥❤ t❤✉ë❝ (0, 1)✳ ❑❤✐ ➜②✱ ✈ỵ✐ ✤✐➸♠ ❜❛♥ ✤➛✉ tũ ỵ x1 E {xk } ✤à♥❤ ❜ð✐ k k xk+1 = − λk (1 − αi )βik I αi βik Ti (xk ), + i=1 k ≥ 1, i=1 ❤ë✐ tư ♠↕♥❤ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ p∗ ∈ C ❦❤✐ k → ∞ ✈➔ p∗ t❤ä❛ ♠➣♥ ✭✷✳✶✾✮✳ ▼➺♥❤ ✤➲ ✷✳✽✳ ❈❤♦ E ✱ {Ai}✱ αi✱ λk ✈➔ si ✤÷đ❝ ❣✐↔ t❤✐➳t t÷ì♥❣ tü ▼➺♥❤ ✤➲ ✷✳✻✳ ●✐↔ sû a ❧➔ sè t❤ü❝ ❝è ✤à♥❤ t❤✉ë❝ (0, 1)✳ ❑❤✐ ➜②✱ ✈ỵ✐ ✤✐➸♠ ❜❛♥ tũ ỵ x1 E {xk } ①→❝ ✤à♥❤ ❜ð✐ k xk+1 = − λk k (1 − αi )βik I αi βik J Ai (xk ), + k ≥ 1, i=1 i=1 ❤ë✐ tö ♠↕♥❤ tỵ✐ ❦❤ỉ♥❣ ✤✐➸♠ ❝❤✉♥❣ p∗ ∈ C ❦❤✐ k → ∞ ✈➔ p∗ t❤ä❛ ♠➣♥ ✭✷✳✶✾✮✳ ▼➺♥❤ ✤➲ ✷✳✾✳ ❈❤♦ E ✱ {Ti}✱ αi✱ a✱ λk ✈➔ si ✤÷đ❝ ❣✐↔ tt tữỡ tỹ ợ u ∈ E ❝è ✤à♥❤ ✈➔ ✈ỵ✐ ✤✐➸♠ ❜❛♥ ✤➛✉ tũ ỵ x1 E {xk } ✤à♥❤ ❜ð✐ k xk+1 = λk u + − λk k (1 − i=1 αi )βik I αi βik Ti (xk ), + k ≥ 1, i=1 ❤ë✐ tö ♠↕♥❤ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ p∗ ∈ C ❦❤✐ k → ∞ ✈➔ p∗ t❤ä❛ ♠➣♥ ✭✷✳✷✹✮✳ ✶✺ ▼➺♥❤ ✤➲ ✷✳✶✵✳ ❈❤♦ E ✱ {Ai}✱ αi✱ a✱ λk ✈➔ si ✤÷đ❝ ❣✐↔ t❤✐➳t t÷ì♥❣ tü ▼➺♥❤ ✤➲ ✷✳✽✳ ❑❤✐ ➜②✱ ✈ỵ✐ ♠å✐ u ∈ E ❝è ✤à♥❤ ✈➔ ✈ỵ✐ tũ ỵ x1 E {xk } ①→❝ ✤à♥❤ ❜ð✐ k k (1 − xk+1 = λk u + − λk αi )βik I αi βik J Ai (xk ), + i=1 k ≥ 1, i=1 ❤ë✐ tư ♠↕♥❤ tỵ✐ ❦❤ỉ♥❣ ✤✐➸♠ ❝❤✉♥❣ p∗ ∈ C ❦❤✐ k → ∞ ✈➔ p∗ t❤ä❛ ♠➣♥ ✭✷✳✷✹✮✳ ✷✳✸✳ P❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ❞ò♥❣ →♥❤ ①↕ S k ✷✳✸✳✶ ◆ë✐ ❞✉♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❳✉➜t t tứ x1 tũ ỵ tở E ❧➦♣ ❤✐➺♥ {xk } ✤÷đ❝ t❤✐➳t ❦➳ ♥❤÷ s❛✉✿ xk+1 = (I − λk F )S k (xk ), tr♦♥❣ ✤â S k = αI + (1 − α)T k ✈ỵ✐ T k = 1, 2, 3, k k (si /˜ sk )Ti := Sk = ✭✷✳✸✶✮ ✈➔ α ∈ (0, 1) ❧➔ ♠ët sè t❤ü❝ ❝è i=1 ✤à♥❤✱ si ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✹✮✱ s˜k = k si ✈➔ λk t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭▲✶✮ ✈➔ ✭▲✷✮✳ ✷✳✸✳✷ ❙ü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✣à♥❤ ❧➼ ✷✳✺✳ ✭✸✮ ✸ i=1 ❈❤♦ E ❧➔ ❦❤æ♥❣ tỹ ỗ t õ ✈✐ ●➙t❡❛✉① ✤➲✉✳ ❈❤♦ F : E → E ❧➔ →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈ỵ✐ ❤➺ sè η ✈➔ γ ✲❣✐↔ ❝♦ ❝❤➦t ✈ỵ✐ η + γ > 1✳ ❈❤♦ ∞ {Ti } ❧➔ ❤å ✈æ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr➯♥ E ✈ỵ✐ C := Fix(Ti ) = ∅✳ ▲➜② ♠ët ❣✐→ trà i=1 ❝è ✤à♥❤ α ∈ (0, 1)✳ ●✐↔ sû λk ✈➔ si t÷ì♥❣ ù♥❣ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭▲✶✮✱ ✭▲✷✮ ✈➔ ✭✷✳✹✮✳ ❑❤✐ ➜②✱ ❞➣② {xk } ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✸✶✮ ❤ë✐ tư ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t x∗ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✷✮ ❦❤✐ k → ∞✳ ✷✳✸✳✸ ▼ët sè ❤➺ q✉↔ ▼➺♥❤ ✤➲ ✷✳✶✶✳ ❈❤♦ E ✱ α✱ λk ✈➔ si ✤÷đ❝ ❣✐↔ t❤✐➳t t÷ì♥❣∞ tü ✣à♥❤ ❧➼ ✷✳✺✳ ❈❤♦ {Ai} ❧➔ ❤å ✈ỉ ❤↕♥ ❝→❝ →♥❤ ①↕ ❥✲✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr➯♥ E ✈ỵ✐ C := Zer(Ai ) = ∅✳ ợ i=1 tũ ỵ x1 ∈ E ✱ ❞➣② {xk } ①→❝ ✤à♥❤ ❜ð✐ k (si /˜ sk )J Ai (xk ), xk+1 = (I − λk F ) (1 − α)I + α k ≥ 1, i=1 ❤ë✐ tư ♠↕♥❤ tỵ✐ ❦❤ỉ♥❣ ✤✐➸♠ ❝❤✉♥❣ x∗ ∈ C ✈➔ x∗ ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✷✮ ❦❤✐ k → ∞✳ ✸ ❍❛✱ ◆❣✳ ❙✳✱ ❇✉♦♥❣✱ ◆❣✳✱ ❚❤✉② ◆❣✳ ❚✳ ❚✳ ✭✷✵✶✽✮✱ ✧❆ ♥❡✇ s✐♠♣❧❡ ♣❛r❛❧❧❡❧ ✐t❡r❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ❛ ❝❧❛ss ♦❢ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✧✱ ✱ ✹✸✱ ♣♣✳ ✷✸✾✲✷✺✺ ❱✐❡t♥❛♠✳ ❆❝t❛ ▼❛t❤✳ ✶✻ ◆➠♠ ✷✵✵✼✱ ✤➸ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝❤♦ ♠ët ❤å ✈æ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ {Ti} tr➯♥ ♠ët t➟♣ ỗ õ Q E ỹ ữủ ỗ ữ s xk+1 = k u + (1 − λk ) (1 − α)I + α tr♦♥❣ ✤â✱ ∞ k ≥ 1, ✭✷✳✸✽✮ i=1 tò② þ✱ {si} ❧➔ ❞➣② ❝→❝ sè t❤ü❝ ❞÷ì♥❣ t❤ä❛ ♠➣♥ si = s˜ < ✈➔ λk ∈ [0, 1] t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭▲✶✮ ✈➔ ✭▲✷✮✳ i=1 ◆➠♠ ✷✵✵✾✱ ✤➸ t➻♠ ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ❝→❝ →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ Ai : E → E ✱ ❖❢♦❡❞✉ ✈➔ ❙❤❡❤✉ ✤➣ ✤➲ ①✉➜t t❤✉➟t t♦→♥✿ u ∈ Q ❝è ✤à♥❤✱ (si /˜ s)Ti (xk ), x1 ∈ Q ∞ σi,k (1 − δ)I + δJ Ai (xk ), xk+1 = λk u + k ≥ 1, ✭✷✳✸✾✮ i=1 ∞ ð ✤➙②✱ σi,k = − λk ✈➔ < γ1 ≤ δ ≤ γ2 < 1✳ ❈→❝ ❞➣② t❤❛♠ sè ❧➦♣ λk ✈➔ σi,k t÷ì♥❣ i=1 ù♥❣ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭▲✶✮✱ ✭▲✷✮ ✈➔ ∞ |σi,k+1 − σi,k | = 0, lim k→∞ ∞ i=1 σi,k (1 − δ)xk + δJ Ai xk − xk /λk = lim k→∞ i=1 ◆❤➟♥ ①➨t ✷✳✶✸✳ ó r ữợ q t t t t↕✐ ♠é✐ ✈á♥❣ ❧➦♣✱ ❝❤✉é✐ ❝→❝ t♦→♥ tû tr♦♥❣ ❝→❝ t❤✉➟t t♦→♥ ✭✷✳✸✽✮✱ ✭✷✳✸✾✮ ❝❤♦ t❤➜② ♥➳✉ ❝❤ó♥❣ t❛ ❦❤ỉ♥❣ ❝â ❝→❝ t❤æ♥❣ t✐♥ t✐➯♥ ♥❣❤✐➺♠ ✈➲ tê♥❣ ❝→❝ ❝❤✉é✐ ♥➔② t❤➻ t❤✐➳t ❦➳ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✤â ❦❤ỉ♥❣ t❤➸ →♣ ❞ö♥❣✳ ❱➻ t❤➳✱ ❝→❝ ❦➳t q✉↔ tr➯♥ ❧➔ ❦❤â ✤➸ ♥❤➟♥ ❜✐➳t ✈➔ ❦❤â ❝â t❤➸ ❧➟♣ tr➻♥❤ t➼♥❤ t♦→♥ tr➯♥ ♠→② t➼♥❤✳ ◆❤÷ ✈➟②✱ ♠ët ✈➜♥ ✤➲ tü ♥❤✐➯♥ ✤➦t r❛ ❧➔ ❧✐➺✉ ❝❤ó♥❣ t❛ ❝â t❤➸ t❤❛② t❤➳ ❝→❝ tê♥❣ ✈ỉ ❤↕♥ tr♦♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✸✽✮ ✈➔ ✭✷✳✸✾✮ ❜ð✐ ❝→❝ tê♥❣ r✐➯♥❣ t÷ì♥❣ ù♥❣ ❤❛② ❦❤æ♥❣❄ ◆❤ú♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② t✐➳♣ t❤❡♦✱ ♥❣♦➔✐ ✈✐➺❝ ❦❤➢❝ ♣❤ö❝ ♥❤ú♥❣ ❦❤â ❦❤➠♥ ✤➣ ♣❤➙♥ t➼❝❤ ❧➔ tr↔ ❧í✐ ❝❤♦ ❝➙✉ ❤ä✐ ♥➯✉ tr➯♥✳ ◆❤➟♥ ①➨t ✷✳✶✹✳ ❇➡♥❣ ✈✐➺❝ sû ❞ư♥❣ ❝→❝ ❦➼ ❤✐➺✉ ✈➔ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tỹ ữ t t ợ x1 tũ ỵ tở E t ụ ữủ ữợ xk+1 = λk αI + (1 − α)T k (xk ), k ≥ 1, ✭✷✳✹✵✮ ✈➔ k ξik J Ai (xk ), k ≥ xk+1 = − λk αI + (1 − α) i=1 ✭✷✳✹✶✮ tr♦♥❣ ✤â λk = (1 − a)λk ✳ ❍❛✐ ♠➺♥❤ ✤➲ s❛✉ ❧➔ ❤➺ q✉↔ trü❝ t✐➳♣ ❝õ❛ ✣à♥❤ ❧➼ ✷✳✺✳ ▼➺♥❤ ✤➲ ✷✳✶✷✳ ❈❤♦ E ✱ {Ti}✱ α✱ λk ✈➔ si ✤÷đ❝ ❣✐↔ t❤✐➳t t÷ì♥❣ tü ✣à♥❤ ❧➼ ✷✳✺✳ ●✐↔ sû a ❧➔ sè t❤ü❝ ❝è ✤à♥❤ t❤✉ë❝ (0, 1)✳ ❑❤✐ ➜②✱ ❞➣② {xk } ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✹✵✮ ❤ë✐ tư ♠↕♥❤ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ p∗ ∈ C ❦❤✐ k → ∞ ✈➔ p∗ t❤ä❛ ♠➣♥ ✭✷✳✶✾✮✳ ✶✼ ▼➺♥❤ ✤➲ ✷✳✶✸✳ ❈❤♦ E ✱ {Ai}✱ α✱ λk ✈➔ si ✤÷đ❝ ❣✐↔ t❤✐➳t t÷ì♥❣ tü ▼➺♥❤ ✤➲ ✷✳✶✶✳ ●✐↔ sû ❧➔ sè t❤ü❝ ❝è ✤à♥❤ t❤✉ë❝ (0, 1)✳ ❑❤✐ ➜②✱ ❞➣② {xk } ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✹✶✮ ❤ë✐ tư ♠↕♥❤ tỵ✐ ❦❤ỉ♥❣ ✤✐➸♠ ❝❤✉♥❣ p∗ ∈ C ❦❤✐ k → ∞ ✈➔ p∗ t❤ä❛ ♠➣♥ ✭✷✳✶✾✮✳ ◆❤➟♥ ①➨t ✷✳✶✺✳ P❤÷ì♥❣ ♣❤→♣ ✭✷✳✸✶✮ ✈➔ ❝→❝ ❤➺ q✉↔ trü❝ t✐➳♣ ❝õ❛ ♥â t❤➸ ❤✐➺♥ ✤÷đ❝ ♠ët sè ✤✐➸♠ ✈÷đt trë✐✳ ✐✮ ❈➜✉ tró❝ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✸✶✮ ❧➔ ✤ì♥ ❣✐↔♥ ❤ì♥ ✭✷✳✶✮ ✈➔ ✭✷✳✷✺✮✳ ❇➯♥ ❝↕♥❤ ✤â✱ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ✤➣ ❧➔♠ ❣✐↔♠ sè t❤➔♥❤ ♣❤➛♥ t ộ ữợ t õ ❝➛♥ ➼t t❤í✐ ❣✐❛♥ t➼♥❤ t♦→♥ ❤ì♥ tr➯♥ ♠→② t➼♥❤ ✭①❡♠ t❤➯♠ ❱➼ ❞ư ✸✳✹ tr♦♥❣ ▼ư❝ ✸✳✷ ❝õ❛ ❈❤÷ì♥❣ ✸✮✳ ✐✐✮ ❈→❝ t❤✉➟t t♦→♥ ✭✷✳✹✵✮ ✈➔ ✭✷✳✹✶✮ sû ❞ö♥❣ ❞➣② ❝→❝ tê♥❣ r✐➯♥❣ ❝õ❛ ❝❤✉é✐ ❤➔♠ ❧➔ ✤ì♥ ❣✐↔♥ ❤ì♥✱ ❞➵ ♥❤➟♥ ❜✐➳t ❤ì♥ ✈➔ ❝â t❤➸ t➼♥❤ t♦→♥ tr➯♥ ♠→② t➼♥❤✳ ❚r♦♥❣ ❦❤✐ ✤â✱ ✤è✐ ✈ỵ✐ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❖❢♦❡❞✉ ✈➔ ❙✉③✉❦✐ ❧➔ ❦❤ỉ♥❣ t❤ü❝ ❤✐➺♥ ✤÷đ❝✳ ◆❤➟♥ ①➨t ✷✳✶✻✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣✱ ♥➳✉ Ti ❧➔ →♥❤ ổ tr ởt t ỗ õ Q ❝õ❛ E t❤➻ Ti : Q → Q ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝ 1✲▲✐♣s❝❤✐t③✳ ◆➳✉ Q ❝❤ù❛ ♣❤➛♥ tû ❣è❝ ❝õ❛ E t❤➻ xk+1 ∈ Q✳ ❉♦ ✤â ▼➺♥❤ ✤➲ ✷✳✶✷ ✈➝♥ ✤ó♥❣ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✳ ❚✐➳♣ t❤❡♦✱ ♥➳✉ Q ❦❤æ♥❣ ❝❤ù❛ ♣❤➛♥ tû ❣è❝ ❝õ❛ E t❤➻ t❛ ①➨t f := aI + (1 − a)u ✈ỵ✐ u ∈ Q ❧➔ ♣❤➛♥ tû ❝è ✤à♥❤✳ ❑❤✐ ✤â✱ t❤❛② ✈➻ ✭✷✳✹✵✮ t❛ ♥❤➟♥ ✤÷đ❝ ♣❤÷ì♥❣ ♣❤→♣ ❝↔✐ ❜✐➯♥ ❦✐➸✉ ❍❛❧♣❡r♥✿  a x ∈ E, xk+1 = λ u + − λ αI + (1 − α)T k (xk ), k k k ≥ ✭✷✳✹✷✮ ❚❛ õ t q tờ qt ỡ ữợ ✤➙②✳ ▼➺♥❤ ✤➲ ✷✳✶✹✳ ❈❤♦ E ✱ α✱ a✱ λk ✈➔ si ✤÷đ❝ ❣✐↔ t❤✐➳t t÷ì♥❣ tü ▼➺♥❤ ✤➲ ✷✳✶✷✳ ∞❈❤♦ {Ti} ❧➔ ❤å ✈æ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ tr ởt t ỗ õ Q E ✈ỵ✐ C := Fix(Ti) = ∅✱ i=1 tr♦♥❣ ✤â Fix(Ti) := {x ∈ Q : x = Ti(x)}✳ ❑❤✐ ➜②✱ ❞➣② {xk } ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✹✷✮ ❤ë✐ tö ♠↕♥❤ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ p∗ ∈ C ❦❤✐ k → ∞ ✈➔ p∗ t❤ä❛ ♠➣♥ ✭✷✳✷✹✮✳ ❱ỵ✐ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tü ♥❤÷ tr♦♥❣ ♥❤➟♥ ①➨t ð tr➯♥✱ ❝❤ó♥❣ tỉ✐ ụ ữủ ữỡ ợ ởt ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❍❛❧♣❡r♥ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ❝→❝ →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳ ▼➺♥❤ ✤➲ ✷✳✶✺✳ ❈❤♦ E ✱ α✱ a✱ λk ✈➔ si ✤÷đ❝ ❣✐↔ t❤✐➳t t÷ì♥❣ tü ▼➺♥❤ ✤➲ ✷✳✶✷✳ ❈❤♦ Ai : Q → E ❧➔ ❤å ✈æ ❤↕♥ ❝→❝ →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr➯♥ ♠ët t➟♣ ỗ õ Q E ợ C := Zer(Ai ) = ∅✳ ❑❤✐ ➜②✱ ✈ỵ✐ ♠å✐ u ∈ Q ố x1 tũ ỵ tở E ❞➣② {xk } i=1 ①→❝ ✤à♥❤ ❜ð✐ k xk+1 = λk u + − λk αI + (1 − α) i=1 si A i J (xk ), k ≥ 1, s˜k ❤ë✐ tư ♠↕♥❤ tỵ✐ ❦❤ỉ♥❣ ✤✐➸♠ ❝❤✉♥❣ p∗ ∈ C ❦❤✐ k → ∞ ✈➔ p∗ t❤ä❛ ♠➣♥ ✭✷✳✷✹✮✳ ✭✷✳✹✸✮ ❈❤÷ì♥❣ ✸ ▼ët ❜➔✐ t♦→♥ t❤ü❝ t➳ ✈➔ ❦➳t q✉↔ t➼♥❤ t♦→♥ sè 18 ✸✳✶✳ ❇➔✐ t♦→♥ ♣❤➙♥ ♣❤è✐ ❜➠♥❣ t❤æ♥❣ ❳➨t ♠ët ♠↕♥❣ ✭■✐❞✉❦❛ ✈➔ ❯❝❤✐❞❛✱ ✷✵✶✶✮ ỗ S = {1, 2, , S} t ỗ = {1, 2, , L} t ❝→❝ ❧✐➯♥ ❦➳t tr♦♥❣ ♠↕♥❣✳ ❱ỵ✐ ♠é✐ ❧✐➯♥ ❦➳t l ∈ L ˜ ❝â ❞✉♥❣ ❧÷đ♥❣ ❧➔ L cl > tt r ộ ởt ỗ õ t ũ ♥❤✐➲✉ ✤÷í♥❣ ❞➝♥✳ ❑➼ ❤✐➺✉ Ps ❧➔ t➟♣ ˜ ❝→❝ ữớ ữủ sỷ ỗ s L (p) s ⊂ L ❧➔ t➟♣ ❝→❝ ❧✐➯♥ ❦➳t ♠➔ t❤æ♥❣ q✉❛ ♥â ✤÷í♥❣ ❞➝♥ p ∈ Ps ✤✐ q✉❛ ✈➔ ns ❧➔ sè ❝→❝ ♣❤➛♥ tû ❝õ❛ Ps✱ N = ns✳ ●✐↔ sû x(p) s ❧➔ tè❝ s∈S ✤ë tr✉②➲♥ t ỗ s ữớ p Ps q õ tố tr t ỗ s ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ❜ð✐ xs = x(p) s ✳ p∈Ps ❚➟♣ r➔♥❣ ❜✉ë❝ ✈➲ ❞✉♥❣ ❧÷đ♥❣ ❦➼ ❤✐➺✉ ❧➔ C õ r ữủ ố ợ ♠é✐ ❧✐➯♥ ❦➳t s❛♦ ❝❤♦ tê♥❣ tè❝ ✤ë tr✉②➲♥ t↔✐ ỗ ũ ởt t õ ❧➔ ♥❤ä ❤ì♥ ❤♦➦❝ ❜➡♥❣ ❞✉♥❣ ❧÷đ♥❣ ❝õ❛ ❧✐➯♥ ❦➳t✳ ❚➟♣ C ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ C := Cl = ∅ ✈ỵ✐ ˜ l∈L Cl := (x(p) s )p∈Ps s∈S˜ (p) ∈ RN + : x(p) s Is,l ≤ cl , ˜ s∈S,p∈P s ♥➳✉ l ∈ L˜ (p) s , ð ✤➙② tr♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ ❦❤→❝ ●✐↔ sû ỗ s õ ởt tố tr t↔✐ ✤➸ s❛♦ ❝❤♦ tè❝ ✤ë tè✐ t❤✐➸✉ ♣❤↔✐ ❧➔ rs > 0✳ ❑➼ ❤✐➺✉ t➟♣ r➔♥❣ ❜✉ë❝ ✤✐➲✉ ❝❤➾♥❤ ❧➔ D := Ds✱ tr♦♥❣ ✤â Ds ❧➔ t➟♣ ❜❛♦ ỗ (p) Is,l = sS tố ữ ố ợ ỗ s ✈➔ Ds := (x(p) s )p∈Ps s∈S˜ ∈ RN + : x(p) s ≥ rs p∈Ps ❇➔✐ t♦→♥ ♣❤➙♥ ♣❤è✐ ❜➠♥❣ t❤æ♥❣ ❝â t❤➸ q✉② ✈➲ ❜➔✐ t♦→♥ ❝ü❝ ✤↕✐ tr➯♥ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ♠ët →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ s❛✉ ✤➙②✿ ❚➻♠ x∗ ∈ ❋✐①(T ) s❛♦ ❝❤♦ : U (x∗) = ❋✐① max U (x), ✭✸✳✸✮ (T ) tr♦♥❣ ✤â U : RN → R ❧➔ ❤➔♠ t✐➺♥ ➼❝❤ ✤÷đ❝ ❣✐↔ t❤✐➳t ❧➔ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ✈➔ T : RN → RN ①→❝ ✤à♥❤ ❜ð✐ T (x) := (1/2)(x + Tˆ(x)) ✈ỵ✐ Tˆ(x) := PRN+ ∩Cl0 vl P Cl + ˜ l∈L,l=l us PDs (x) s∈S˜ ✶✾ ❚❛ ❝â t❤➸ sû ❞ư♥❣ ❜❛ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✮✱ ✭✷✳✷✺✮ ✈➔ ✭✷✳✸✶✮ ✤➸ t➻♠ ♥❣❤✐➺♠ ①➜♣ ①➾ ❝õ❛ ❜➔✐ t♦→♥ ✭✸✳✸✮✳ ❈ö t❤➸✱ ❝❤♦ E := RN ✱ F := −∇U ✈➔ ❝❤å♥ Ti := T ✈ỵ✐ ♠å✐ i ∈ N ❇❛ ♠➺♥❤ ✤➲ s❛✉ t÷ì♥❣ ù♥❣ ❧➔ ❤➺ q✉↔ trü❝ t✐➳♣ ❝õ❛ ❝→❝ ✣à♥❤ ❧➼ ✷✳✶✱ ✣à♥❤ ❧➼ ✷✳✹ ✈➔ ✣à♥❤ ❧➼ ✷✳✺✳ ▼➺♥❤ ✤➲ ✸✳✶✳ ❈❤♦ U : RN → R ❧➔ ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ❝â −∇U : RN → RN ❧➔ →♥❤ ①↕ η ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ γ ✲❣✐↔ ❝♦ ❝❤➦t ✈ỵ✐ η + γ > 1✳ ❈❤♦ T : RN → RN ❧➔ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ✈ỵ✐ ❋✐①(T ) = ∅✳ ❈❤♦ αi ∈ (0, 1)✳ ●✐↔ sû λk ∈ (0, 1) ✈➔ si t÷ì♥❣ ù♥❣ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭▲✶✮✱ ✭▲✷✮ ✈➔ ✭✷✳✹✮✳ ❑❤✐ ✤â✱ ✈ỵ✐ ✤✐➸♠ tũ ỵ x1 RN {xk } ①→❝ ✤à♥❤ ❜ð✐ k xk+1 = (I + λk ∇U ) k (1 − αi )ξik I αi ξik T (xk ), + i=1 k ≥ 1, ✭✸✳✻✮ i=1 ❤ë✐ tư tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ✭✸✳✸✮✱ tr♦♥❣ ✤â ξik := si/˜sk ▼➺♥❤ ✤➲ ✸✳✷✳ ❈❤♦ U ✱ T ✈➔ αi ✤÷đ❝ ❣✐↔ t❤✐➳t t÷ì♥❣ tü ▼➺♥❤ ✤➲ ✸✳✶✳ ●✐↔ sû λk ∈ (0, 1) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭▲✶✮✱ ✭▲✷✮ ✈➔ {si} ❧➔ ❞➣② sè t❤ü❝ ❞÷ì♥❣ ❣✐↔♠ ♥❣➦t✱ ❤ë✐ tư ✈➲ 0✳ ❑❤✐ õ ợ tũ ỵ x1 RN ✱ ❞➣② {xk } ①→❝ ✤à♥❤ ❜ð✐ k xk+1 = (I + λk ∇U ) k (1 − i=1 αi )βik I αi βik T (xk ), + k ≥ 1, ✭✸✳✼✮ i=1 ❤ë✐ tư tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ✭✸✳✸✮✱ tr♦♥❣ ✤â βik := (si−1 − si)/(s0 − sk ) ▼➺♥❤ ✤➲ ✸✳✸✳ ❈❤♦ U ✈➔ T ✤÷đ❝ ❣✐↔ t❤✐➳t t÷ì♥❣ tü ▼➺♥❤ ✤➲ ✸✳✶✳ ●✐↔ sû α ❧➔ ♠ët sè t❤ü❝ ❝è ✤à♥❤ t❤✉ë❝ (0, 1) ✈➔ λk ∈ (0, 1) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭▲✶✮ ✈➔ ✭▲✷✮✳ ❑❤✐ ✤â✱ ✈ỵ✐ ✤✐➸♠ ❜❛♥ ✤➛✉ tò② þ x1 ∈ RN ✱ ❞➣② {xk } ①→❝ ✤à♥❤ ❜ð✐ xk+1 = (I + λk ∇U )((1 − α)I + αT )(xk ), k ≥ 1, ✭✸✳✽✮ ❤ë✐ tö tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ✭✸✳✸✮✳ ◆❤➟♥ ①➨t ✸✳✶✳ ◆➳✉ αi := α ∈ (0, 1) ❝è ✤à♥❤ t❤➻ ✭✸✳✻✮ ✈➔ ✭✸✳✼✮ s➩ ❝â ❞↕♥❣ ✭✸✳✽✮✳ ▼➦t ❦❤→❝✱ t❛ ❧↕✐ ❝â ❋✐①(T ) = ❋✐①(Tˆ) ♥➯♥ ✈ỵ✐ α = 1/2 t❛ ❝â t❤➸ ❝❤å♥ Ti := Tˆ ✈ỵ✐ ♠å✐ i ∈ N ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ❞➣② {xk } trð t❤➔♥❤ xk+1 = (I + λk ∇U )T (xk ), k ≥ 1, ✭✸✳✾✮ ✈➔ ❞➣② ❧➦♣ ✭✸✳✾✮ ❤ë✐ tư tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ✭✸✳✸✮ ợ tt tữỡ tỹ ❱➼ ❞ư sè ♠✐♥❤ ❤å❛ ❈→❝ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❞↕♥❣ ❤✐➺♥ ♠ỵ✐ ❝õ❛ ❝❤ó♥❣ tỉ✐ ❝â t❤➸ →♣ ❞ư♥❣ ✤➸ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❝ü❝ trà✿ ❚➻♠ x∗ ∈ C s❛♦ ❝❤♦ : ∞ ϕ(x∗ ) = ϕ(x), x∈C C := Ci , i=1 ✭✸✳✶✵✮ ✷✵ tr♦♥❣ ✤â ỗ õ (x) ❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③✱ ✤ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ Rn Ci t ỗ õ Rn ✤÷đ❝ ❝❤♦ ❜ð✐ Ci = {x ∈ Rn : ai1 u1 + ai2 u2 + · · · + ain un ≤ bi }, ✭✸✳✶✶✮ ❤♦➦❝ n n (uj − aij )2 ≤ ri2 }, Ci = {x ∈ R : ✭✸✳✶✷✮ ri > 0, j=1 ð ✤➙② aij , bi, ri ∈ R (1 ≤ j ≤ n) ❱➼ ❞ư ✸✳✶✳ ❳➨t ❜➔✐ t♦→♥ ✭✸✳✶✵✮✲✭✸✳✶✶✮ tr♦♥❣ tr÷í♥❣ ❤đ♣ n = 2✳ ❍➔♠ ♠ö❝ t✐➯✉ ϕ : R2 → R ❝â ❞↕♥❣ ϕ(x) := x = u21 + u22 ✈ỵ✐ x = (u1 , u2 ) ❈→❝ t➟♣ Ci ✤÷đ❝ ❝❤♦ ❜ð✐ Ci = {x ∈ R2 : ai1u1 + ai2u2 ≤ bi} ✈ỵ✐ ai1 = 1/i, ai2 = −1 ✈➔ bi = ✈ỵ✐ ♠å✐ i ≥ 1✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ❞➵ t❤➜② x∗ = (0; 0) ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥✳ ⑩♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✮ ❝❤♦ ✈➼ ❞ư ♥➔② ✈ỵ✐ F (x) = ∇ϕ(x) ✈➔ Ti = PC ✳ ❈❤å♥ ✤✐➸♠ ❜❛♥ ✤➛✉ x1 = (2.0; −3.0) ✈➔ ❝→❝ ❞➣② t❤❛♠ sè t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❤ë✐ tö ❝õ❛ ✣à♥❤ ❧➼ ✷✳✶ ❧➔ λk = 1/(k + 2) ✈➔ si = αi = 1/i(i + 1) ữợ t ữủ ❦➳t q✉↔ t➼♥❤ t♦→♥ x100 = (−0.000100272; −0.000040995)✳ ❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ tỉ✐ →♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✶✳✼✮ ❝õ❛ ■❡♠♦t♦ ✈➔ ✤t❣ ❝❤♦ ❝ò♥❣ ❜➔✐ t♦→♥ tr➯♥✳ ❚❛ ❝❤å♥ ❝→❝ t❤❛♠ sè t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❤ë✐ tö ❝õ❛ ✣à♥❤ ❧➼ ✶✳✺ ❧➔ λk = 1/(k+2), αk = 1/100+1/k(k+1) ✈➔ ρ = 1/20 t q t t ố ợ ữỡ ợ ũ số ữợ ❧➔ x100 = (−0.335041279; −0.149090066)✳ ◆➳✉ t❛ ❝❤å♥ ρ = 1/3 t❤➻ ❦➳t q✉↔ ♥❤➟♥ ✤÷đ❝ ♥❤÷ s❛✉ x100 = (−0.037590156; −0.016727249)✳ ❇➙② ❣✐í✱ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✶✳✽✮ ❝õ❛ ❨❛♦ ✈➔ ✤t❣✳ ❈→❝ t❤❛♠ sè ✤÷đ❝ ❝❤å♥ t❤ä❛ ♠➣♥ ✣à♥❤ ❧➼ ✶✳✻ ❧➔ λk = 1/(k + 2), αk = 1/100 + 1/k(k + 1) ✈➔ γk = 1/100 t q t t ữỡ ợ ũ số ữợ ữủ x100 = (0.000210945; −0.000385873)✳ ◆➳✉ t❛ ❝❤å♥ γk = 1/1000 t❤➻ ❦➳t q✉↔ ♥❤➟♥ ✤÷đ❝ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② ❧➔ x100 = (0.000373078; 0.000568259) ữợ tữỡ q ✈➲ s❛✐ sè t➼♥❤ t♦→♥ s♦ ✈ỵ✐ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✭✶✳✼✮✱ ✭✶✳✽✮ ✈➔ ✭✷✳✶✮✳ P❤÷ì♥❣ ♣❤→♣ k xk − x∗ ❚❤í✐ ❣✐❛♥ ✭❣✐➙②✮ ✭✶✳✼✮ ✭✈ỵ✐ ρ = 1/3 4.1143 ì 102 ợ = 1/20 3.6671 ì 101 ợ k = 1/100 4.3976 ì 104 ợ k = 1/1000✮ ✶✵✵ 6.7978 × 10−4 ✵✳✵✹✽✵ ✭✷✳✶✮ ✶✵✵ 1.0833 × 10−4 ✵✳✵✸✶✵ i ✷✶ ◆❤➟♥ ①➨t ✸✳✷✳ ❈â t❤➸ t❤➜② r➡♥❣✱ tr♦♥❣ ♠é✐ ♣❤÷ì♥❣ ♣❤→♣ ✭✶✳✼✮✱ ✭✶✳✽✮ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✮ ❝õ❛ ❝❤ó♥❣ tỉ✐ ✤➲✉ ❝â ✸ ❞➣② ❝→❝ t❤❛♠ sè ❧➦♣✳ ❚❤❛♠ sè t❤ù ♥❤➜t✱ λk ð ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭▲✶✮ ✈➔ ✭▲✷✮ ♥❤÷ ♥❤❛✉ ✈➔ ✤÷đ❝ ❝❤å♥ ❣✐è♥❣ ♥❤❛✉ ❧➔ λk = 1/(k +2)✳ ❚❤❛♠ sè t❤ù ❤❛✐ αi t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❦❤→❝ ♥❤❛✉ ✤↔♠ ❜↔♦ sü ❤ë✐ tö✱ ð ✤â ♥â ✤÷đ❝ ❝❤å♥ s❛♦ ❝❤♦ ❝→❝ ❜✐➳♥ t❤➸ ❝õ❛ ♥â tr♦♥❣ ♥❤ú♥❣ t❤✉➟t t♦→♥ ♥➔② ❧➔ ♥❤÷ ♥❤❛✉✳ ❚❤❛♠ sè ρ tr♦♥❣ ✭✶✳✼✮✱ γk tr♦♥❣ ✭✶✳✽✮ ✈➔ si tr♦♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✮ ❝õ❛ ❝❤ó♥❣ tỉ✐ ✤â♥❣ ✈❛✐ trá ❦❤→❝ ♥❤❛✉✱ ♥â ❝❤♦ ♣❤➨♣ t❤ü❝ ❤✐➺♥ ❝→❝ q✉② t➢❝ r✐➯♥❣ ❜✐➺t tr♦♥❣ t❤✐➳t ❦➳ ❝õ❛ ♠é✐ t❤✉➟t t♦→♥✳ ❘ã r➔♥❣✱ tr♦♥❣ ✈➼ ❞ư tr➯♥✱ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✮ ❝õ❛ ❝❤ó♥❣ tỉ✐ ✤➲ ①✉➜t ❝â tè❝ ✤ë ❤ë✐ tư ♥❤❛♥❤ ❤ì♥ ✈➔ ❝➛♥ ➼t t❤í✐ ❣✐❛♥ t➼♥❤ t♦→♥ ❤ì♥ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✭✶✳✼✮✱ ✭✶✳✽✮✳ ❱➼ ❞ö ✸✳✷✳ ❳➨t ❜➔✐ t♦→♥ ✭✸✳✶✵✮✲✭✸✳✶✷✮ tr♦♥❣ tr÷í♥❣ ❤đ♣ n = 2✳ ❍➔♠ ♠ư❝ t✐➯✉ ϕ : R2 → R ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ϕ(x) = (u1 − 1)2 + (u2 − 2)2 ✈ỵ✐ x = (u1 , u2 ) ❈→❝ t➟♣ Ci ✤÷đ❝ ❝❤♦ ❜ð✐ Ci = {x ∈ R2 : (u1 − ai1 )2 + (u2 − ai2 )2 ≤ ri2 } √ ✈ỵ✐ ri = 1✱ ai1 = + 1/i ✈➔ ai2 = ợ i r trữớ ủ x∗ = (1.5; 0.75) ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t ữỡ ợ F (x) = ∇ϕ(x) ✈➔ Ti = PC ✳ ❈❤å♥ ✤✐➸♠ ❜❛♥ ✤➛✉ ❧➔ x1 = (3.0; 3.0) ✈➔ ❞➣② ❝→❝ t❤❛♠ sè t÷ì♥❣ tü ♥❤÷ tr♦♥❣ ❱➼ ❞ư ✸✳✶✳ ❑❤✐ ✤â✱ ❦➳t q t t ữợ t ữủ ♥❣❤✐➺♠ ①➜♣ ①➾ ❧➔ (1.54118986; 0.88877202)✳ ❚r♦♥❣ ❦❤✐ ✤â✱ ❝ò♥❣ ữợ ữ tr ữỡ ✈ỵ✐ ρ = 1/3 t❤➻ ♥❣❤✐➺♠ ①➜♣ ①➾ ❧➔ (1.552771131; 0.894458825) sỷ ữỡ ợ k = 1/100 t❤➻ t❛ ♥❤➟♥ ✤÷đ❝ ♥❣❤✐➺♠ ①➜♣ ①➾ ❧➔ (1.548117716; 0.903764265)✳ ❇↔♥❣ t÷ì♥❣ q✉❛♥ ✈➲ s❛✐ sè t➼♥❤ t♦→♥ s♦ ợ ữỡ ✈➔ ✭✷✳✶✮ tr♦♥❣ ✈➼ ❞ư ♥➔② ❧➔✿ P❤÷ì♥❣ ♣❤→♣ k xk − x∗ ❚❤í✐ ❣✐❛♥ ✭❣✐➙②✮ ✭✶✳✼✮ ✭✈ỵ✐ ρ = 1/3 5994 ì 102 ợ k = 1/100✮ ✹✻✵✵✵ 6.1152 × 10−2 ✹✵✶✼✳✽✷✵✵ ✭✷✳✶✮ ✹✻✵✵✵ 4.7053 × 10−2 ✽✽✷✳✼✼✹✵ ◆❤➟♥ ①➨t ✸✳✸✳ ❚r♦♥❣ ✈➼ ❞ư ♥➔②✱ ❝❤ó♥❣ t❛ ❝ơ♥❣ t❤➜② ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✮ ❝â tè❝ ✤ë ❤ë✐ tư ♥❤❛♥❤ ❤ì♥ ✈➔ ❝➛♥ ➼t t❤í✐ ❣✐❛♥ t➼♥❤ t♦→♥ ❤ì♥ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✭✶✳✼✮ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ✭✶✳✽✮✳ i ❱➼ ❞ư ✸✳✸✳ ❚❛ ①➨t ❜➔✐ t♦→♥ ✭✸✳✶✵✮✲✭✸✳✶✶✮ tr♦♥❣ tr÷í♥❣ ❤đ♣ n = ✈➔ ❤➔♠ ♠ư❝ t✐➯✉ ϕ : R2 → R ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ϕ(x) = xT Ax + bT x + c ✈ỵ✐ x = (u1 , u2 ), ✷✷ tr♦♥❣ ✤â A= ,b = −4 −6 ✈➔ c = 13 ❈→❝ t➟♣ Ci ✤÷đ❝ ❝❤♦ ❜ð✐ Ci = {x ∈ R2 : ai1u1 + ai2u2 ≥ bi} ✈ỵ✐ ai1 = 1✱ ai2 = i ✈➔ bi = ✈ỵ✐ ♠å✐ i ≥ 1✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② x∗ = (2.0; 3.0) ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥✳ ⑩♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✷✺✮ ❝❤♦ ✈➼ ❞ư ♥➔② ✈ỵ✐ F (x) = ∇ϕ(x) ✈➔ Ti = PC ✳ ❈❤å♥ ✤✐➸♠ ❜❛♥ ✤➛✉ x1 = (−3.0; −3.0) ✈➔ ❝→❝ t❤❛♠ sè t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❤ë✐ tö ❝õ❛ ✣à♥❤ ❧➼ ✷✳✹ ❧➔ λk = 1/k + 2, si = 1/(i + 1)(i + 2) ✈ỵ✐ i ≥ 0✱ αi = 1/i(i + 1) ✈ỵ✐ i ≥ ❙❛✉ ✶✵✵✵ ✈á♥❣ ❧➦♣✱ t❛ ❝â x1000 = (1.999975551; 2.999969617) ◆➳✉ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✶✳✼✮ ✈ỵ✐ ❝ò♥❣ ✤✐➸♠ ①✉➜t ♣❤→t ✈➔ ❝❤å♥ ❝→❝ t❤❛♠ sè ❧➦♣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❤ë✐ tö ❝õ❛ ✣à♥❤ ❧➼ ✶✳✺ ❧➔ λk = 1/(k + 2), αk = 1/100 + 1/k(k + 1) ✈➔ ρ = 1/20 t❤➻ ❦➳t q t t ố ợ ữỡ ữợ ❧➦♣ t❤ù 1000 ❧➔ x1000 = (−0.003777417; 0.004757678)✳ ◆❣❤✐➺♠ ♥➔② ❝á♥ s❛✐ sè r➜t ❧ỵ♥ s♦ ✈ỵ✐ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❝õ❛ ❜➔✐ t♦→♥✳ ◆➳✉ sû ❞ư♥❣ ✭✶✳✽✮ ✈ỵ✐ ❝ò♥❣ ✤✐➸♠ ①✉➜t ♣❤→t ✈➔ ❝❤å♥ ❝→❝ t❤❛♠ sè ❧➦♣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❤ë✐ tö ❝õ❛ ✣à♥❤ ❧➼ ✶✳✻ ❧➔ λk = 1/(k + 2), αk = 1/100 + 1/k(k + 1) ✈➔ γk = 1/2 t❤➻ ❦➳t q✉↔ ð ❝ò♥❣ số ữợ x1000 = (1.999988011; 2.999986013) sỷ ữỡ ợ ũ t t ❝→❝ t❤❛♠ sè ❧➦♣ ✤÷đ❝ ❝❤å♥ t÷ì♥❣ tü ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✷✺✮ t❤➻ t❛ ♥❤➟♥ ✤÷đ❝ x1000 = (1.999993006; 2.999991008)✳ tữỡ q s số t t s ợ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✭✶✳✼✮✱ ✭✶✳✽✮✱ ✭✷✳✶✮ ✈➔ ✭✷✳✷✺✮ tr♦♥❣ ✈➼ ❞ư ♥➔② ❧➔✿ P❤÷ì♥❣ ♣❤→♣ k xk − x∗ ❚❤í✐ ❣✐❛♥ ✭❣✐➙②✮ ✭✶✳✼✮ ✶✵✵✵ 3.603692620 ✶✳✾✹✶✵ ✭✶✳✽✮ ✶✵✵✵ 1.8420 × 10−5 ✶✳✼✺✶✵ ✭✷✳✶✮ ✶✵✵✵ 1.1390 × 10−5 ✵✳✼✾✹ ✭✷✳✷✺✮ ✶✵✵✵ 3.8998 × 10−5 ✵✳✽✸✸ ◆❤➟♥ ①➨t ✸✳✹✳ ❚r♦♥❣ ✈➼ ❞ö ♥➔②✱ t❛ t❤➜② tè❝ ✤ë ❤ë✐ tư ✈➔ t❤í✐ ❣✐❛♥ t➼♥❤ t♦→♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✮ ❧➔ ♥❤❛♥❤ ♥❤➜t tr♦♥❣ sè ❜è♥ ♣❤÷ì♥❣ ♣❤→♣✳ ❚è❝ ✤ë ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✷✺✮ ❧➔ ♥❤❛♥❤ ❤ì♥ ♣❤÷ì♥❣ ữ ỡ s ợ ữỡ ✭✶✳✽✮✳ ❚✉② ♥❤✐➯♥✱ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✷✺✮ ❧↕✐ ❝â t❤í✐ ❣✐❛♥ t t s ợ ữỡ ✭✶✳✽✮ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✳ i ❱➼ ❞ư ✸✳✹✳ ❙û ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✸✶✮ ❝õ❛ ❝❤ó♥❣ tỉ✐ ❝❤♦ ❜➔✐ t♦→♥ ợ tt tữỡ tỹ ữ tr ❞ư ✸✳✶✳ ❱ỵ✐ ❝ò♥❣ ✤✐➸♠ ❜❛♥ ✤➛✉ x1 = (2.0; −3.0)✱ ❝❤å♥ α = 0.5 ✈➔ ❣✐→ trà ❝õ❛ ❝→❝ t❤❛♠ sè ❧➦♣ ❦❤→❝ ✤÷đ❝ ❝❤å♥ ❣✐è♥❣ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✮ ð ❱➼ ❞ö ✸✳✶ ❧➔ λk = 1/(k + 2) ✈➔ si = 1/i(i + 1) t❤➻ s❛✉ ✶✵✵ ữợ ú tổ ữủ x100 = (0.000078416; 0.000004588) ✷✸ ❇➙② ❣✐í✱ sû ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✭✶✳✶✼✮✱ ✭✶✳✶✽✮ ❝õ❛ ◆❣✉②➵♥ ❇÷í♥❣ ✈➔ ✤t❣✳ ❈❤å♥ ❝→❝ t❤❛♠ sè λk , si ♥❤÷ tr➯♥ ✈➔ γk = 0.5✳ ❑❤✐ ✤â✱ ✈ỵ✐ ❝ò♥❣ ✤✐➸♠ ❜❛♥ ✤➛✉✱ t❛ ❝â ♥❣❤✐➺♠ ①➜♣ ①➾ ũ số ữợ tữỡ ự ợ ữỡ ♣❤→♣ ❧➔ x100 = (−0.00539367; −0.00032443) ✈➔ x100 = (−0.01002674; −0.00057259)✳ ❇↔♥❣ t÷ì♥❣ q✉❛♥ ✈➲ s❛✐ sè t➼♥❤ t♦→♥ s♦ ợ ữỡ ✭✷✳✶✮ ✈➔ ✭✷✳✸✶✮ tr♦♥❣ ✈➼ ❞ư ♥➔② ❧➔✿ P❤÷ì♥❣ ♣❤→♣ k xk − x∗ ❚❤í✐ ❣✐❛♥ ✭❣✐➙②✮ ✭✶✳✶✼✮ ✶✵✵ 5.4034 × 10−3 ✵✳✵✹✼✵ ✭✶✳✶✽✮ ✶✵✵ 1.0004 × 10−2 ✵✳✵✸✷✵ ✭✷✳✶✮ ✶✵✵ 1.0833 × 10−5 ✵✳✵✸✶✵ ✭✷✳✸✶✮ ✶✵✵ 1.5868 × 10−6 t sỷ ữỡ ố ợ t ợ tt tữỡ tỹ ữ tr♦♥❣ ❱➼ ❞ư ✸✳✷✳ ❑❤✐ ✤â✱ ✈ỵ✐ ❝ò♥❣ ✤✐➸♠ ❜❛♥ ✤➛✉ x1 = (3.0; 3.0)✱ ❝❤å♥ α = 0.5 ✈➔ ❣✐→ trà ❝õ❛ ❝→❝ t❤❛♠ sè ❧➦♣ ❦❤→❝ ✤÷đ❝ ❝❤å♥ ố ữ tr t t ữợ 45000 ①➜♣ ①➾ ❝õ❛ ❜➔✐ t♦→♥ ❧➔ (1.5034141156; 0.8682249753)✳ ❙û ❞ö♥❣ ữỡ ợ k = 1/50 k ❝❤➤♥ ❝á♥ γk = 1/100 ♥➳✉ k ❧➫ t❤➻ t❛ ❝â ♥❣❤✐➺♠ ①➜♣ ①➾ t÷ì♥❣ ù♥❣ ❧➔ x46000 = (1.709749782; 0.707411290) ✈➔ x46000 = (1.578254678; 0.816731616)✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② t❛ ❝â ❜↔♥❣ t÷ì♥❣ q✉❛♥✿ P❤÷ì♥❣ ♣❤→♣ k xk − x∗ ❚❤í✐ ❣✐❛♥ ✭❣✐➙②✮ ✭✶✳✶✼✮ ✹✻✵✵✵ 0.262970355 ✾✵✹✳✾✻✽✵ ✭✶✳✶✽✮ ✹✻✵✵✵ 9.2486 × 10−2 ✾✵✸✳✶✺✼✵ ✭✷✳✶✮ ✹✻✵✵✵ 4.7053 × 10−2 ✽✽✷✳✼✼✹✵ ✭✷✳✸✶✮ ✹✺✵✵✵ 4.0613 × 10−3 ✽✻✹✳✾✾✶✵ ◆❤➟♥ ①➨t ✸✳✺✳ ❚r♦♥❣ ✈➼ ❞ư ♥➔②✱ ❝❤ó♥❣ t❛ ❝â t❤➸ t❤➜② ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✸✶✮ ❝â tè❝ ✤ë ❤ë✐ tư ♥❤❛♥❤ ❤ì♥ ✈➔ tè♥ ➼t t❤í✐ ❣✐❛♥ t➼♥❤ t♦→♥ ❤ì♥ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✭✶✳✶✼✮✱ ✭✶✳✶✽✮ ✈➔ ✭✷✳✶✮✳ ❇➯♥ ❝↕♥❤ ✤â✱ ♥â t❤➸ ❤✐➺♥ t➼♥❤ ✈÷đt trë✐ ❤ì♥ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ tr♦♥❣ ❝→❝ ✈➼ ❞ö ✤➣ tr➻♥❤ ❜➔② ð tr➯♥✳ ✷✹ ❑➌❚ ▲❯❾◆ ❱⑨ ❑■➌◆ ◆●❍➚ ❚r♦♥❣ ❧✉➟♥ →♥✱ ❝❤ó♥❣ tỉ✐ ✤➣ ✤➲ ①✉➜t ❜❛ ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ ✤÷í♥❣ ❞è❝ ♥❤➜t ①➜♣ ①➾ ♥❣❤✐➺♠ ❝❤♦ ♠ët ❧ỵ♣ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr➯♥ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr➯♥ ổ tỹ ỗ t õ ❝❤✉➞♥ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉✳ ▲✉➟♥ →♥ ✤➣ ✤↕t ✤÷đ❝ ❝→❝ ❦➳t q✉↔ s❛✉✿ ✲ ✣➲ ①✉➜t ✤÷đ❝ ❜❛ ♣❤÷ì♥❣ ợ ữỡ ữỡ ♣❤→♣ ✭✷✳✷✺✮ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✸✶✮ ✤➸ ①➜♣ ①➾ ♥❣❤✐➺♠ ❝❤♦ ❧ỵ♣ ❜➔✐ t♦→♥ ♥❣❤✐➯♥ ❝ù✉✳ ◆ë✐ ❞✉♥❣ ✈➔ sü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✭✣à♥❤ ❧➼ ✷✳✶✱ ✣à♥❤ ❧➼ ✷✳✹ ✈➔ ✣à♥❤ ❧➼ ✷✳✺✮ ✤÷đ❝ tr➻♥❤ ❜➔② tt ữỡ ợ õ t ❞ö♥❣ ❝❤♦ ❜➔✐ t♦→♥ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ❤å ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ❤♦➦❝ t➻♠ ❦❤æ♥❣ ✤✐➸♠ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ✈æ ❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ →♥❤ ①↕ j ✲✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳ ✲ ❳➙② ❞ü♥❣ ✤÷đ❝ ❜è♥ ✈➼ ❞ư sè ❝ư t❤➸ ♠✐♥❤ ❤å❛ ❝❤♦ ❝→❝ t❤✉➟t t♦→♥ ♠ỵ✐ ✤➲ ①✉➜t ✈➔ tữỡ q ợ ởt số ữỡ ởt số ữợ ự t t ❦➳t q✉↔ ❝õ❛ ❧✉➟♥ →♥ ♥❤÷ s❛✉✿ ✭■✮ ◆❣❤✐➯♥ ❝ù✉ t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣✳ ✭■■✮ ◆❣❤✐➯♥ ❝ù✉ ❝→❝ t✐➯✉ ❝❤✉➞♥ ❞ø♥❣ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✤➣ ✤➲ ①✉➜t tø ✤â ❝â t❤➯♠ ❝ì sð ✤➸ s♦ s→♥❤ tè❝ ✤ë ❤ë✐ tư ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✤➣ ✤➲ ①✉➜t s♦ ✈ỵ✐ ❝→❝ ❦➳t q✉↔ ❝õ❛ ♠ët sè t→❝ ❣✐↔ ❦❤→❝✳ ✭■■■✮ ◆❣❤✐➯♥ ❝ù✉ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ t→❝❤ ✭❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ♥❤✐➲✉ ❝➜♣✮✳ ❉❆◆❍ ▼Ö❈ ❈⑩❈ ❈➷◆● ❚❘➐◆❍ ✣❶ ❈➷◆● ❇➮ ▲■➊◆ ◗❯❆◆ ✣➌◆ ▲❯❾◆ ⑩◆ ✭✶✮ ❇✉♦♥❣ ◆❣✳✱ ❍❛ ◆❣✳ ❙✳✱ ❚❤✉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✧✱ ◆♦♥❧✐♥❡❛r ❋✉♥❝t✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✷✶✱ ♣♣✳ ✷✼✸✲✷✽✼✳ ✭✸✮ ❍❛ ◆❣✳ ❙✳✱ ❇✉♦♥❣ ◆❣✳✱ ❚❤✉② ◆❣✳ ❚✳ ❚✳ ✭✷✵✶✽✮✱ ✧❆ ♥❡✇ s✐♠♣❧❡ ♣❛r❛❧❧❡❧ ✐t❡r❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ❛ ❝❧❛ss ♦❢ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✧✱ ❆❝t❛ ▼❛t❤✳ ❱✐❡t♥❛♠✳✱ ✹✸✱ ♣♣✳ ✷✸✾✲✷✺✺✳