❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖ ❱■➏◆ ❍⑨◆ ▲❹▼ ❑❍❖❆ ❍➴❈ ❱⑨ ❈➷◆● ◆●❍➏ ❱■➏❚ ◆❆▼ ❍➴❈ ❱■➏◆ ❑❍❖❆ ❍➴❈ ❱⑨ ❈➷◆● ◆●❍➏ ✳✳✳✳✳✳✳✳✳✳✳✳✯✯✯✳✳✳✳✳✳✳✳✳✳✳✳ P❍❆▼ ❚❍➚ ❚❍❯ ❍❖⑨■ ▼❐❚ ❙➮ P❍×❒◆● P❍⑩P ●■❷■ ❇⑨■ ❚❖⑩◆ ❚➐▼ ❑❍➷◆● ✣■➎▼ ❈Õ❆ ❚❖⑩◆ ❚Û ✣❒◆ ✣■➏❯ ❈Ü❈ ✣❸■ ❱⑨ ❇⑨■ ❚❖⑩◆ ❈❍❻P ◆❍❾◆ ❚⑩❈❍ ◆❍■➋❯ ❚❾P ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ö♥❣ ▼➣ sè✿ ✾ ✹✻ ✵✶ ✶✷ ❚➶▼ ❚➁❚ ▲❯❾◆ ⑩◆ ❚■➌◆ ❙➒ ❚❖⑩◆ ❍➴❈ ❍➔ ◆ë✐ ✲ ✷✵✷✷ ❈ỉ♥❣ tr➻♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐✿ ❍å❝ ✈✐➺♥ ❑❤♦❛ ❤å❝ ✈➔ ❈æ♥❣ ♥❣❤➺ ✲ ❱✐➺♥ ❍➔♥ ❧➙♠ ❑❤♦❛ ❤å❝ ✈➔ ❈æ♥❣ ♥❣❤➺ ❱✐➺t ◆❛♠ ữớ ữợ ữớ P ✶✿ P❤↔♥ ❜✐➺♥ ✷✿ P❤↔♥ ❜✐➺♥ ✸✿ ▲✉➟♥ →♥ s➩ ữủ trữợ ỗ t s➽✱ ❤å♣ t↕✐ ❍å❝ ✈✐➺♥ ❑❤♦❛ ❤å❝ ✈➔ ❈æ♥❣ ♥❣❤➺ ✲ ❱✐➺♥ ❍➔♥ ❧➙♠ ❑❤♦❛ ❤å❝ ✈➔ ❈æ♥❣ ♥❣❤➺ ❱✐➺t ỗ t ✷✵✷✷ ❈â t❤➸ t➻♠ ❤✐➸✉ ❧✉➟♥ →♥ t↕✐✿ ✲ ❚❤÷ ✈✐➺♥ ❍å❝ ✈✐➺♥ ❑❤♦❛ ❤å❝ ✈➔ ❈ỉ♥❣ ♥❣❤➺ ✲ ❚❤÷ ✈✐➺♥ ◗✉è❝ ❣✐❛ ❱✐➺t ◆❛♠ ▼ð ✤➛✉ ◆❤✐➲✉ ❜➔✐ t♦→♥ tr♦♥❣ ❦❤♦❛ ❤å❝ ❦ÿ t❤✉➟t ✈➔ tr♦♥❣ ✤í✐ sè♥❣ ❞➝♥ ✤➳♥ ❜➔✐ t♦→♥ tê♥❣ q✉→t ❧➔ t➻♠ ❝ü❝ t✐➸✉ ❝õ❛ ♠ët ♣❤✐➳♠ ❤➔♠ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❤♦➦❝ ✈æ ❤↕♥ ❝❤✐➲✉✳ ❈❤♦ ✤➳♥ ♥❛②✱ ❝â ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ✤÷đ❝ ✤➲ ①✉➜t ✤➸ t➻♠ ❝ü❝ t✐➸✉ ❝õ❛ ♠ët ♣❤✐➳♠ ❤➔♠✳ ▼ët ♣❤÷ì♥❣ ♣❤→♣ ✤➦❝ ❜✐➺t q✉❛♥ trå♥❣ ✤➸ t➻♠ ♠ët ỹ t ỗ ữỡ ❣➛♥ ❦➲ ✤÷đ❝ ✤➲ ①✉➜t ❜ð✐ ▼❛rt✐♥❡t ✈➔♦ ♥➠♠ ✶✾✼✵✳ ◆➠♠ ✶✾✼✻✱ ❘♦❝❦❛❢❡❧❧❛r ✤➣ ♠ð rë♥❣ ♣❤÷ì♥❣ ♣❤→♣ tr➯♥ ❝❤♦ ❜➔✐ t♦→♥ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ ♠ët t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ T tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✱ tù❝ ❧➔✿ ❚➻♠ ♣❤➛♥ tû p∗ ∈ H s❛♦ ❝❤♦ ∈ T p∗ ✭✵✳✶✮ ❚→❝ ❣✐↔ ✤➣ ①➙② ❞ü♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ xk+1 = Jk xk + ek ❤♦➦❝ xk+1 = Jk (xk + ek ), k ≥ 1, ✭✵✳✷✮ tr♦♥❣ ✤â Jk = (I + rk T )−1 ❧➔ t♦→♥ tû ❣✐↔✐ ❝õ❛ T ✈ỵ✐ t❤❛♠ sè rk > 0✱ ek ❧➔ ✈➨❝ tì s❛✐ sè ✈➔ I ❧➔ →♥❤ ①↕ ✤ì♥ ✈à tr➯♥ H ✳ ➷♥❣ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✵✳✷✮ ❤ë✐ tư ②➳✉ tỵ✐∞ ♠ët ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ T ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ t➟♣ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ T ❦❤→❝ ré♥❣✱ ∥ek ∥ < ∞ ✈➔ rk ≥ ε > ✈ỵ✐ ♠å✐ k ≥ k=1 uăr r r ữỡ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❝❤➾ ✤↕t ✤÷đ❝ sü ❤ë✐ tư ②➳✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈æ ❤↕♥ ❝❤✐➲✉✳ ✣➸ t❤✉ ✤÷đ❝ sü ❤ë✐ tư ♠↕♥❤✱ ♠ët sè ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ✤➣ ✤÷đ❝ ✤÷❛ r❛ ♥❤÷✿ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❤✐➺✉ ❝❤➾♥❤ ❚✐❦❤♦♥♦✈ ❝õ❛ ▲❡❤❞✐❤✐ ✈➔ ▼♦✉❞❛❢✐ ✭✶✾✾✻✮ ✈➔ ✤÷đ❝ ♠ð rë♥❣ ❜ð✐ ❳✉ ✭✷✵✵✻✮✱ ❇♦✐❦❛♥②♦ ✈➔ ▼♦r♦s❛♥✉ ✭✷✵✶✷✮❀ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❝♦ ❝õ❛ ❑❛♠✐♠✉r❛ ✈➔ ❲✳❚❛❦❛❤❛s❤✐ ✭✷✵✵✵✮ ✈➔ ✤÷đ❝ tê♥❣ q✉→t ❜ð✐ ❨❛♦ ✈➔ ◆♦♦r ✭✷✵✵✽✮❀ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ♠➲♠ ❝õ❛ ❲✳❚❛❦❛❤❛s❤✐ ✭✷✵✵✼✮✳ ❚r♦♥❣ ❤➛✉ ❤➳t ❝→❝ ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❝ơ♥❣ ♥❤÷ ❜↔♥ t❤➙♥ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ t❤❛♠ sè rk ❝õ❛ t♦→♥ tû ❣✐↔✐ ữợ ởt số ợ ỡ ✵✳ ●➛♥ ✤➙②✱ ♥➠♠ ✷✵✶✼✱ ◆✳ ❇÷í♥❣✱ P✳❚✳❚✳ ❍♦➔✐ ✈➔ ◆✳❉✳ ◆❣✉②➵♥ ✤➣ tr➻♥❤ ❜➔② ♠ët sè ❝↔✐ ❜✐➯♥ ♠ỵ✐ ✷ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❝❤♦ tr÷í♥❣ ❤đ♣ rk ❞➛♥ tỵ✐ ✵✱ ❝ư t❤➸ rk t❤♦↔ ∞ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (A8) ❧➔ rk < +∞✳ ▼ët ❝➙✉ ❤ä✐ ✤÷đ❝ ✤➦t r❛ ✤➸ ♥❣❤✐➯♥ k=1 ❝ù✉ ❧➔ ❧✐➺✉ ❝â tỗ t ởt ữỡ ❦➲ ❤ë✐ tư ♠➔ sü ❤ë✐ tư ♠↕♥❤ t❤✉ ✤÷đ❝ ✈ỵ✐ ❞➣② {rk } ❧➔ ♠ët ❞➣② sè ❜➜t ❦ý tr♦♥❣ (0, ∞)❄ ❑❤✐ ♣❤✐➳♠ ❤➔♠ ❝ü❝ t✐➸✉ ❧➔ tê♥❣ ỗ t ❜➔✐ t♦→♥ t➻♠ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ tê♥❣ ❤❛✐ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ A, B, ✤â ❧➔ ❜➔✐ t♦→♥✿ ❚➻♠ ♣❤➛♥ tû p∗ ∈ H s❛♦ ❝❤♦ ∈ (A + B)p∗ ✭✵✳✸✮ ❇➔✐ t♦→♥ ✭✵✳✸✮ t❤✉ ❤ót ✤÷đ❝ sỹ ú ỵ ự õ ❧➔ ❝èt ❧ã✐ ❝õ❛ ♥❤✐➲✉ ❜➔✐ t♦→♥ ♥❤÷✿ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✱ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤✱ ❜➔✐ t♦→♥ ❝ü❝ t✐➸✉ ❤â❛ ✈ỵ✐ ❝→❝ ù♥❣ ❞ư♥❣ tr♦♥❣ ❤å❝ ♠→②✱ ỷ ỵ t ữủ t t ❜✐➳t r➡♥❣✱ ♥➳✉ tê♥❣ ❆✰❇ ❝ô♥❣ ❧➔ ♠ët t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✱ t❤➻ ❝â t❤➸ →♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✵✳✷✮ ✈ỵ✐ ❚❂❆✰❇ ✤➸ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ tê♥❣✳ ❚✉② ♥❤✐➯♥✱ ♥❤✐➲✉ ❦❤✐ ❚ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ❝❤♦ ❞ị ❆ ✈➔ ❇ ❧➔ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳ ❉♦ ✤â✱ ❝❤➾ ❝â t❤➸ ①➙② ❞ü♥❣ ♠ët ♣❤➨♣ ❧➦♣ ❞ü❛ ✈➔♦ t♦→♥ tû ❣✐↔✐ ❝õ❛ tø♥❣ t♦→♥ tû ❆ ✈➔ ❇✳ ✣✐➲✉ ♥➔② ❝ơ♥❣ ❧đ✐ t❤➳✱ ♥❣❛② ❝↔ ❦❤✐ ❚ ❧➔ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✱ ♥❤÷♥❣ ✈✐➺❝ t➼♥❤ ❣✐→ trà ❝õ❛ t♦→♥ tû ❣✐↔✐ ❝õ❛ ❚ ❦❤â ❤ì♥ ✈✐➺❝ t➼♥❤ ♥â ❝❤♦ tø♥❣ ❆ ✈➔ ❇✳ ❇ð✐ ✈➟②✱ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ ❝❤♦ ❣✐↔✐ ❜➔✐ t♦→♥ ✭✵✳✸✮ ❝❤➼♥❤ ❧➔ sû ❞ö♥❣ t♦→♥ tû ❣✐↔✐ JrA, JrB ❝õ❛ A ✈➔ B t❤❛② ❝❤♦ ❞ò♥❣ t♦→♥ tû ❣✐↔✐ JrA+B ❝õ❛ A + B ✳ P❤÷ì♥❣ ♣❤→♣ t→❝❤ ❝ê ✤✐➸♥ ❝õ❛ P❡❛❝❡♠❛♥✲❘❛❝❤❢♦r❞✱ ❉♦✉❣❧❛s✲❘❛❝❤❢♦r❞ ✤÷đ❝ ✤➲ ①✉➜t ✈➔♦ ♥❤ú♥❣ ♥➠♠ ✶✾✺✵ ❝❤♦ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❦❤✐ ❝↔ A ✈➔ B ✤➲✉ ❧➔ t♦→♥ tû t✉②➳♥ t➼♥❤ ✤ì♥ trà✳ ◆➠♠ ✶✾✼✾✱ ▲✐♦♥s rr rở sỡ ỗ t sr trữớ ủ ợ A B t tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ✤❛ trà✳ ▼ët ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ t❤ỉ♥❣ ❞ư♥❣ ❦❤→❝ ✤÷đ❝ ✤÷❛ r❛ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ✭✵✳✸✮ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ t✐➳♥ ❧ị✐✳ P❤÷ì♥❣ ♣❤→♣ ♥➔② ✤÷đ❝ ✤➲ ①✉➜t ❜ð✐ ▲✐♦♥s ✈➔ ▼❡r❝✐❡r✱ P❛sst② ✈➔♦ ợ xk ữủ xk+1 = Jk (I − rk A)xk , k ≥ 1, ✭✵✳✹✮ tr♦♥❣ ✤â A, B ❧➔ ❝→❝ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr➯♥ H ✱ Jk = (I + rk B)−1 ❧➔ t♦→♥ tû ❣✐↔✐ ❝õ❛ B ✱ {rk } ❧➔ ❞➣② sè ❞÷ì♥❣✳ ❚✉② ♥❤✐➯♥✱ ❞➣② ❧➦♣ xk ①→❝ ✸ ✤✐♥❤ ❜ð✐ ✭✵✳✹✮ ❝ơ♥❣ ❝❤➾ ❤ë✐ tư ②➳✉ tỵ✐ ♠ët ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ A + B ✳ ✣➸ t❤✉ ✤÷đ❝ sü ❤ë✐ tư ♠↕♥❤✱ ♠ët sè ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ t✐➳♥ ❧ị✐ ✤➣ ✤÷đ❝ ✤÷❛ r ợ ữủ ỹ t ủ ợ ❝õ❛ ▼❛♥♥❀ ❍❛❧♣❡r♥❀ ▼❛♥♥✲❍❛❧♣❡r♥✳ ❙ü ❤ë✐ tö ♠↕♥❤ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ t❤✉ ✤÷đ❝ ✤➲✉ ❝➛♥ ✤✐➲✉ ❦✐➺♥ t❤❛♠ sè rk t tỷ ữợ ♠ët ❤➡♥❣ sè ❧ỵ♥ ❤ì♥ ✵✳ ▼ët ✈➜♥ ✤➲ ❝ơ♥❣ ♥❤÷ tr➯♥ ♥↔② s✐♥❤ ❧➔ ❧✐➺✉ ❝â t❤➸ ①➙② ❞ü♥❣ ✤÷đ❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ ✭✵✳✸✮ ♠➔ sü ❤ë✐ tử t ữủ ợ rk tợ ✵✱ ❤♦➦❝ ✤✐➲✉ ❦✐➺♥ tê♥❣ q✉→t ❤ì♥ ❝❤♦ ❞➣② t❤❛♠ sè ❝õ❛ t♦→♥ tû ❣✐↔✐ ✤â ❧➔ {rk } ❧➔ ♠ët ❞➣② sè ❜➜t ❦ý tr♦♥❣ (0, α) ✈ỵ✐ α > ❦❤ỉ♥❣❄ ✣➸ tr↔ ❧í✐ ❝❤♦ ❝➙✉ ❤ä✐ ♥➔②✱ tr♦♥❣ ❈❤÷ì♥❣ ✸ ❝õ❛ ❧✉➟♥ →♥✱ ❝❤ó♥❣ tỉ✐ ✤➲ ①✉➜t ♣❤÷ì♥❣ ♣❤→♣ ❞↕♥❣ t→❝❤ t✐➳♥ ❧ị✐ ❝❤♦ ❜➔✐ t♦→♥ ✭✵✳✸✮ sỹ tử t ữủ ợ ♥❤÷ ✤➣ ♥➯✉ tr➯♥ ❝❤♦ ❞➣② t❤❛♠ sè {rk } ❝õ❛ t♦→♥ tû ❣✐↔✐✳ ❇➔✐ t♦→♥ t✐➳♣ t❤❡♦ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❧✉➟♥ →♥ ♥➔② ❧➔ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ♥❤✐➲✉ t➟♣ ✭▼❙❙❋P✮✿ ❚➻♠ x ∈ C := Ci, s❛♦ ❝❤♦ Ax ∈ Q := Qj , ✭✵✳✺✮ i∈J1 j∈J2 tr♦♥❣ ✤â {Ci}i∈J ✈➔ {Qj }j∈J t÷ì♥❣ ù♥❣ ❧➔ ổ ữủ t ỗ ✤â♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H1 ✈➔ H2✱ A : H1 → H2 ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥✳ ◆➠♠ ✶✾✾✹✱ ❈❡♥s♦r ✈➔ ❊❧❢✈✐♥❣ ✤÷❛ r❛ ❜➔✐ t♦→♥ ✭✵✳✺✮ tr♦♥❣ tr÷í♥❣ ❤đ♣ J1 ✈➔ J2 ❧➔ ❤ú✉ ❤↕♥✳ ❈→❝ t→❝ ❣✐↔ ✤➲ ①✉➜t ♣❤÷ì♥❣ ♣❤→♣ ❈◗ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ♥➔② tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉✳ ❇➔✐ t♦→♥ ✭✵✳✺✮ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ t❤ü❝ t➳✱ ❝❤➥♥❣ ❤↕♥ ♥❤÷ ✤➸ ♠ỉ ❤➻♥❤ ❤♦→ ❜➔✐ t♦→♥ ♥❣÷đ❝✱ ❜➔✐ t s tứ ỗ tr t→✐ t↕♦ ❤➻♥❤ ↔♥❤ ② t➳✳ ●➛♥ ✤➙②✱ ♥❣÷í✐ t❛ ❝á♥ ♣❤→t ❤✐➺♥ r❛ r➡♥❣ ▼❙❙❋P ❝ơ♥❣ ❝â t❤➸ ✤÷đ❝ →♣ ❞ư♥❣ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ✤✐➲✉ ❝❤➳ ❝÷í♥❣ ✤ë ①↕ trà✳ P❤÷ì♥❣ ♣❤→♣ ❈◗ ❝❤➾ t❤✉ ✤÷đ❝ sü ❤ë✐ tư ②➳✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈æ ❤↕♥ ❝❤✐➲✉✳ ✣➸ t❤✉ ✤÷đ❝ sü ❤ë✐ tư ♠↕♥❤✱ ♠ët sè ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❈◗ ✤÷đ❝ ✤÷❛ r❛ ❜ð✐ ❳✉ ✭✷✵✵✻✮✱ ❍❡ ❝ò♥❣ ❝ë♥❣ sü ✭✷✵✶✺✮✱ ❲❡♥ ❝ò♥❣ ❝ë♥❣ sü ✭✷✵✶✺✮✳ ◆➠♠ ✷✵✶✼✱ ◆✳ ❇÷í♥❣ ✤➣ ♠ð rë♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❈◗ ❝❤♦ tr÷í♥❣ ❤đ♣ J1 ✈➔ J2 ❧➔ ❝→❝ ❤å ✈ỉ ❤↕♥ ữủ õ t ộ ữợ t ❝❤➾ ❞ò♥❣ ❤ú✉ ❤↕♥ ❝→❝ t➟♣ ❝õ❛ ❝→❝ ❤å {Ci} ✈➔ {Qj }✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ✤ì♥ ❣✐↔♥ J1 = J2 = {1}✱ ❳✉ ✭✷✵✶✵✮ ✤➲ ①✉➜t ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❝❤♦ ❜➔✐ t♦→♥✳ ❉♦ ✤â✱ ♠ö❝ t✐➯✉ t❤ù ❜❛ tr♦♥❣ ❧✉➟♥ →♥ ♥➔② ❧➔ ①➙② ❞ü♥❣ ✹ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❝❤♦ ❣✐↔✐ ❜➔✐ t♦→♥ ✭✵✳✺✮ tr♦♥❣ tr÷í♥❣ ❤đ♣ J1 ✈➔ J2 ❧➔ ❝→❝ ❤å ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝✳ ▲✉➟♥ →♥ ♥➔② ♥❤➡♠ ❣✐↔✐ q✉②➳t ❜❛ ♠ư❝ t✐➯✉ ♥➯✉ tr➯♥ ✈➔ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❜è♥ ❝❤÷ì♥❣✳ ❚r♦♥❣ ❈❤÷ì♥❣ ✶✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à q✉❛♥ trå♥❣ ❝❤♦ ✈✐➺❝ tr➻♥❤ t q ữỡ s ỗ ởt số ỡ t ỗ ởt sè ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✱ tê♥❣ ❤❛✐ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✱ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ♥❤✐➲✉ t➟♣ ✈➔ ❝→❝ ữỡ r ữỡ ợ t ởt ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ✤➸ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ sü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❦❤ỉ♥❣ ❝➛♥ t❤➯♠ ✤✐➲✉ ❦✐➺♥ ♥➔♦ ❦❤→❝ ❧➯♥ t❤❛♠ sè ❝õ❛ t♦→♥ tû ❣✐↔✐ ❝õ❛ t♦→♥ tû ✤➣ ❝❤♦✳ ❚r♦♥❣ ❈❤÷ì♥❣ ✸✱ ❝❤ó♥❣ tỉ✐ ✤➣ ♥❤➟♥ ✤÷đ❝ ♠ët ❦➳t q✉↔ t÷ì♥❣ tü ❝❤♦ ❜➔✐ t♦→♥ ❜❛♦ ❤➔♠ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✤ì♥ ✤✐➺✉✳ ❚r♦♥❣ ❈❤÷ì♥❣ ✹✱ ✤÷đ❝ ❞➔♥❤ ✤➸ ✤➲ ①✉➜t ♠ët ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✈ỵ✐ ❤❛✐ ❤å ✈ỉ ❤↕♥ t õ ỗ q trồ ữỡ ộ ữợ ũ ỳ ❝→❝ t➟♣ ❝õ❛ ❤❛✐ ❤å tr➯♥✳ ✣➸ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❣✐↔✐ q✉②➳t ❝→❝ ♠ư❝ t✐➯✉ ✤➦t r❛✱ ❝❤ó♥❣ tỉ✐ ✤➣ sû ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✈➔ ❝ỉ♥❣ ❝ư ❤✐➺♥ ✤↕✐ t t ỗ ỵ tt tố ÷✉ ✈➔ ❝→❝ ❦➳t q✉↔ ✤➣ ❝â ✈➲ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♥❤ú♥❣ ❜➔✐ t♦→♥ ♥➯✉ tr➯♥✳ ❈❤÷ì♥❣ ✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ❜➔✐ t♦→♥ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❝ì ❜↔♥ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ❝➛♥ t❤✐➳t ♥❤➡♠ ♣❤ö❝ ✈ö ❝❤♦ ✈✐➺❝ tr➻♥❤ ❜➔② ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ →♥ ð ❝→❝ ữỡ s trú ữỡ ỗ ử ✶✳✶ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ t ỗ t tỷ ỡ ỡ ❝ü❝ ✤↕✐✱ tê♥❣ ❝õ❛ ❤❛✐ t♦→♥ tû ✤ì♥ ✤✐➺✉✳ ▼ư❝ ợ t tờ q ởt số ữỡ t ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ✈➔ tê♥❣ ❝õ❛ ❤❛✐ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳ ▼ư❝ ✶✳✸ tr➻♥❤ ❜➔② ✈➲ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ♥❤✐➲✉ t➟♣ ✈➔ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐✳ ▼ư❝ ✶✳✹ ❧➔ ♠ët sè ❜ê ✤➲ ✤÷đ❝ sû ❞ư♥❣ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ ✤↕t ✤÷đ❝ tr♦♥❣ ❝→❝ ❝❤÷ì♥❣ t✐➳♣ t❤❡♦ ❝õ❛ ❧✉➟♥ →♥✳ ❈❤÷ì♥❣ ✷ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❝❤♦ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❈❤÷ì♥❣ tr ữỡ ợ t❤❛♠ sè ❜➜t ❦ý tr♦♥❣ ✤â sü ❤ë✐ tö ♠↕♥❤ t ữủ ợ tờ qt t sè {rk } ❝õ❛ t♦→♥ tû ❣✐↔✐ ❧➔ ♠ët ❞➣② sè ❜➜t ❦ý tr♦♥❣ (0, ∞)✳ ❙❛✉ ✤â ✤÷❛ r❛ ✈➼ ❞ö sè ♠✐♥❤ ❤å❛ ❝❤♦ ❝→❝ ❦➳t q✉↔ ✤↕t ✤÷đ❝✳ ❈→❝ ❦➳t q✉↔ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ tr➻♥❤ ❜➔② ❞ü❛ ✈➔♦ ❝→❝ ❝æ♥❣ tr➻♥❤ [3] ✈➔ [5] tr♦♥❣ ❉❛♥❤ ♠ư❝ ❝→❝ ❝ỉ♥❣ tr➻♥❤ ✤➣ ❝ỉ♥❣ ❜è✳ ✷✳✶✳ P❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ✈ỵ✐ ❞➣② t❤❛♠ sè ❜➜t ❦ý ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ①➨t ❜➔✐ t♦→♥✿ ❚➻♠ ♣❤➛♥ tû p∗ ∈ H s❛♦ ❝❤♦ ∈ T p∗ ✭✷✳✶✮ ✈ỵ✐ H ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ T : H → 2H ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳ ▼ët tr♦♥❣ ♥❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝ê ✤✐➸♥ ✤➸ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲✳ ❚✉② ♥❤✐➯♥ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ❝❤➾ ✤↕t ✤÷đ❝ sü ❤ë✐ tư ②➳✉ ♠➔ ❦❤ỉ♥❣ ❤ë✐ tư ♠↕♥❤ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỉ ❤↕♥ ❝❤✐➲✉✳ ✣➸ t❤✉ ✤÷đ❝ sü ❤ë✐ tư ♠↕♥❤✱ ♠ët sè ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ✤➣ ✤÷đ❝ ✤÷❛ r❛✳ ❙ü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ❝→❝ ❝↔✐ ❜✐➯♥ ✤➲✉ ❞➝♥ ✤➳♥ t❤❛♠ sè rk ❝õ❛ t tỷ ữợ ởt sè ❧ỵ♥ ❤ì♥ ✵✳ ◆➠♠ ✷✵✶✼✱ ❝❤ó♥❣ tỉ✐ ✤➣ ✤➲ t ợ ữỡ ❦➲ ❝❤♦ ❜➔✐ t♦→♥ ✭✷✳✶✮ ❝â ❞↕♥❣ ❣✐è♥❣ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❤✐➺✉ ❝❤➾♥❤ ❚✐❦❤♦♥♦✈ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❝♦✱ ✤â ❧➔ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣✿ xk+1 = J k (tk u + (1 − tk )xk + ek ), k ≥ 1, z k+1 = tk u + (1 − tk )J k z k + ek , k ≥ 1, ✼ tr♦♥❣ ✤â J k = J1J2 · · · Jk ❧➔ ❤ñ♣ ❝õ❛ k t♦→♥ tû ❣✐↔✐ Ji = (I + riT )−1, i = 1, 2, , k ❙ü ❤ë✐ tö ♠↕♥❤ ❝õ❛ ữỡ ữủ ữ r ợ (A8) ❧➔ ∞ t❤❛♠ sè rk t❤♦↔ ♠➣♥ rk < +∞✱ tù❝ ❧➔ rk ❞➛♥ tỵ✐ ✵✳ ✣➙② ❧➔ ✤✐➲✉ ❦✐➺♥ k=1 ❤♦➔♥ t♦➔♥ ❦❤→❝ ✈ỵ✐ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝❤♦ t❤❛♠ sè rk ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✤➣ ♥➯✉✳ ❚✉② ♥❤✐➯♥✱ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ❦❤→ ♣❤ù❝ t↕♣ ✈➻ ð ♠é✐ ữợ ũ t tỷ t➼♥❤ t♦→♥ s➩ trð ♥➯♥ ❦❤â ❦❤➠♥ ❤ì♥ ✈➔ ✤✐➲✉ ❦✐➺♥ (A8) ❧➔ ❦❤→ ❤↕♥ ❝❤➳✳ ✣➸ ❦❤➢❝ ♣❤ö❝ ✤✐➲✉ ú tổ ợ t ữỡ ✈ỵ✐ ❞➣② t❤❛♠ sè ❜➜t ❦ý ❝❤♦ ❣✐↔✐ ❜➔✐ t♦→♥ ✭✷✳✶✮ ❜➡♥❣ ❝→❝❤ t❤❛② ❤ñ♣ J k = J1J2 · · · Jk ❜ð✐ ❞↕♥❣ ✤ì♥ ❣✐↔♥ ❤ì♥ ✤â ❧➔ ũ t tỷ t ộ ữợ ✈➔ t❤❛② ✤✐➲✉ ❦✐➺♥ (A8) ❜ð✐ ✤✐➲✉ ❦✐➺♥ (A8′ ) ✤â ❧➔ {rk } ❧➔ ❞➣② sè ❜➜t ❦ý tr♦♥❣ ❦❤♦↔♥❣ (0, ∞)✳ ❚r♦♥❣ ❝ỉ♥❣ tr➻♥❤ [3]✱ ❝❤ó♥❣ tỉ✐ ✤÷❛ r❛ ♠ët sè ❝→❝ ❞➣② ❧➦♣ xk ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐✿ xk+1 = Jk Jc (tk u + (1 − tk )xk ) + ek , ✭✷✳✷✮ xk+1 = tk u + (1 − tk )(Jk Jc xk + ek ), ✭✷✳✸✮ xk+1 = t′k u + βk′ Jc xk + γk′ Jk xk + ek , ✭✷✳✹✮ ð ✤â Jc = (I + cT )−1 ✈➔ c ❧➔ sè t❤ü❝ ❞÷ì♥❣ ❝è ✤à♥❤ ❜➜t ❦ý✳ ❈❤ó♥❣ tỉ✐ ❝❤➾ r❛ r➡♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✷✮✲✭✷✳✹✮ ❧➔ ❝→❝ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ z k+1 = Jk Jc (I − tk µF )z k + ek , ✭✷✳✺✮ z k+1 = (1 − βk )(I − tk µF )Jc z k + βk Jk z k + ek , ✭✷✳✻✮ ✤➸ t➻♠ ♥❣❤✐➺♠ p∗ ∈ C ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ p∗ ∈ C : ⟨F p∗ , p∗ − p⟩ ≤ 0, ∀p ∈ C ✭✷✳✼✮ ✈ỵ✐ C := ZerT ✱ F : H → H ❧➔ →♥❤ ①↕ η✲ ✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ tử L st ợ , L > số ố tở (0, 2/L2) ỵ ỵ ữợ t q ❤ë✐ tư ♠↕♥❤ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ tr➯♥✳ ′ ′ ỵ T t tỷ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H s❛♦ ❝❤♦ ZerT ̸= Ø✱ F ❧➔ →♥❤ ①↕ η ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ ❧✐➯♥ tư❝ L✲▲✐♣s❝❤✐t③ tr➯♥ H ✈ỵ✐ L số ữỡ sè ❝è ✤à♥❤ t❤✉ë❝ ✽ ❦❤♦↔♥❣ (0, 2η/L2 )✳ ●✐↔ sû tk ✱ rk ✈➔ ek t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥✿ ∥ek ∥ (A1) ∥e ∥ < ∞ ❤♦➦❝ (A1 ) lim = 0✱ k→∞ tk k=1 ∞ ′ k ∞ (A5) tk ∈ (0, 1), ∀k ≥ 1, lim tk = 0, k→∞ tk = ∞✱ k=1 (A8′ ) {rk } ❧➔ ❞➣② sè ❜➜t ❦➻ tr♦♥❣ ❦❤♦↔♥❣ (0, ∞)✳ ❑❤✐ ✤â✱ ❞➣② z k ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✺✮ ❤ë✐ tư ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t p∗ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✷✳✼✮ ❦❤✐ k → ∞✳ ✣à♥❤ ỵ sỷ H, F, A, tk , rk ek ữ tr ỵ tt t❤❛♠ sè βk t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥✿ (A6) βk ∈ [a, b] ⊂ (0, 1)✳ ❑❤✐ ✤â✱ ❞➣② {z k } ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✻✮ ❤ë✐ tư ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t p∗ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✷✳✼✮ ❦❤✐ k → ∞✳ ◆❤➟♥ ①➨t ✷✳✶ ◆❤➟♥ ①➨t ♥➔② tr➻♥❤ ❜➔② ❝→❝❤ ❝❤å♥ →♥❤ ①↕ F ✤➸ tø ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✺✮ t❛ t❤✉ ✤÷đ❝ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✷✮✱ ✭✷✳✸✮ ✈➔ tø ✭✷✳✻✮ t❤✉ ✤÷đ❝ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✹✮✳ ❉ü❛ tr➯♥ ❦➳t q✉↔ ❝õ❛ ❳✉ ✭✷✵✵✷✮✱ ❈❡♥❣ ❝ò♥❣ ❝ë♥❣ sü ✭✷✵✵✽✮✱ ❝❤ó♥❣ tỉ✐ t✐➳♣ tư❝ ✤➲ ①✉➜t ❝↔✐ ❜✐➯♥ ♠ỵ✐ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❣➛♥ ❦➲ ❝❤♦ ❜➔✐ t♦→♥ ✭✷✳✶✮✱ ð õ ộ ữợ ỗ ữợ t ỹ tử ữỡ ợ ụ ữủ ❝❤ù♥❣ ♠✐♥❤ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ tê♥❣ q✉→t ❝❤♦ ❞➣② t❤❛♠ sè {rk } ❝õ❛ t♦→♥ tû ❣✐↔✐ ✤â ❧➔ ✤✐➲✉ ❦✐➺♥ (A8′)✳ ❑➳t q✉↔ s❛✉ ✤÷đ❝ tr➻♥❤ ❜➔② ❞ü❛ ✈➔♦ ❝ỉ♥❣ tr➻♥❤ [5] tr♦♥❣ ❞❛♥❤ ♠ư❝ ❝→❝ ❝ỉ♥❣ tr➻♥❤ ✤➣ ổ ố ỵ H ổ ❍✐❧❜❡rt t❤ü❝ ✈➔ T ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr♦♥❣ H s❛♦ ❝❤♦ ZerT ̸= Ø✱ c ❧➔ sè t❤ü❝ ❞÷ì♥❣ ❝è ✤à♥❤ ❜➜t ❦ý✱ u ∈ H ✳ ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t❤ä❛ ♠➣♥ (A5)✱ (A8′ )✱ (A1) ❤♦➦❝ ∞ (A1′′ ) ∥ek+1 ∥ ≤ ηk ∥˜ xk+1 − xk ∥ ✈ỵ✐ k=1 ηk < ∞ ❑❤✐ ✤â✱ ❞➣② {xk } ①→❝ ✤à♥❤ ❜ð✐ x˜k+1 = J (xk + ek+1 ), k xk+1 = t u + (1 − t )J x˜k+1 , k k ✭✷✳✽✮ c ❤ë✐ tư ♠↕♥❤ tỵ✐ ✤✐➸♠ p∗ = PZerT u✱ ❧➔ ❤➻♥❤ ❝❤✐➳✉ ♠❡tr✐❝ ❝õ❛ u ❧➯♥ t➟♣ ZerT ❦❤✐ k → ∞✳ ✶✵ ❇↔♥❣ ✷✳✷✳ ❑➳t q t t ũ ữỡ ợ ek ≡ k xk+1 xk+1 k xk+1 xk+1 ✶✵ ✶✳✵✾✵✾✵✾✵✾✵✾ ✶✳✾✸✾✻✼✼✹✵✺✹ ✶✵✵ ✶✳✵✵✾✾✵✵✾✾✵✶ ✶✳✹✻✻✵✵✸✶✵✹✶ ✷✵ ✶✳✵✹✼✻✶✾✵✹✼✻ ✶✳✽✻✼✾✶✼✼✾✾✾ ✷✵✵ ✶✳✵✵✹✾✼✺✶✷✹✹ ✶✳✶✸✶✾✽✸✹✽✵✺ ✸✵ ✶✳✵✸✷✷✺✽✵✻✹✺ ✶✳✽✵✼✸✽✶✹✽✽✽ ✸✵✵ ✶✳✵✵✸✸✷✷✷✺✾✶ ✵✳✾✵✽✽✻✸✺✷✸✵ ✹✵ ✶✳✵✷✹✸✾✵✷✹✸✾ ✶✳✼✺✵✾✼✸✽✽✷✼ ✹✵✵ ✶✳✵✵✷✹✾✸✼✻✺✻ ✵✳✼✺✸✶✶✸✻✹✷✽ ✺✵ ✶✳✵✶✾✻✵✼✽✹✸✶ ✶✳✼✾✼✺✼✷✵✹✸✺ ✺✵✵ ✶✳✵✵✶✾✾✻✵✵✽✵ ✵✳✻✹✵✵✷✾✵✼✻✷ ❈→❝ ❦➳t q✉↔ t➼♥❤ t♦→♥ sè t❤✉ ✤÷đ❝ tr♦♥❣ ❝→❝ ❜↔♥❣ tr➯♥ ❝❤♦ t❤➜② ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✷✮ ❝õ❛ ú tổ s ữợ t ữủ ①➾ ❣➛♥ ♥❣❤✐➺♠ ✤ó♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✾✮ ❤ì♥ s♦ ợ ữỡ ụ ♥â✐ t❤➯♠ r➡♥❣ ♥❤ú♥❣ s♦ s→♥❤ ♥➔② ❝❤➾ ❞ü❛ tr➯♥ ❝→❝ ❦➳t q✉↔ t❤ü❝ ♥❣❤✐➺♠✳ ❈→❝ ❦➳t q✉↔ t➼♥❤ t♦→♥ ữủ tr r Ps ợ t➼♥❤ ■♥t❡❧✭❘✮ ❈♦r❡✭❚▼✮ ✐✺✲✺✷✵✵❯ ❈P❯ ❅ ✷✳✷✵●❍③✱ ✷✳✷✵●❍③✱ ✹✳✵✵●❇ ♦❢ ❘❆▼✳ ❈❤÷ì♥❣ ✸ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ tê♥❣ ❤❛✐ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❈❤÷ì♥❣ ✸✱ ❜➡♥❣ ❝→❝❤ ❦➳t ❤đ♣ ♣❤÷ì♥❣ ♣❤→♣ ữớ ố t ợ ữỡ t tũ ú tổ ✤➲ ①✉➜t ♣❤÷ì♥❣ ♣❤→♣ ❞↕♥❣ t→❝❤ t✐➳♥✲❧ị✐ ❝❤♦ ❜➔✐ t♦→♥ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ tê♥❣ ❤❛✐ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ❙ü ❤ë✐ tư ♠↕♥❤ ữỡ ữủ ữ r ợ ❞➣② t❤❛♠ sè ❝õ❛ t♦→♥ tû ❣✐↔✐ ❧➔ rk ❞➛♥ tỵ✐ ✵ ❤♦➦❝ ✤✐➲✉ ❦✐➺♥ tê♥❣ q✉→t ❤ì♥ ❧➔ {rk } ❧➔ ♠ët ❞➣② sè ❜➜t ❦ý tr♦♥❣ (0, α) ợ > số ữủ ữ r ✤➸ ♠✐♥❤ ❤å❛ ❝❤♦ ♣❤÷ì♥❣ ♣❤→♣ ✤➣ ✤➲ ①✉➜t✳ ❈→❝ ❦➳t q✉↔ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ ✈✐➳t tr➯♥ ❝ì sð ❝→❝ ❝ỉ♥❣ tr➻♥❤ [1] ✈➔ [4] tr♦♥❣ ❉❛♥❤ ♠ư❝ ❝→❝ ❝æ♥❣ tr➻♥❤ ✤➣ ❝æ♥❣ ❜è ❧✐➯♥ q✉❛♥ ✤➳♥ ❧✉➟♥ →♥✳ ✸✳✶✳ P❤÷ì♥❣ ♣❤→♣ ❞↕♥❣ t→❝❤ t✐➳♥ ❧ị✐ ◆➠♠ ✷✵✶✷✱ ❲✳❚❛❦❛❤❛s❤✐✱ ❲♦♥❣ ✈➔ ❨❛♦ ✤➣ ✤÷❛ r❛ ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ t✐➳♥ ❧ò✐ ❝❤♦ ❜➔✐ t♦→♥✿ ❚➻♠ ♣❤➛♥ tû p∗ ∈ H s❛♦ ❝❤♦ ∈ (A + B)p∗, ợ ữủ ỹ r õ ❞↕♥❣ xk+1 = tk u + (1 − tk )Jk (I − rk A)xk , ✭✸✳✷✮ tr♦♥❣ ✤â u ∈ H, {tk } ⊂ (0, 1)✱ Jk = (I + rk B)−1✱ {rk } ⊂ (0, ∞)✳ ❈→❝ t→❝ ❣✐↔ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ❞➣② xk ❤ë✐ tư ♠↕♥❤ tỵ✐ ♣❤➛♥ tû PΩu✱ ❧➔ ❤➻♥❤ ❝❤✐➳✉ ♠➯tr✐❝ ❝õ❛ u ❧➯♥ t➟♣ Ω ❧➔ t➟♣ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ A + B ♥➳✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t❤ä❛ ♠➣♥✿ ∞ (B5) tk ∈ (0, 1), ∀k ≥ 1✱ lim tk = 0✱ tk = ∞ ✱ k→∞ k=0 ✶✷ ∞ (B6) k=0 |tk+1 − tk | < ∞✱ (B7) < ε ≤ rk ≤ 2α✱ ∞ k=1 |rk+1 − rk | < ∞✳ ◆❤➡♠ ❣✐↔♠ ♥❤➭ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤➦t ❧➯♥ ❝→❝ ❞➣② t❤❛♠ sè tr♦♥❣ ❦➳t q✉↔ tr➯♥✱ ♥➠♠ ✷✵✶✽✱ tr♦♥❣ ❝ỉ♥❣ tr➻♥❤ [1]✱ ❜➡♥❣ ❝→❝❤ ❦➳t ❤đ♣ ♣❤÷ì♥❣ ♣❤→♣ ✤÷í♥❣ ố t ợ ữỡ t tũ ú tổ ①✉➜t ♣❤÷ì♥❣ ♣❤→♣ ❞↕♥❣ t→❝❤ t✐➳♥✲❧ị✐ ❝❤♦ ❜➔✐ t♦→♥ ✭✸✳✶✮✱ ✤â ❧➔ ❝→❝ ❞➣② ❧➦♣ xk = (I − tk F )T k xk , ✭✸✳✸✮ z ∈ H, z k+1 = T k (I − tk F )z k + ek , ✭✸✳✹✮ ✤➸ t➻♠ ♥❣❤✐➺♠ p∗ ∈ Ω t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ⟨F p∗ , p∗ − p⟩ ≤ ∀p ∈ Ω, ✭✸✳✺✮ tr♦♥❣ ✤â T k = T1T2 · · · Tk ✈ỵ✐ Ti = Ji(I − riA)✱ ≤ i ≤ k✱ F : H → H ❧➔ →♥❤ ①↕ η✲ ✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ γ˜✲❣✐↔ ❝♦ ❝❤➦t ✈ỵ✐ η + γ˜ > 1✳ ❙❛✉ ✤â✱ tø ✭✸✳✹✮✱ ❜➡♥❣ ❝→❝❤ ❝❤å♥ →♥❤ ①↕ F t❤➼❝❤ ❤đ♣ ❝❤ó♥❣ tỉ✐ t❤✉ ✤÷đ❝ ♠ët sè ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ t✐➳♥✲❧ị✐ tr♦♥❣ ✤â ❝â ♣❤÷ì♥❣ ♣❤→♣ ✤÷đ❝ ✤➲ ①✉➜t ❜ð✐ ❲✳❚❛❦❛❤❛s❤✐✱ ❲♦♥❣ ✈➔ ❨❛♦✳ ❙♦ ✈ỵ✐ ❦➳t q✉↔ ❝õ❛ ❲✳❚❛❦❛❤❛s❤✐✱ ❲♦♥❣ ✈➔ ❨❛♦ sü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❝❤ó♥❣ tỉ✐ t❤✉ ✤÷đ❝ ✤➣ ❧♦↕✐ ❜ä ✤✐➲✉ ❦✐➺♥ (B6) ✈➔ t❤❛② ✤✐➲✉ ❦✐➺♥ (B7) ❜ð✐ ✤✐➲✉ ❦✐➺♥ ♠ỵ✐ ❝❤♦ ❞➣② t❤❛♠ sè {rk } ❝õ❛ t♦→♥ tû ❣✐↔✐ ✤â ❧➔ ✤✐➲✉ ❦✐➺♥ ∞ (B9′ ) rk ∈ (0, α) ✈ỵ✐ ∀k ≥ ✈➔ rk < +∞✱ k=1 ❉➵ t❤➜②✱ ♥➳✉ ✤✐➲✉ ❦✐➺♥ (B9′) t❤ä❛ ♠➣♥ t❤➻ rk → ❦❤✐ k → ∞✳ ✣➙② ❧➔ ✤✐➲✉ ❦✐➺♥ ❝❤♦ t❤❛♠ sè rk ❝õ❛ t♦→♥ tû ❣✐↔✐ ❤♦➔♥ t♦➔♥ ❦❤→❝ ✈ỵ✐ ❝→❝ ✤✐➲✉ ❦✐➺♥ tr♦♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✤➣ ♥➯✉✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✤↕t ✤÷đ❝✱ ❝❤ó♥❣ tỉ✐ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❝→❝ ❦➳t q✉↔ ❝➛♥ t❤✐➳t s❛✉✳ ▼➺♥❤ ✤➲ ✸✳✶ ❈❤♦ H ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✱ F : H → H ❧➔ →♥❤ ①↕ η ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ γ ✲❣✐↔ ❝♦ ❝❤➦t ✈ỵ✐ η + γ > ✈➔ T ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr➯♥ H s❛♦ ❝❤♦ F ix(T ) ̸= Ø✳ ❑❤✐ ✤â✱ ✈ỵ✐ ❞➣② ❜à ❝❤➦♥ {xk } ❜➜t ❦ý tr♦♥❣ H s❛♦ ❝❤♦ limk→∞ ∥xk − T xk ∥ = 0✱ t❛ ❝â lim sup ⟨F p∗ , p∗ − xk ⟩ ≤ 0, k→∞ p∗ ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✸✳✺✮✳ ✭✸✳✻✮ ✶✸ ❇ê ✤➲ ✸✳✶ ❈❤♦ H ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✱ C t ỗ õ H A ❧➔ →♥❤ ①↕ α✲♥❣÷đ❝ ✤ì♥ ✤✐➺✉ ♠↕♥❤ tø C ✈➔♦ H ✱ α > 0✱ B ❧➔ →♥❤ ①↕ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr♦♥❣ H s❛♦ ❝❤♦ D(B) ⊆ C ✳ ●✐↔ sû Ω := Zer(A+B) ̸= ∅✱ rk ∈ (0, α) ✈➔ →♥❤ ①↕ T k ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ T k = T1 T2 · · · Tk ✈ỵ✐ Ti = Ji (I − ri A)✱ ≤ i ≤ k ✳ ❑❤✐ ✤â✱ F ix(T k ) = Ω ❇ê ✤➲ ✸✳✷ ❈❤♦ H, C, A, B, Ω ✈➔ Ti ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✸✳✶✳ ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t❤ä❛ ♠➣♥✿ (B9′ ) ✈➔ (B12) ∥Ax∥ ✈➔ |Bx| ≤ φ(∥x∥), ð ✤â |Bx| = inf{∥y∥ : y ∈ Bx} ✈➔ φ(t) ❧➔ ❤➔♠ ❦❤æ♥❣ ➙♠ ✈➔ ❦❤ỉ♥❣ ❣✐↔♠ ✈ỵ✐ ∀t ≥ 0✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ ✤✐➸♠ ❝è ✤à♥❤ x ∈ C ✈➔ i < k t limk Tik x tỗ t Tik = Ti · · · Tk ✳ ❇ê ✤➲ ✸✳✸ ❈❤♦ H, C, A, B ✈➔ Ω ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✸✳✶✳ ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ (B9′ ) ✈➔ (B12) t❤ä❛ ♠➣♥✳ ❑❤✐ ✤â✱ F ix(T ∞ ) = ỵ H, A, B, rk ữ tr ợ D(A) = H ✱ F ❧➔ →♥❤ ①↕ η ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ γ˜ ✲❣✐↔ ❝♦ ❝❤➦t tr➯♥ H s❛♦ ❝❤♦ η + γ˜ > 1✳ ❑❤✐ ✤â✱ ❞➣② {xk } ①→❝ ✤à♥❤ ❜ð✐ ✭✸✳✸✮✱ ✈ỵ✐ tk ∈ (0, 1) ✈➔ tk → 0✱ ❤ë✐ tư ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t p∗ ∈ Ω t❤♦↔ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✸✳✺✮ ❦❤✐ k → ∞✳ ▼➺♥❤ ✤➲ ✸✳✷ ❈❤♦ F, H, A, B, , rk tk ữ tr ỵ ✸✳✶✳ ❑❤✐ ✤â✱ ✈ỵ✐ ❜➜t ❦ý ❞➣② ❜à ❝❤➦♥ {xk } ⊂ H ✱ t❤ä❛ ♠➣♥ limk→∞ ∥T m xk − xk ∥ = ✈ỵ✐ sè ♥❣✉②➯♥ ❝è ✤à♥❤ t ý m t õ ỵ ✸✳✷ ❈❤♦ H, A, B✱ Ω ✈➔ F ♥❤÷ tr♦♥❣ ỵ sỷ s tọ ♠➣♥✿ (B5)✱ (B9′ )✱ (B12)✱ (B3) ❤♦➦❝ ∥ek ∥ = 0✳ k→∞ tk ❑❤✐ ✤â✱ ❞➣② {z k } ①→❝ ✤à♥❤ ❜ð✐ ✭✸✳✹✮ ❤ë✐ tư ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t (B3′ ) lim p∗ ∈ Ω t❤♦↔ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✸✳✺✮ ❦❤✐ k → ∞✳ ◆❤➟♥ ①➨t ✸✳✶ ◆❤➟♥ ①➨t ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝❤ ❝❤å♥ →♥❤ ①↕ F t❤➼❝❤ ❤đ♣ ✤➸ tø ♣❤÷ì♥❣ ♣❤→♣ ✭✸✳✹✮ t❤✉ ✤÷đ❝ ♠ët sè ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ t✐➳♥✲❧ị✐ tr♦♥❣ ✤â ❝â ♣❤÷ì♥❣ ♣❤→♣ ✤÷đ❝ ✤÷❛ r❛ ❜ð✐ ❲✳❚❛❦❛❤❛s❤✐✱ ❲♦♥❣ ✈➔ ❨❛♦✱ ✤â ❧➔✿ ✶✹ ✭✸✳✼✮ y k+1 = tk u + (1 − tk )T k y k ✭✸✳✽✮ ❉➵ t❤➜②✱ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✭✸✳✼✮ ✈➔ ✭✸✳✽✮ ❧➔ ❦❤→ ♣❤ù❝ t↕♣ ❦❤✐ k ✤õ ❧ỵ♥✱ ❜ð✐ ✈➻ sè ❝→❝ t♦→♥ tû t✐➳♥✲❧ò✐ t➠♥❣ ❧➯♥ q✉❛ ♠é✐ ữợ ỡ ỳ tự ố ợ ❞➣② t❤❛♠ sè ❝õ❛ t♦→♥ tû ❣✐↔✐ {rk } tr♦♥❣ (B9′) ✈➔ ✤✐➲✉ ❦✐➺♥ (B12) ❧➔ ❦❤→ ❤↕♥ ❝❤➳✳ ✣➸ ❦❤➢❝ ♣❤ư❝ ✤✐➲✉ ✤â✱ tr♦♥❣ ❝ỉ♥❣ tr➻♥❤ [4]✱ ❝❤ó♥❣ tỉ✐ t tử ợ t ởt số ữỡ t tũ tr ộ ữợ ự ❤❛✐ t♦→♥ tû t✐➳♥✲❧ị✐✳ ❙ü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ♣❤÷ì♥❣ t ữủ ố ợ t❤❛♠ sè rk ✈➔ ❧♦↕✐ ❜ä ✤✐➲✉ ❦✐➺♥ (B12)✳ ❈❤ó♥❣ tỉ✐ t✐➳♣ tư❝ ✤➲ ①✉➜t ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ s❛✉✿ xk+1 = Tk Tc (t′k u + (1 − t′k )xk + ek ), ✭✸✳✾✮ xk+1 = t′k u + (1 − t′k )Tk Tc xk + ek , ✭✸✳✶✵✮ xk+1 = t′k u + βk′ Tc xk + γk′ Tk xk + ek , ✭✸✳✶✶✮ xk+1 = t′k f (Tc xk ) + βk′ Tc xk + γk′ Tk xk + ek , ✭✸✳✶✷✮ tr♦♥❣ ✤â Tk = Jk (I − rk A)✱ Tc = (I + cB)−1(I − cA)✱ rk > 0✱ c ❧➔ sè t❤ü❝ ❞÷ì♥❣ ❜➜t ❦ý ✤õ ♥❤ä t❤ä❛ ♠➣♥ < c < α✳ ❈❤ó♥❣ tỉ✐ ❝❤➾ r❛ r➡♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✭✸✳✾✮✱ ✭✸✳✶✵✮ ✈➔ ✭✸✳✶✶✮✱ ✭✸✳✶✷✮ t÷ì♥❣ ù♥❣ ❧➔ ❝→❝ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ xk+1 = Tk Tc (I − tk F )xk + ek , ✭✸✳✶✸✮ xk+1 = βk (I − tk F )Tc xk + (1 − βk )Tk xk + ek , ✭✸✳✶✹✮ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ t➻♠ ♠ët ✤✐➸♠ p∗ ∈ Ω s❛♦ ❝❤♦ t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✸✳✺✮ ✈ỵ✐ F : H → H ❧➔ →♥❤ ①↕ η✲ ✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ γ˜✲ ❣✐↔ ❝♦ ❝❤➦t✱ η + γ˜ > 1✳ xk+1 = T k (tk u + (1 − tk )xk ), ỵ H ổ rt tỹ > 0✱ A ❧➔ →♥❤ ①↕ α✲♥❣÷đ❝ ✤ì♥ ✤✐➺✉ ♠↕♥❤ tø H ✈➔♦ H ✈ỵ✐ D(A) = H ✱ B ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr➯♥ H ✈➔ →♥❤ ①↕ F : H → H ❧➔ η ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ γ˜ ✲❣✐↔ ❝♦ ❝❤➦t s❛♦ ❝❤♦ η + γ˜ > 1✳ ●✐↔ sû Ω ̸= Ø ✈➔ ❞➣② ❧➦♣ z k ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝ỉ♥❣ t❤ù❝✿ z k+1 = Tk Tc (I − tk F )z k , ✭✸✳✶✺✮ ✶✺ ✈ỵ✐ Tk = Jk (I − rk A)✱ Tc = (I + cB)−1 (I − cA)✱ rk > 0✱ c > 0✳ ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t❤ä❛ ♠➣♥ (B5) ✈➔ (B9′′ ) c, rk ∈ (0, α) ✈ỵ✐ ♠å✐ k ≥ 1✳ ❑❤✐ ✤â✱ ❞➣② zk ❤ë✐ tư ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t p∗ ∈ Ω t❤♦↔ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✸✳✺✮ k ỵ H, A, B, F ữ tr ỵ sû ❞➣② {xk } ①→❝ ✤à♥❤ ❜ð✐ ✭✸✳✶✹✮ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ (B3) ❤♦➦❝ (B3′ )✱ (B5)✱ (B8)✱ (B9′′ )✳ ❑❤✐ ✤â✱ ❞➣② xk ❤ë✐ tư ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t p∗ ∈ Ω t❤♦↔ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✸✳✺✮ ❦❤✐ k → ∞✳ ✸✳✷✳ ❱➼ ❞ö sè ♠✐♥❤ ❤å❛ ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ✈➼ ❞ư sè ♠✐♥❤ ❤å❛ ❝❤♦ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✭✸✳✽✮ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❝õ❛ ❲✳❚❛❦❛❤❛s❤✐✱ ❲♦♥❣ ✈➔ ❨❛♦✳ ❈❤ó♥❣ tỉ✐ →♣ ❞ö♥❣ ❝→❝ t❤✉➟t t♦→♥ tr➯♥ ❝❤♦ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✿ t➻♠ ♠ët ✤✐➸♠ p∗ ∈ C s❛♦ ❝❤♦ ⟨Ap∗ , p∗ − p⟩ ≤ 0, ∀p C, õ C t ỗ ✤â♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ A ❧➔ →♥❤ ①↕ α✲♥❣÷đ❝ ✤ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥ H ✳ ❚❛ ❜✐➳t r➡♥❣✱ p∗ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✸✳✶✻✮ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥â ❧➔ ♠ët ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ ❜❛♦ ❤➔♠ t❤ù❝ ∈ (A + B)x ✈ỵ✐ B ❧➔ ♥â♥ ♣❤→♣ t✉②➳♥ ❝õ❛ C ✱ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ NC x = {w ∈ H : ⟨w, v − x⟩ ≤ 0, ∀v ∈ C} ●✐↔ sû χC ❧➔ ❤➔♠ ❝❤➾ ❝õ❛ C ✱ tù❝ ❧➔ 0, x ∈ C, χC = +∞, x ∈ / C ❑❤✐ ✤â✱ χC ❧➔ ❤➔♠ ỗ tữớ ỷ tử ữợ H (, ] ữợ C ①↕ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ✈➔ ∂χC = NC ✱ C JrBk = Jr∂χ = PC ✳ k ✣➸ t➼♥❤ t♦→♥✱ ❝❤ó♥❣ tỉ✐ ①➨t ✈ỵ✐ n n (xj − aj )2 ≤ r2 }, C = {x ∈ E : j=1 ✭✸✳✶✼✮ ✶✻ ð ✤â aj , r ∈ (−∞; +) ợ j n t trữớ ❤đ♣ ✤ì♥ ❣✐↔♥ ✈ỵ✐ n = 2, a1 = a2 = 2, r = ✈➔ Ax = φ′(x) ð ✤â φ(x) = (x1 − 1.5)2/2✳ ▲➜② u = (2.0; 1.5) ∈ C t❛ ♥❤➟♥ ✤÷đ❝ p∗ = PΩu = (1.5; 1.5) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✸✳✶✻✮✲✭✸✳✶✼✮ ð ✤â Ω = {(1.5; (−∞, ∞))} ∩ C ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✤➣ ♥➯✉✳ ❑➳t q✉↔ t➼♥❤ t♦→♥ ❦❤✐ ❞ị♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✸✳✽✮ ✈ỵ✐ ✤✐➸♠ ❜➢t ✤➛✉ x1 = (2.7; 2.7) ∈ C ✱ tk = 1/(k + 1)✱ rk = 1/(k(k + 1)) ữủ ữợ ❑➳t q✉↔ t➼♥❤ t♦→♥ ❦❤✐ →♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✸✳✽✮✳ k y1k y2k k y1k y2k ✶✵ ✶✳✺✻✼✺✵✺✼✶✾✶ ✶✳✺✹✹✶✵✷✸✹✼✸ ✷✵✵ ✶✳✺✵✸✺✹✹✸✸✸✺ ✶✳✺✵✷✶✶✸✺✹✷✻ ✷✵ ✶✳✺✸✹✺✽✶✾✻✷✻ ✶✳✺✷✶✺✹✹✽✼✼✺ ✸✵✵ ✶✳✺✵✷✸✻✺✶✹✽✽ ✶✳✺✵✶✹✵✽✵✸✽✹ ✸✵ ✶✳✺✷✸✷✺✾✷✸✽✽ ✶✳✺✶✹✷✻✵✹✶✸✶ ✹✵✵ ✶✳✺✵✶✼✼✹✼✶✶✻ ✶✳✺✵✶✵✺✺✻✺✼✼ ✹✵ ✶✳✺✶✼✺✷✸✽✸✼✵ ✶✳✺✶✵✻✺✼✹✸✵✶ ✺✵✵ ✶✳✺✵✶✹✷✵✶✼✽✵ ✶✳✺✵✵✽✹✹✸✹✽✶ ✺✵ ✶✳✺✶✹✵✺✼✽✻✽✸ ✶✳✺✵✽✺✵✼✽✾✸✹ ✶✵✵✵ ✶✳✺✵✵✼✶✵✹✾✽✺ ✶✳✺✵✵✹✷✶✾✾✻✵ ✶✵✵ ✶✳✺✵✼✵✻✽✺✵✵✶ ✶✳✺✵✹✷✸✻✵✶✵✶ ✷✵✵✵ ✶✳✺✵✵✸✺✺✸✺✶✽ ✶✳✺✵✵✷✶✵✾✺✸✺ ❈ị♥❣ t❤í✐ ❣✐❛♥ ✤â✱ ♥➳✉ ❞ị♥❣ t❤✉➟t t♦→♥ ✭✸✳✷✮ ✈ỵ✐ rk = 0.2 + 1/(k(k + 1)) ∈ (0.2, 1) ✈➔ ❝→❝ t❤❛♠ sè ❦❤→❝ ❦❤ỉ♥❣ ✤ê✐✱ ❝❤ó♥❣ tỉ✐ t❤✉ ✤÷đ❝ ❜↔♥❣ sè s❛✉✳ ❇↔♥❣ ✸✳✷✳ ❑➳t q✉↔ t➼♥❤ t♦→♥ ❦❤✐ →♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✸✳✷✮✳ ❦ xk1 xk2 ❦ xk1 xk2 ✶✵ ✶✳✻✾✷✽✹✻✶✻✵✻ ✶✳✻✵✻✸✺✼✷✶✹✻ ✷✵✵ ✶✳✺✶✷✹✸✻✶✾✼✾ ✶✳✺✶✷✹✸✻✶✽✶✸ ✷✵ ✶✳✻✶✺✹✵✼✾✽✺✻ ✶✳✻✶✹✽✸✾✻✷✷✻ ✸✵✵ ✶✳✺✵✽✸✵✺✶✼✺✶ ✶✳✺✵✽✸✵✺✶✼✶✾ ✸✵ ✶✳✺✼✾✾✵✺✷✷✸✺ ✶✳✺✼✾✽✷✼✺✸✹✶ ✹✵✵ ✶✳✺✵✻✷✸✹✷✶✺✻ ✶✳✺✵✻✷✸✹✹✷✶✹✻ ✹✵ ✶✳✺✻✵✼✷✷✷✾✹✾ ✶✳✺✻✵✼✵✺✷✸✺✹ ✺✵✵ ✶✳✺✵✹✾✽✾✾✶✽✼ ✶✳✺✵✹✾✽✾✾✶✽✸ ✺✵ ✶✳✺✹✽✾✵✵✾✾✺✶ ✶✳✺✹✽✽✾✺✸✶✵✶ ✶✵✵✵ ✶✳✺✵✷✹✾✼✹✽✾✾ ✶✳✺✵✷✹✾✼✹✽✾✾ ✶✵✵ ✶✳✺✷✹✼✸✾✵✾✷✻ ✶✳✺✷✹✼✸✽✽✵✽✻ ✷✵✵✵ ✶✳✺✵✷✹✾✼✹✽✽✾ ✶✳✺✵✷✹✾✼✹✽✽✾ ❈→❝ ❦➳t q✉↔ t➼♥❤ t♦→♥ ✤÷đ❝ ❝❤↕② tr➯♥ ♣❤➛♥ ♠➲♠ ❋r❡❡ P❛s❝❛❧ ■❉❊ ✈ỵ✐ ♠→② t➼♥❤ ■♥t❡❧✭❘✮ ❈♦r❡✭❚▼✮ ✐✺✲✺✷✵✵❯ ❈P❯ ❅ ✷✳✷✵●❍③✱ ✷✳✷✵●❍③✱ ✹✳✵✵●❇ ♦❢ ❘❆▼✳ ❈❤÷ì♥❣ ✹ P❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❝❤♦ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ♥❤✐➲✉ t➟♣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❈❤÷ì♥❣ ✹ ❝õ❛ ❧✉➟♥ →♥ ✤÷đ❝ ❞➔♥❤ ✤➸ tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❝❤♦ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ♥❤✐➲✉ t➟♣ tr♦♥❣ tr÷í♥❣ ❤ñ♣ ❤❛✐ t➟♣ ❝❤➾ sè J1, J2 ❧➔ ❝→❝ ❤å ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝✳ ❙❛✉ ✤â✱ ♥❤÷ ♥❤ú♥❣ ❤➺ q✉↔✱ ú tổ ữủ ởt số t q ợ ❝→❝ tr÷í♥❣ ❤đ♣ ♠ët tr♦♥❣ ❝→❝ t➟♣ J1, J2 ❤♦➦❝ ❝↔ ❤❛✐ ✤➲✉ ❤ú✉ ❤↕♥✳ ❚✐➳♣ t❤❡♦ ❧➔ ✈➼ ❞ö ✈➔ ❝→❝ ❦➳t q✉↔ t➼♥❤ t♦→♥ sè ♠✐♥❤ ❤å❛ ❝❤♦ ♣❤÷ì♥❣ ♣❤→♣ tr➯♥✳ ❑➳t q✉↔ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ tr➻♥❤ ❜➔② ❞ü❛ ✈➔♦ ❝ỉ♥❣ tr➻♥❤ [2] tr♦♥❣ ❞❛♥❤ ♠ư❝ ❝→❝ ❝ỉ♥❣ tr➻♥❤ ✤➣ ❝ỉ♥❣ ❜è✳ ✹✳✶✳ P❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ✈➔ ♥❣❤✐➺♠ ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t ❳➨t ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✭❙❋P✮ x ∈ C s❛♦ ❝❤♦ Ax ∈ Q, tữỡ ữỡ ợ t ỹ t ❉ü❛ ✈➔♦ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❚✐❦❤♦♥♦✈✱ ♥➠♠ ✷✵✶✵✱ ❳✉ ①➨t ❜➔✐ t♦→♥ ❤✐➺✉ ❝❤➾♥❤ ✭✹✳✸✮ fα (x) := ||Ax − PQ Ax||2 + α||x||2 , x∈C ð ✤â α > ❧➔ t❤❛♠ sè ❤✐➺✉ ❝❤➾♥❤✳ ❚→❝ ❣✐↔ ❝❤➾ r❛ r➡♥❣✱ ❜➔✐ t♦→♥ ✭✹✳✸✮ ❝â ♥❣❤✐➺♠ t ỵ x P õ t ợ lim x tỗ t ✈➔ ❧➔ ♥❣❤✐➺♠ ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t ❝õ❛ ❙❋P✳ ❈ô♥❣ tr♦♥❣ ❜➔✐ ❜→♦ ♥➔②✱ t→❝ ❣✐↔ ✤➣ ✤➲ ①✉➜t ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❝õ❛ ❇❛❦✉s❤✐♥s❦② ✭✶✾✼✼✮ ✈➔ ❇r✉❝❦ ✭✶✾✼✹✮ ❝â ❞↕♥❣ xk+1 = PC [I − γk (A∗ (I − PQ )A + αk I)]xk , k ≥ 1, ✭✹✳✹✮ f (x) := ||Ax − PQ Ax||2 x∈C ✶✽ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❞➣② {xk } ❤ë✐ tư ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t ❝õ❛ ❙❋P ✭✹✳✶✮ ♥➳✉ ❝→❝ ❞➣② t❤❛♠ sè{αk }, {γk } t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ αk , ∀k ✤õ ❧ỵ♥ (C4) < γk ≤ ||A||2 + αk (C5) αk → ✈➔ γk → 0, ∞ αk γk = ∞, (C6) k=1 (C7) |γk+1 − γk | + γk |αk+1 − αk | → (αk+1 γk+1 )2 ◆➠♠ ✷✵✶✷✱ ❨❛♦ ❝ị♥❣ ❝→❝ ❝ë♥❣ sü ❝ơ♥❣ ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tư ♠↕♥❤ ❝❤♦ t❤✉➟t t♦→♥ ✭✹✳✹✮ tỵ✐ ♥❣❤✐➺♠ ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t ❝õ❛ ❙❋P ✭✹✳✶✮ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✤➦t ❧➯♥ ❝→❝ ❞➣② t❤❛♠ sè {αk }, {γk } ②➳✉ ❤ì♥✳ ❈❤✉❛♥❣ ✭✷✵✶✸✮ ❝á♥ ❧➔♠ ②➳✉ ❤ì♥ ♥ú❛ ✈ỵ✐ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤➦t ❧➯♥ ❝→❝ ❞➣② t❤❛♠ sè {αk }, {γk } ♠➔ ✈➝♥ t❤✉ ✤÷đ❝ sü ❤ë✐ tư ♠↕♥❤ ❝❤♦ t❤✉➟t t♦→♥ ✭✹✳✹✮✳ ✹✳✷✳ P❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❝❤♦ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ♥❤✐➲✉ t➟♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❳➨t ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ♥❤✐➲✉ t➟♣ ✭▼❙❙❋P✮✿ ❚➻♠ x ∈ C := Ci s❛♦ ❝❤♦ Ax ∈ Q := i∈J1 ✭✹✳✺✮ Qj , j∈J2 ð ✤â {Ci}i∈J ✈➔ {Qj }j∈J t÷ì♥❣ ù♥❣ ❧➔ ❤❛✐ ❤å ❝→❝ t➟♣ ỗ õ tr ổ rt tỹ H1 H2✱ A : H1 → H2 ❧➔ →♥❤ ①↕ t✉②➳♥ t ỵ t ▼❙❙❋P ✭✹✳✺✮✳ ❉ü❛ tr➯♥ ❦➳t q✉↔ ❝õ❛ ❳✉ ✭✷✵✶✵✮ ✈➔ ❨❛♦ ❝ị♥❣ ❝→❝ ❝ë♥❣ sü ✭✷✵✶✷✮ ❝❤ó♥❣ tỉ✐ ♠ð rë♥❣ tt t P ợ trữớ ủ J1 ✈➔ J2 ❧➔ ❝→❝ ❤å ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝✱ tù❝ ❧➔ J1 = J2 = N+ ❧➔ t➟♣ t➜t ❝↔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ❚❤✉➟t t♦→♥ ❝õ❛ ❝❤ó♥❣ tỉ✐ ✤÷đ❝ ①➙② ❞ü♥❣ ♥❤÷ s❛✉✿ xk+1 = Uk Tγ ,α xk , x1 ∈ H1 , ✭✹✳✻✮ ð ✤â k Uk = β˜k k βi PCi , Tγk ,αk i=1 k = I − γk (A (I − Vk )A + αk I), Vk = η˜k k ∗ ηj PQj , j=1 ✭✹✳✼✮ ✶✾ β˜k = β1 + · · · + βk ✱ η˜k = η1 + · · · + ηk ✱ ❝→❝ t❤❛♠ sè βi✱ ηj ✱ αk ✈➔ γk t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥✿ (C10) βi > ✈ỵ✐ ♠å✐ i ∈ N+ ✈➔ ∞ i=1 βi = 1❀ (C11) ηj > ✈ỵ✐ ♠å✐ j ∈ N+ ✈➔ ∞ j=1 ηj = 1❀ (C12) αk ∈ (0, 1)✱ k ∈ N+ s❛♦ ❝❤♦ limk→∞ αk = ✈➔ ∞ k=1 αk = ∞❀ (C13) γk ∈ (ε0 , 2/(∥A∥2 + 2)) ✈ỵ✐ ♠å✐ k ∈ N+ ✱ ε0 ❧➔ ♠ët sè ❞÷ì♥❣ ọ ỵ r tr tt t ú tổ t ộ ữợ ũ tờ ỳ ✈✐➺❝ t➼♥❤ t♦→♥ s➩ ❞➵ ❞➔♥❣ ❤ì♥✳ ❙ü ❤ë✐ tư tt t ữủ ự ợ ✤✐➲✉ ❦✐➺♥ (C10), (C11), (C12) ✈➔ (C13)✳ ❚ø ✤â✱ ♥❤÷ ♥❤ú♥❣ ❤➺ q✉↔✱ ❝❤ó♥❣ tỉ✐ ✤➣ ♥❤➟♥ ✤÷đ❝ ♠ët sè ❦➳t q✉↔ ❝❤♦ ❝→❝ tr÷í♥❣ ❤đ♣ ♠ët tr♦♥❣ ❤❛✐ t➟♣ J1, J2 ❤♦➦❝ ❝↔ ❤❛✐ ✤➲✉ ❤ú✉ ❤↕♥✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ ❝❤➼♥❤ ✤↕t ✤÷đ❝✱ ❝❤ó♥❣ tỉ✐ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❜ê ✤➲ s❛✉✳ ❇ê ✤➲ ✹✳✶ ❈❤♦ H1 ✈➔ H2 ❧➔ ❤❛✐ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✱ Tj ✈ỵ✐ ♠é✐ j ∈ J2 ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ H2 s❛♦ ❝❤♦ ∩j∈J2 ❋✐①(Tj ) ̸= ∅ ✈➔ A ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥ tø H1 ✈➔♦ H2 ✳ ❑❤✐ ✤â✱ ∩j∈J2 A−1 ❋✐①(Tj ) = ∩j∈J2 ❋✐①(I − γA∗ (I − Tj )A) = A−1 (∩j∈J2 (Tj )), ợ số ữỡ ❈❤♦ H1, H2, A ✈➔ γ ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✹✳✶ ✈➔ Tj ✈ỵ✐ ♠é✐ j ∈ N+ ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ H2 s❛♦ ❝❤♦ ∩∞ j=1 ❋✐①(Tj ) ̸= ∅✳ ❑❤✐ ✤â C˜ := ∩j∈N+ ❋✐①(I − γA∗ (I − Tj )A) = ❋✐①(T∞ ), ð ✤â T∞ = I − γA∗ (I − V∞ )A✱ V∞ = ∞ j=1 ηj Tj ✈➔ ηj t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (C11)✳ ❇ê ✤➲ ✹✳✸ ❈❤♦ H ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ✈➔ Si ✈ỵ✐ ♠é✐ i ∈ N+ ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t tr♦♥❣ H ✳ ●✐↔ sû ✤✐➲✉ ❦✐➺♥ (C10) t❤ä❛ ♠➣♥✳ ❑❤✐ ✤â✱ ❝→❝ →♥❤ ①↕ S∞ := ∞ i=1 βi Si ✈➔ I − S∞ ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t✳ ❇ê ✤➲ ✹✳✹ ❈❤♦ H1✱ H2 ✈➔ A ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✹✳✶✳ ❑❤✐ õ ợ số ố tũ ỵ (0, 2/(∥A∥2 + 2α))✱ →♥❤ ①↕ Tγ,α := I − γ(A∗ (I − V )A + αI) ❧➔ ❝♦ ✈ỵ✐ ❤➡♥❣ sè − γα✱ ð ✤â V ❧➔ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ❝❤➦t ✈➔ α ❧➔ sè ❞÷ì♥❣ tr♦♥❣ ❦❤♦↔♥❣ (0, 1)✳ ❑❤✐ α = 0✱ Tγ := I − γA∗ (I − V )A ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ỵ H1, H2 A ữ tr♦♥❣ ❇ê ✤➲ ✹✳✶✱ {Ci}i∈N ✈➔ {Qj }j∈N+ + t÷ì♥❣ ự ổ t ỗ ✤â♥❣ tr♦♥❣ H1 ✈➔ H2 ✳ ●✐↔ sû Γ ̸= Ø ✈➔ ❝→❝ ✤✐➲✉ ❦✐➺♥ (C10), (C11), (C12), (C13) t❤ä❛ ♠➣♥✳ ❑❤✐ ✤â✱ ❞➣② {xk } ①→❝ ✤à♥❤ ❜ð✐ ✭✹✳✻✮✲ ✭✹✳✼✮ ❤ë✐ tư ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t ❝õ❛ ▼❙❙❋P ✭✹✳✺✮ ✈ỵ✐ J1 = J2 = N+ ✳ ỵ H1, H2 A ữ tr ỵ {Ci}Ni=1 {Qj }jN + tữỡ ự t ỗ õ tr H1 ✈➔ H2 ✳ ●✐↔ sû Γ ̸= Ø✱ ❞➣② ❧➦♣ xk ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐✿ x1 ∈ H1 , xk+1 = U Tγk ,αk xk , N i=1 βi PCi ✱ ð ✤â U = ✭✹✳✽✮ ∀k ≥ 1, Tγk ,αk = I−γk (A∗ (I−Vk )A+αk I)✱ Vk = k j=1 ηj PQj ✱ η˜k ✈ỵ✐ ❝→❝ t❤❛♠ sè βi ✱ ηj ✱ αk ✈➔ γk t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ (C11), (C12), (C13) ✈➔ (C10′ ) βi > ✈ỵ✐ ≤ i ≤ N s❛♦ ❝❤♦ N βi = 1✳ i=1 ❑❤✐ ✤â✱ ❞➣② xk ❤ë✐ tö ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ỵ H1, H2 A ữ tr ỵ {Ci}iN + {Qj }M j=1 tữỡ ự t ỗ õ tr♦♥❣ H1 ✈➔ H2 ✳ ●✐↔ sû Γ ̸= Ø✱ ❞➣② ❧➦♣ xk ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐✿ x1 ∈ H1 , xk+1 = Uk (I − γk (A∗ (I − V )A + αk I))xk , ð ✤â Uk = β˜k k i=1 βi PCi ✱ V = M ∀k ≥ 1, ✭✹✳✾✮ ηj PQj ✈ỵ✐ ❝→❝ t❤❛♠ sè βi ✱ ηj ✱ αk ✈➔ γk t❤ä❛ i=1 ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ (C10), (C12), (C13) ✈➔ (C11′ ) ηj > ✈ỵ✐ ≤ j ≤ M s❛♦ ❝❤♦ M ηj = 1✳ j=1 ❑❤✐ ✤â✱ ❞➣② xk ❤ë✐ tư ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t ❝õ❛ ❜➔✐ t ỵ H1, H2 A ữ tr ỵ {Ci}Mi=1 {Qj }Nj=1 tữỡ ự t ỗ õ tr H1 ✈➔ H2 ✳ ●✐↔ sû Γ ̸= Ø✱ ❞➣② ❧➦♣ xk ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐✿ x1 ∈ H1 , xk+1 = U (I − γk (A∗ (I − V )A + αk I))xk , ∀k ≥ 1, ✭✹✳✶✵✮ ✷✶ ð ✤â U = N i=1 βi PCi ✱ M ηj PQj ✈ỵ✐ ❝→❝ t❤❛♠ sè βi ✱ ηj ✱ αk ✈➔ γk t❤ä❛ V = i=1 ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ (C10′ ), (C11′ ), (C12), (C13)✳ ❑❤✐ ✤â✱ ❞➣② xk ❤ë✐ tư ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ✭✹✳✺✮✳ ✹✳✸✳ ❱➼ ❞ö sè ♠✐♥❤ ❤å❛ ❈❤ó♥❣ tỉ✐ ①➨t ▼❙❙❋P ✭✹✳✺✮ ✈ỵ✐ C := ❛♥❞ Ci Q := i∈J1 Qj j∈J2 ð ✤â ✭✹✳✶✶✮ Ci = {x ∈ En : ai1 x1 + ai2 x2 + · · · + ain xn ≤ bi }, ✈ỵ✐ ail, bi ∈ (−∞; +∞)✱ ≤ l ≤ n ✈➔ i ∈ J1✱ m ✭✹✳✶✷✮ (yl − ajl )2 ≤ Rj }, Rj > 0, m Qj = {y ∈ E : l=1 ✈ỵ✐ ajl ∈ (−∞; +∞)✱ ≤ l ≤ m✱ j ∈ J2✱ ✈➔ A ❧➔ ♠❛ tr➟♥ ❝ï m × n✳ ❚r♦♥❣ ✈➼ ❞ư t❤ù ♥❤➜t✱ ❝❤ó♥❣ tỉ✐ ①➨t tr÷í♥❣ ❤đ♣ m = n = 2, A ❧➔ ♠❛ tr➟♥ ✤ì♥ ✈à✱ ai1 = 1/i, ai2 = −1✱ bi = 0✱ ∀i ≥ 1✱ Rj = ✈➔ aj = (1/j, 0)✱ ∀j ≥ 1✳ ❑❤✐ ✤â✱ x∗ = (0; 0) ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t ❝õ❛ ✭✹✳✶✶✮✲✭✹✳✶✷✮✳ ❚ø A = I ✱ ♣❤÷ì♥❣ ♣❤→♣ ✭✹✳✻✮✲ ✭✹✳✼✮ ❝â ❞↕♥❣ xk+1 = Uk ((1 − γk (1 + αk ))xk + γk Vk xk ) ũ ữỡ ợ i = ηi = 1/(i(i + 1))✱ αk = 1/k✱ γk = 1/(1 + 0.05 + (1/k) ✈➔ ✤✐➸♠ ❜➢t ✤➛✉ x1 = (−3.0; 3.0)✱ ❝❤ó♥❣ tỉ✐ ♥❤➟♥ ✤÷đ❝ ❜↔♥❣ ❦➳t q✉↔ sè s❛✉✳ ❇↔♥❣ ✹✳✶✳ ❑➳t q✉↔ t➼♥❤ t♦→♥ ❦❤✐ ❞ò♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✹✳✶✸✮✳ k ✶ ✶✵ ✷✵ ✸✵ ✹✵ xk+1 ✵✳✵✷✹✸✾✵✷✹✸✾ ✵✳✵✶✵✷✺✺✸✷✼✹ ✵✳✵✵✺✺✸✹✹✾✽✷ ✵✳✵✵✸✽✶✽✵✹✷✽ ✵✳✵✵✷✾✷✹✾✽✻✷ xk+1 ✵✳✸✻✺✽✺✸✻✺✽✺ ✵✳✵✻✾✹✼✾✹✾✻✽ ✵✳✵✸✼✹✾✻✵✸✼✻ ✵✳✵✷✺✽✻✼✶✶✶✷ ✵✳✵✶✾✽✶✻✻✽✷✼ k ✶✵✵ ✺✵✵ ✶✵✵✵ ✷✵✵✵ ✸✵✵✵ xk+1 ✵✳✵✵✶✷✸✾✵✺✵✺ ✵✳✵✵✵✷✻✾✺✸✹✼ ✵✳✵✵✵✶✸✾✹✶✾✷ ✵✳✵✵✵✵✼✷✵✽✷✹ ✵✳✵✵✵✵✹✽✾✾✾✹ xk+1 ✵✳✵✵✽✸✾✹✺✷✺✶ ✵✳✵✵✶✽✷✻✵✽✽✽ ✵✳✵✵✵✾✹✹✺✻✵✻ ✵✳✵✵✵✹✽✽✸✺✺✽ ✵✳✵✵✵✵✸✸✶✾✻✾ ✷✷ ❚r♦♥❣ ✈➼ ❞ư t❤ù ❤❛✐✱ ✈ỵ✐ Ci, βi, ηj , Rj , γk , αk ✈➔ ✤✐➸♠ ❜➢t ✤➛✉ x1 ❣✐è♥❣ ♥❤÷ ✈➼ ❞ư tr➯♥✱ ❝❤ó♥❣ tỉ✐ ①➨t t➟♣ ♠ỵ✐ Qj = {y ∈ E3 : ∥y − aj ∥ ≤ 1} ð ✤â aj = (1/(j + 1); 1/(j + 1); 1/(j + 1)) ✈➔ A tr ù ì ợ ai1 = 1✱ i = 1, 2, 3✱ ❝→❝ ♣❤➛♥ tû ❝á♥ ❧↕✐ ❜➡♥❣ ✵✳ ❑❤✐ ✤â✱ x∗ = (0; 0) ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t✳ ❑➳t q✉↔ t➼♥❤ t♦→♥ ❦❤✐ ❞ị♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✹✳✻✮✲ ✭✹✳✼✮ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❜↔♥❣ sè s❛✉✳ ❇↔♥❣ ✹✳✷✳ ❑➳t q✉↔ t➼♥❤ t♦→♥ ❦❤✐ ❞ị♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✹✳✻✮✲ ✭✹✳✼✮ k xk+1 xk+1 k xk+1 xk+1 ✶ ✵✳✻✵✶✾✸✽✽✷✼✹ ✶✳✺✸✻✺✽✸✸✻✺✾ ✶✵✵ ✵✳✵✶✹✷✵✹✼✹✶✺ ✵✳✵✸✻✸✵✵✾✽✺✷ ✶✵ ✵✳✶✶✼✻✾✾✹✾✽✶ ✵✳✸✵✵✹✺✹✻✻✶✵ ✺✵✵ ✵✳✵✵✸✵✾✸✹✷✻✽ ✵✳✵✵✼✽✾✻✻✼✸✹ ✷✵ ✵✳✵✻✸✺✶✽✾✺✶✻ ✵✳✶✻✷✶✹✻✺✷✾✵ ✶✵✵✵ ✵✳✵✵✶✻✵✵✶✵✷✹ ✵✳✵✵✹✵✽✹✻✷✹✹ ✸✵ ✵✳✵✹✸✽✶✾✸✹✹✸ ✵✳✶✶✶✽✺✽✽✶✸✾ ✷✵✵✵ ✵✳✵✵✵✽✷✼✷✽✸✹ ✵✳✵✵✷✶✶✶✽✷✽✹ ✹✵ ✵✳✵✸✺✻✾✽✶✶✹✵ ✵✳✵✽✺✻✾✹✺✺✻✻ ✸✵✵✵ ✵✳✵✵✵✺✻✷✸✻✶✺ ✵✳✵✵✶✹✸✺✺✺✺✸ ❈→❝ ❦➳t q✉↔ t➼♥❤ t♦→♥ ✤÷đ❝ ❝❤↕② tr➯♥ ♣❤➛♥ ♠➲♠ ❋r❡❡ P❛s❝❛❧ ■❉❊ ✈ỵ✐ ♠→② t➼♥❤ ■♥t❡❧✭❘✮ ❈♦r❡✭❚▼✮ ✐✺✲✺✷✵✵❯ ❈P❯ ❅ ✷✳✷✵●❍③✱ ✷✳✷✵●❍③✱ ✹✳✵✵●❇ ♦❢ ❘❆▼✳ ◗✉❛ ❝→❝ ❦➳t q✉↔ t➼♥❤ t♦→♥ sè t❤✉ ✤÷đ❝ tr♦♥❣ ❇↔♥❣ ✹✳✶ ✈➔ ❇↔♥❣ ✹✳✷ ❝❤♦ t❤➜② ữỡ ú tổ s ữợ t❤✉ ✤÷đ❝ ♥❣❤✐➺♠ ①➜♣ ①➾ ❦❤→ ❣➛♥ ♥❣❤✐➺♠ ✤ó♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✭✹✳✺✮✳ ✷✸ ❑➌❚ ▲❯❾◆ ▲✉➟♥ →♥ ✤➣ ✤↕t ✤÷đ❝ ❝→❝ ❦➳t q✉↔ s❛✉✿ ✭✶✮✣➸ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ú tổ ợ t ởt ữỡ ✤✐➸♠ ❣➛♥ ❦➲✱ sü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❦❤ỉ♥❣ ❝➛♥ t❤➯♠ ✤✐➲✉ ❦✐➺♥ ♥➔♦ ❦❤→❝ ❧➯♥ t❤❛♠ sè ❝õ❛ t♦→♥ tû ❣✐↔✐ ❝õ❛ t♦→♥ tû ✤➣ ❝❤♦✳ ✭✷✮ ✣➣ ♥❤➟♥ ✤÷đ❝ ♠ët ❦➳t q✉↔ t÷ì♥❣ tü ❝❤♦ ❜➔✐ t♦→♥ ❜❛♦ ❤➔♠ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✤ì♥ ✤✐➺✉✳ ✭✸✮ ✣➣ ✤➲ ①✉➜t ♠ët ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✈ỵ✐ ❤❛✐ ổ t õ ỗ q trồ ữỡ ộ ữợ ❞ò♥❣ ❤ú✉ ❤↕♥ ❝→❝ t➟♣ ❝õ❛ ❤❛✐ ❤å tr➯♥✳ ✭✹✮ ✣÷❛ r❛ ❝→❝ ✈➼ ❞ư sè ♠✐♥❤ ❤♦↕ ❝❤♦ ❝→❝ ữỡ t ữợ ự t t ✭✶✮ ✣➲ ①✉➜t ✈➔ ♥❣❤✐➯♥ ❝ù✉ sü ❤ë✐ tö ❝õ❛ ữỡ ợ t ổ t♦→♥ tû ❞↕♥❣ ✤ì♥ ✤✐➺✉✱ ❝õ❛ tê♥❣ ❤❛✐ t♦→♥ tû ✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ ❇❛♥❛❝❤✳ ✭✷✮ ✣→♥❤ tố ở tử tợ ữỡ ♣❤→♣ ❧➦♣ ✤➸ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✱ tê♥❣ ❤❛✐ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✤➣ ✤↕t ✤÷đ❝ ð ❈❤÷ì♥❣ ✷✱ ❈❤÷ì♥❣ ✸✳ ✭✸✮ ❚✐➳♣ tư❝ ♥❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❝❤♦ ❣✐↔✐ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ♥❤✐➲✉ t➟♣ tr♦♥❣ tr÷í♥❣ ❤đ♣ t❤❛♠ sè ❧➦♣ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ❝❤✉➞♥ ❝õ❛ t♦→♥ tû ❝❤✉②➸♥ A✳ ✷✹ ❉❆◆❍ ▼Ö❈ ❈⑩❈ ❈➷◆● ❚❘➐◆❍ ✣❶ ❈➷◆● ❇➮ [1] ◆✳ ❇✉♦♥❣✱ P✳❚✳❚✳ ❍♦❛✐✱ ■t❡r❛t✐✈❡ ♠❡t❤♦❞s ❢♦r ③❡r♦s ♦❢ ❛ ♠♦♥♦t♦♥❡ ✈❛r✐✲ ❛t✐♦♥❛❧ ✐♥❝❧✉s✐♦♥ ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✱ ❈❛❧❝♦❧♦✱ ✷✵✶✽✱ ✺✺✱ ❛rt✿✼ ✭❙❈■❊✱ ◗✶✮✳ [2] ◆✳ ❇✉♦♥❣✱ P✳❚✳❚✳ ❍♦❛✐✱ ❑✳❚✳ ❇✐♥❤✱ ■t❡r❛t✐✈❡ ❘❡❣✉❧❛r✐③❛t✐♦♥ ▼❡t❤♦❞s ❢♦r t❤❡ ▼✉❧t✐♣❧❡✲❙❡ts ❙♣❧✐t ❋❡❛s✐❜✐❧✐t② Pr♦❜❧❡♠ ✐♥ ❍✐❧❜❡rt ❙♣❛❝❡s✱ ❆❝t❛ ❆♣♣❧ ▼❛t❤✱ ✷✵✶✾✱ ✶✻✺✱ ✶✽✸✲✶✾✼ ✭❙❈■❊✱ ◗✷✮✳ [3] ◆✳❚✳❚✳ ❚❤✉②✱ P✳❚✳❚✳ ❍♦❛✐✱ ◆✳❚✳❚✳ ❍♦❛✱ ❊①♣❧✐❝✐t ✐t❡r❛t✐✈❡ ♠❡t❤♦❞s ❢♦r ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦rs ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✱ ◆♦♥❧✐♥❡❛r ❋✉♥❝t✐♦♥✲ ♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✷✵✷✵✱ ✷✺✭✹✮✱ ✼✺✸✲✼✻✼ ✭❙❈❖P❯❙✮✳ [4] ◆✳❚✳◗✳ ❆♥❤✱ P✳❚✳❚✳ ❍♦❛✐✱ ▼♦❞✐❢✐❡❞ ❢♦r✇❛r❞✲❜❛❝❦✇❛r❞ s♣❧✐tt✐♥❣ ♠❡t❤✲ ♦❞s ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✱ ❊❛st✲❲❡st ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✷✵✷✵✱ ✷✷✭✶✮✱ ✶✸✲ ✷✾✳ [5] ◆✳ ❇✉♦♥❣✱ ◆✳❚✳❚✳ ❍♦❛✱ P✳❚✳❚✳ ❍♦❛✐✱ ■t❡r❛t✐✈❡ ♠❡t❤♦❞s ✇✐t❤ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦rs ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✱ ❑✛ ②➳✉ ❍ë✐ t❤↔♦ q✉è❝ ❣✐❛ ❧➛♥ t❤ù ❳❳■■■✿ ▼ët sè ✈➜♥ ✤➲ ❝❤å♥ ❧å❝ ❝õ❛ ❈æ♥❣ ♥❣❤➺ t❤æ♥❣ t✐♥ ✈➔ tr✉②➲♥ t❤æ♥❣✱ ◗✉↔♥❣ ◆✐♥❤✱ ♥❣➔② ✺✲✻ t❤→♥❣ ✶✶ ♥➠♠ ✷✵✷✵✱ ◆❤➔ ①✉➜t ❜↔♥ ❑❤♦❛ ❤å❝ ✈➔ ❦ÿ t❤✉➟t✱ ✷✵✷✵✱ ✶✺✽✲✶✻✹✳