✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ◆●❯❨➍◆ ❚❍➚ ❚❘❆◆● P❍×❒◆● P❍⑩P ❍■➏❯ ❈❍➓◆❍ ▲➄P ●■❷■ ❇⑨■ ❚❖⑩◆ ❈❍❻P ◆❍❾◆ ❚⑩❈❍ ✣❆ ❚❾P ❚❘❖◆● ❑❍➷◆● ●■❆◆ ❍■▲❇❊❘❚ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ✕ ✷✵✷✵ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ◆●❯❨➍◆ ❚❍➚ ❚❘❆◆● P❍×❒◆● P❍⑩P ❍■➏❯ ❈❍➓◆❍ ▲➄P ●■❷■ ❇⑨■ ❚❖⑩◆ ❈❍❻P ◆❍❾◆ ❚⑩❈❍ ✣❆ ❚❾P ❚❘❖◆● ❑❍➷◆● ●■❆◆ ❍■▲❇❊❘❚ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ö♥❣ ▼➣ sè✿ ✽ ✹✻ ✵✶ ✶✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈ ●❙✳❚❙✳ ◆●❯❨➍◆ ❇×❮◆● ❚❍⑩■ ◆●❯❨➊◆ ✕ ✷✵✷✵ ▲í✐ ❝↔♠ ì♥ ❚ỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ữớ ữớ t t ữợ ú ✤ï tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ♥❣❤✐➯♥ ❝ù✉ ✤➸ tæ✐ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❍✐➺✉✱ ❝→❝ t❤➛② ❣✐→♦✱ ❝ỉ ❣✐→♦ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✕ ❚✐♥✱ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✕✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ t↕✐ ❚r÷í♥❣✳ ❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ ❇❛♥ ●✐→♠ ❍✐➺✉ ❚r÷í♥❣ ❚❍❈❙ ❈↔♥❤ ❍÷♥❣ ✲ ♥ì✐ tỉ✐ ✤❛♥❣ ❧➔♠ ✈✐➺❝✱ ũ ỗ t ♠➦t ✤➸ tæ✐ t❤❛♠ ❣✐❛ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➺♥ ❝ù✉✳ ◆❤➙♥ ❞à♣ ♥➔②✱ tỉ✐ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ t tợ trữớ ữ ỗ ữớ t ✈✐➺♥✱ ❦❤➼❝❤ ❧➺✱ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ▼ët số ỵ t tt ữỡ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶✳ ✐✐ ✐✈ ✶ ✷ ❈→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ ❣✐↔✐ t➼❝❤ ❤➔♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✶✳✶✳ ❑❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✶✳✷✳ ởt số t t ỗ ữợ ✈✐ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✹✳ ❚♦→♥ tû tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✶✳✺✳ ✣✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✷✳ P❤→t ❜✐➸✉ ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✸✳ P❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ❈❤÷ì♥❣ ✷✳ P❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❝❤♦ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤❛ t➟♣ ✶✾ ✷✳✶✳ ▼ët sè ❜ê ✤➲ ❝➛♥ t❤✐➳t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷✳ ❚❤✉➟t t♦→♥ ✈➔ sü ❤ë✐ tö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸✳ ❱➼ ❞ö sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ❑➳t ❧✉➟♥ t ởt số ỵ ✈➔ ✈✐➳t t➢t H ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H∗ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝õ❛ N t➟♣ sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠ N∗ t➟♣ sè ♥❣✉②➯♥ ❞÷ì♥❣ R t➟♣ ❤đ♣ ❝→❝ sè tỹ C t õ ỗ ∅ t➟♣ ré♥❣ ∀x ✈ỵ✐ ♠å✐ lim sup xn ❣✐ỵ✐ ❤↕♥ tr➯♥ ❝õ❛ ❞➣② sè {xn } lim inf xn ợ ữợ số {xn } xn x0 ❞➣② {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ xn ❞➣② {xn } ❤ë✐ tö ②➳✉ ✈➲ H H x n→∞ n→∞ x0 F ix(T ) ❤♦➦❝ F (T ) x0 x0 t t f ữợ ỗ PC tr C f T ▼ð ✤➛✉ ❇➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✤â♥❣ ✈❛✐ trá ✤➦❝ ❜✐➺t q✉❛♥ trå♥❣ tr♦♥❣ ✈✐➺❝ ♠æ ❤➻♥❤ ❤â❛ ♥❤✐➲✉ ❜➔✐ t♦→♥ ♥❣÷đ❝ ①✉➜t ❤✐➺♥ tr♦♥❣ t❤ü❝ t➳ ♥❤÷ ❜➔✐ t♦→♥ ♥➨♥ ❤➻♥❤ ↔♥❤✱ ❝❤ư♣ ❤➻♥❤ ❝ë♥❣ ❤÷ð♥❣ tø✱ ❦❤ỉ✐ ♣❤ư❝ ↔♥❤✳ ▼ët tr♦♥❣ ♥❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✤➣ ✈➔ ✤❛♥❣ ✤÷đ❝ ♥❤✐➲✉ t→❝ ❣✐↔ sû ❞ư♥❣ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ tr♦♥❣ ✤â ❝➛♥ ♣❤↔✐ t❤ü❝ ❤✐➺♥ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ ❧➯♥ ❝→❝ t ỗ õ ổ rt ✈✐➺❝ t➼♥❤ ↔♥❤ ❝õ❛ →♥❤ ①↕ ❝❤✐➳✉ ♠➯tr✐❝ tr➯♥ ♠ët t ỗ õ t ý ụ ổ tỹ t ❉♦ ✈➟②✱ ❝➛♥ ①➙② ❞ü♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❤✐➺✉ q✉↔ ❤ì♥✳ ✣➲ t➔✐ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✤❛ t➟♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ✣â ❧➔ ❜➔✐ t♦→♥ t➻♠ ♠ët ✤✐➸♠ t❤✉ë❝ ❣✐❛♦ ✤✐➸♠ ❝õ❛ ♠ët ❤å t➟♣ ✤â♥❣✱ ỗ tr ổ rt õ q ♠ët →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❣✐ỵ✐ ♥ë✐ ♥➡♠ ✈➔♦ ❣✐❛♦ ởt t õ ỗ tr ởt ổ ❣✐❛♥ ❍✐❧❜❡rt ❦❤→❝✳ ✣➙② ❧➔ ♠ët ✤➲ t➔✐ ✈ø❛ ❝â ỵ t ỵ tt ỗ tớ ứ õ þ ♥❣❤➽❛ t❤ü❝ t✐➵♥ ❝❛♦✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ❤❛✐ ❝❤÷ì♥❣ ❝❤➼♥❤✿ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❧✉➟♥ ✈➠♥ ✤➲ ❝➟♣ ✤➳♥ ♠ët sè ✈➜♥ ✤➲ ❝ì ❜↔♥ ❝õ❛ ❣✐↔✐ t➼❝❤ ❤➔♠✱ ♣❤→t ❜✐➸✉ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✤❛ t➟♣✱ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤✱ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣✱ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣✳ ❈❤÷ì♥❣ ✷✳ P❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤❛ t➟♣ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❧✉➟♥ ✈➠♥ t➟♣ tr✉♥❣ tr➻♥❤ ❜➔② ❧↕✐ ♠ët ❝→❝❤ ❝❤✐ t✐➳t ❝→❝ ❦➳t q✉↔ ❝õ❛ ◆✳ ❇✉♦♥❣✱ P✳❚✳❚✳ ❍♦➔✐✱ ❑✳❚ ❇➻♥❤ ❬✸❪ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✤❛ t➟♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❈❤÷ì♥❣ ♥➔② ỗ ử tr ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ ❣✐↔✐ t➼❝❤ ❤➔♠✳ ▼ư❝ ✶✳✷ ♣❤→t ❜✐➸✉ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤✳ ▼ö❝ ✶✳✸ ✤➲ ❝➟♣ ✤➳♥ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ ❬✸✱ ✹❪✳ ✶✳✶✳ ❈→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ ❣✐↔✐ t➼❝❤ ❤➔♠ ✶✳✶✳✶✳ ❑❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ X ❈❤♦ ❦❤æ♥❣ tỡ ữợ tr X tr trữớ số t❤ü❝ R✳ ❚➼❝❤ ✈æ ❧➔ ♠ët →♥❤ ①↕ ·, · : X × X → R (x, y) → x, y t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙②✿ ✭✐✮ ✭✐✐✮ x, x ≥ ✈ỵ✐ ♠å✐ y, x = x, y x ∈ X ✱ x, x = ⇔ x = 0❀ ✈ỵ✐ ♠å✐ x, y ∈ X ❀ ✭✐✐✐✮ x + x , y = x, y + x , y ✭✐✈✮ λx, y = λ x, y ❙è x, y ✤÷đ❝ ❣å✐ ❧➔ ◆❤➟♥ ①➨t ✶✳✶✳ ✭✐✮ ✭✐✐✮ ✭✐✐✐✮ ✈ỵ✐ ♠å✐ ✈ỵ✐ ♠å✐ x, x , y ∈ X ❀ x, y ∈ X ✱ λ ∈ R t ổ ữợ tỡ x, y tr X ✳ ❚ø ✤à♥❤ ♥❣❤➽❛ s✉② r❛ ✈ỵ✐ ♠å✐ x, y + z = x, y + x, z x, λy = λ x, y x, y, z ∈ X, λ ∈ R✱ t❛ ❝â ❀ ❀ x, = ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ R✱ ·, · ❈➦♣ (X, ·, à ) tr õ X t ổ ữợ tr X ✤÷đ❝ ❣å✐ ❧➔ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt t❤ü❝✳ ✸ ▼➺♥❤ ✤➲ ✶✳✶✳ ▼å✐ ❦❤æ♥❣ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt X ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥✱ ✈ỵ✐ ❝❤✉➞♥ ①→❝ ✤à♥❤ ❜ð✐ x = ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ X ◆➳✉ ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt t❤ü❝ ✈➔ ✤➛② ✤õ ✤è✐ ✈ỵ✐ ❝❤✉➞♥ ❝↔♠ s✐♥❤ tø t ổ ữợ t H ợ x X x, x X ữủ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✳ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ❉➣② {xn } ữủ tử tợ tỷ x H ỵ xn x xn − x → ❦❤✐ n → ∞❀ ✭✐✐✮ ❍ë✐ tư ②➳✉ tỵ✐ ♣❤➛♥ tû x ∈ H ✱ ỵ xn n ợ x xn , y x, y y H ú ỵ ✶✳✶✳ ✭✐✮ ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✱ ❤ë✐ tư ♠↕♥❤ ❦➨♦ t❤❡♦ ❤ë✐ tư ②➳✉✱ ♥❤÷♥❣ ✤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❦❤ỉ♥❣ ✤ó♥❣✳ ✭✐✐✮ ▼å✐ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✤➲✉ ❝â t➼♥❤ ❝❤➜t ❑❛❞❡❝✲❑❧❡❡✱ tù❝ ❧➔ ♥➳✉ ❞➣② tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤➻ xn → x ❦❤✐ ✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ H t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ xn → x ✈➔ {xn } xn x n → ∞✳ ❈❤♦ C ❧➔ t➟♣ ❝♦♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H✳ ❑❤✐ ✤â C ✤÷đ❝ ❣å✐ ❧➔ ✭✐✮ ❚➟♣ ✤â♥❣ ♥➳✉ ♠å✐ ❞➣② {xn} ⊂ C t❤ä❛ ♠➣♥ xn → x ❦❤✐ n → ∞✱ t❛ ✤➲✉ ❝â x ∈ C❀ ✭✐✐✮ ❚➟♣ ✤â♥❣ ②➳✉ ✤➲✉ ❝â ✭✐✐✐✮ ♥➳✉ ♠å✐ ❞➣② {xn } ⊂ C t❤ä❛ ♠➣♥ xn x ❦❤✐ n → ∞✱ t❛ x ∈ C❀ ❚➟♣ ❝♦♠♣❛❝t ♥➳✉ ♠å✐ ❞➣② ♣❤➛♥ tû t❤✉ë❝ {xn } ⊂ C ✤➲✉ ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tö ✈➲ ♠ët C❀ ❚➟♣ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ ♥➳✉ ♠å✐ ❞➣② {xn} ⊂ C ✤➲✉ ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tö❀ ✭✈✮ ❚➟♣ ❝♦♠♣❛❝t ②➳✉ ♥➳✉ ♠å✐ ❞➣② {xn } ⊂ C ✤➲✉ ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tö ②➳✉ ✈➲ ✭✐✈✮ ♠ët ♣❤➛♥ tû t❤✉ë❝ C❀ ✹ ✭✈✐✮ ❚➟♣ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ ②➳✉ ♥➳✉ ♠å✐ ❞➣② {xn} ⊂ C ✤➲✉ ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tö ②➳✉✳ ◆❤➟♥ ①➨t ✶✳✷✳ ✭✐✮ ▼å✐ t➟♣ ❝♦♠♣❛❝t ✤➲✉ ❧➔ t➟♣ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐✱ ♥❤÷♥❣ ✤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❦❤ỉ♥❣ ✤ó♥❣✳ ✭✐✐✮ ▼å✐ t➟♣ ✤â♥❣ ②➳✉ ✤➲✉ ❧➔ t➟♣ ✤â♥❣✱ ♥❤÷♥❣ ✤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❦❤ỉ♥❣ ✤ó♥❣✳ ▼➺♥❤ ✤➲ ✶✳✷✳ ❈❤♦ H ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ✈➔ C ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ H ✳ ❑❤✐ ✤â✱ t❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉✿ ✭✐✮ ◆➳✉ C t ỗ õ t C t õ ②➳✉❀ ✭✐✐✮ ◆➳✉ C ❧➔ t➟♣ ❜à ❝❤➦♥ t❤➻ C ❧➔ t➟♣ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ ②➳✉✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ❍✐❧❜❡rt t❤ü❝ tû H✳ PC (x) ∈ C ❈❤♦ C ❧➔ ởt t rộ ỗ õ ổ t r ợ ộ x H tỗ t↕✐ ❞✉② ♥❤➜t ♠ët ♣❤➛♥ t❤ä❛ ♠➣♥ x − PC (x) = inf x − y y∈C P❤➛♥ tû →♥❤ ①↕ PC (x) ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ x ❧➯♥ C ✈➔ t❤➔♥❤ PC (x) ✤÷đ❝ ❣å✐ ❧➔ ♣❤➨♣ ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ PC : H → C ❜✐➳♥ ♠é✐ ♣❤➛♥ tû ❝❤✐➳✉ ♠➯tr✐❝ tø H ❧➯♥ C ✳ x∈H ✣➦❝ tr÷♥❣ ❝õ❛ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ ✤÷đ❝ ữợ C ởt t ỗ õ rộ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H ✳ ❑❤✐ ✤â✱ →♥❤ ①↕ PC : H → C ❧➔ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tø H ❧➯♥ C ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x − PC (x), y − PC (x) ≤ ◆❤➟♥ ①➨t ✶✳✸✳ y ∈ C✱ π α≤ ✳ ❱➲ ♣❤÷ì♥❣ ❞✐➺♥ ❤➻♥❤ ❤å❝✱ ✈ỵ✐ ♠å✐ t↕♦ ❜ð✐ ❝→❝ ✈➨❝tì ❱➼ ❞ư ✶✳✶✳ Rn ✈ỵ✐ ♠å✐ y ∈ C x − PC (x) ✈➔ y − PC (x) t❤➻ ♥➳✉ t❛ ❣å✐ ổ rt tỹ ợ t ổ ữợ n x, y = λk αk k=1 α ❧➔ ❣â❝ ✺ tr♦♥❣ ✤â x = (λ1 , λ2 , , λn )✱ y = (α1 , α2 , , αn ) n x n = x, x = ❑❤ỉ♥❣ ❣✐❛♥ l2 ✱ ✈ỵ✐ |αk |2 αk αk = k=1 ❱➼ ❞ö ✶✳✷✳ ✈➔ ❝❤✉➞♥ ❝↔♠ s✐♥❤ k=1 x = {λk }, y = {αk }✱ t❛ ✤à♥❤ ♥❣❤➽❛ ∞ λk αk x, y = k=1 t Ã, à t ổ ữợ (l2 , ·, · ) ✶✳✶✳✷✳ ▼ët sè t➼♥❤ ❝❤➜t ✣à♥❤ ❧➼ ✶✳✶ X✱ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ✭❇➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲❙❝❤✇❛rt③✮ ✳ ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt ✈ỵ✐ ♠å✐ x, y ∈ X t❛ ❧✉æ♥ ❝â ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ | x, y |2 ≤ x, x y, y ✭✶✳✶✮ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ y = ❜➜t ✤➥♥❣ t❤ù❝ ❤✐➸♥ ♥❤✐➯♥ ✤ó♥❣✳ ●✐↔ sû y = ❦❤✐ ✤â ✈ỵ✐ ♠å✐ sè λ∈R t❛ ✤➲✉ ❝â x + λy, x + λy ≥ tù❝ ❧➔ x, x + λ y, x + λ x, y + |λ|2 y, y ≥ ❈❤å♥ λ=− x, y y, y t❛ ✤÷đ❝ x, x − | x, y |2 ≥ ⇔ | x, y |2 ≤ x, x y, y y, y ỵ ữủ ự ✣à♥❤ ❧➼ ✶✳✷✳ ●✐↔ sû {xn}n, {yn}n ❧➔ ❤❛✐ ❞➣② ❤ë✐ tư ②➳✉ ✤➳♥ a, b tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt t❤ü❝ X ✳ ❑❤✐ ✤â lim xn , yn = a, b n→∞ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû n→∞ lim xn = a✱ lim yn = b tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ X ✳ ❚❛ s➩ ❝❤ù♥❣ n→∞ ♠✐♥❤ lim xn , yn = a, b n→∞ tr♦♥❣ R✳ ❚❤➟t ✈➟②✱ t❛ ❝â | xn , yn − a, b | = | xn , yn + xn , b − xn , b − a, b | ❈❤÷ì♥❣ ✷ P❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❝❤♦ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✤❛ t➟♣ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❝❤♦ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✤❛ t➟♣✳ ❈ư t ỗ ử tr ởt số ❜ê ✤➲ ❝➛♥ t❤✐➳t✱ ♠ö❝ ✷✳✷ tr➻♥❤ ❜➔② t❤✉➟t t♦→♥ ✈➔ sü ❤ë✐ tö✱ ♠ö❝ ✷✳✸ tr➻♥❤ ❜➔② ♠ët sè ✈➼ ❞ö✳ ✷✳✶✳ ▼ët sè ❜ê ✤➲ ❝➛♥ t❤✐➳t ❇ê ✤➲ ✷✳✶✳ ❈❤♦ H1 ✈➔ H2 ❧➔ ❤❛✐ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✱ ❝❤♦ Tj ✈ỵ✐ ♠å✐ j ∈ J2 ❧➔ ♠ët →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ H2 s❛♦ ❝❤♦ ∩j∈J ❋✐①(Tj ) = ∅ ✈➔ ❝❤♦ A ❧➔ ♠ët →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❣✐ỵ✐ ♥ë✐ tø H1 ✈➔♦ H2✳ ❑❤✐ ✤â✱ ∩j∈J2 A−1 ❋✐①(Tj ) = ∩j∈J ❋✐①(I − γA∗(I − Tj )A) = A−1(∩j∈J ❋✐①(Tj )), 2 tr♦♥❣ ✤â γ ❧➔ ♠ët sè ❞÷ì♥❣✳ ❈❤ù♥❣ ♠✐♥❤✳ ❘ã r➔♥❣✱ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ✤➥♥❣ t❤ù❝ t❤ù ♥❤➜t ❝❤➾ ❝➛♥ ❝❤➾ r❛ r➡♥❣ A−1 ✈ỵ✐ ♠å✐ j ∈ J2 ✳ ❚❤➟t ✈➟②✱ ♥➳✉ z − γA∗ (I − Tj )Az = z ✱ ❋✐①(I − γA ❦❤✐ γ > 0✳ ∗ ❋✐①(Tj ) ❦❤✐ z∈ = ❋✐①(I − γA∗ (I − Tj )A), z ∈ A−1 ❋✐①(I ❋✐①(Tj )✱ ❝â ♥❣❤➽❛ ❧➔ − γA∗ (I − Tj )A)✳ Az = Tj Az ✱ t❤➻ ❚ø ❜❛♦ ❤➔♠ t❤ù❝ z∈ (I − Tj )A)✱ ❝â ♥❣❤➽❛ ❧➔ γA∗ (I − Tj )Az = s✉② r❛ A∗ (I − Tj )Az = 0✱ ❉♦ ✤â✱ Tj Az = Az + wj , A∗ wj = ▲➜② ♠ët ♣❤➛♥ tû Tj Ap = Ap✳ p ∈ A−1 ❋✐①(Tj )✳ ❑❤✐ ✤â✱ t❛ ❝â Ap ∈ ❋✐①(Tj )✱ ❝â ♥❣❤➽❛ ❧➔ ❉♦ ✤â✱ Az − Ap ≥ Tj Az − Tj Ap = Az − Ap 2 = Az − Ap + wj + wj 2 + wj , A(z − p) ✷✵ ❱➻ ✈➟②✱ + A ∗ wj , z − p = Az − Ap + wj = Az − Ap + wj wj = 0✳ ❈â ♥❣❤➽❛ ❧➔ Tj Az = Az ✱ ❦❤✐ z ∈ A−1 ❋✐①(Tj )✳ ✣➥♥❣ t❤ù❝ t❤ù ❤❛✐ ✤÷đ❝ s✉② r❛ tø ∩j∈J2 A−1 ❋✐①(Tj ) = A−1 (∩j∈J2 ❋✐①(Tj )) ❙✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❇ê ✤➲ ✷✳✷✳ ❈❤♦ H1, H2, A ✈➔ γ ❣✐è♥❣ ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✷✳✶ ✈➔ ❝❤♦ Tj ✈ỵ✐ ♠å✐ ❧➔ ♠ët →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr♦♥❣ H2 s❛♦ ❝❤♦ ∩∞j=1❋✐①(Tj ) = ∅✳ ❑❤✐ ✤â✱ j ∈ N+ C˜ := ∩j∈N+ ❋✐①(I − γA∗ (I − Tj )A) = ❋✐①(T∞ ), tr♦♥❣ ✤â T∞ = I − γA∗(I − V∞)A✱ V∞ = ∞j=1 ηj Tj ✈➔ ηj t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭η✮✳ ❈❤ù♥❣ ♠✐♥❤✳ ◆❤÷ t❛ ✤➣ ❜✐➳t →♥❤ ①↕ V∞ ❧➔ ❦❤ỉ♥❣ ❣✐➣♥ ✈ỵ✐ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❋✐①(V∞ ) ❱ỵ✐ ♠é✐ ˜ = ∩∞ j=1 (Tj ) rữợ t t ự t❤ù❝ C ⊂ ❋✐①(T∞ )✳ ˜ ✱ t❛ ❝â (I − γA∗ (I − Tj )A)z = z ✈ỵ✐ ♠å✐ j ∈ N+ ✳ ❚ø ✤â✱ t❛ ✤✐➸♠ z ∈ C ❝â ∞ ηj (I − γA∗ (I − Tj )A)z = z, j=1 ❞♦ ✤â I − γA∗ (I − V∞ )A)z = z, ✈➻ I ✈➔ A∗ ❧➔ ❤❛✐ →♥❤ ①↕ t✉②➳♥ t➼♥❤✳ ✣✐➲✉ ✤â ❝â ♥❣❤➽❛ ❧➔ ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ ❋✐①(T∞ ) p∈ ❋✐①(V∞ )✳ ❉♦ z∈ ⊂ C˜ ✳ ❋✐①(T∞ ) ♥➯♥ t❛ ❝â ▲➜② ❤❛✐ ✤✐➸♠ ❜➜t ❦➻ A∗ (I − V∞ )Az = 0✳ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tü ♥❤÷ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ❇ê ✤➲ ✷✳✶ ✈ỵ✐ ❜➡♥❣ ❞♦ ✤â V∞ t❛ ♥❤➟♥ ✤÷đ❝ ✤➥♥❣ t❤ù❝ z ∈ ❋✐①(T∞ )✳ z ∈ ❋✐①(T∞ )✳ V∞ Az = Az ✱ ♥❣❤➽❛ ❧➔ z∈ ❋✐①(T∞ ) ✈➔ ❚✐➳♣ t❤❡♦✱ ❜➡♥❣ J2 = {1} ✈➔ T1 t❤❛② γA∗ (I − V∞ )Az = 0✱ ❙✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❇ê ✤➲ ✷✳✸✳ ❈❤♦ H ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ✈➔ ❝❤♦ Si✱ ✈ỵ✐ ♠å✐ i ∈ N+ ❧➔ ♠ët →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t tr♦♥❣ H ✳ ●✐↔ sû ✤✐➲✉ ❦✐➺♥ ✭β ✮ ✤÷đ❝ t❤ä❛ ♠➣♥✳ ❑❤✐ ✤â✱ ❝→❝ →♥❤ ①↕ S∞ := ∞i=1 βiSi ✈➔ I − S∞ ❝ơ♥❣ ❧➔ ❦❤ỉ♥❣ ❣✐➣♥ ❝❤➦t✳ ❈❤ù♥❣ rữợ t t r r S ổ ❣✐➣♥ ❝❤➦t✳ ❱ỵ✐ ♠ư❝ ✤➼❝❤ ♥➔②✱ t❛ ①➨t →♥❤ ①↕ Sk := k ˜ i=1 (βi /βk )Si ✳ ❱➻ Si ❧➔ ❦❤ỉ♥❣ ❣✐➣♥ ❝❤➦t ✈ỵ✐ ♠å✐ ✷✶ i ∈ N+ x ỗ k (i /k ) Si u − Si v, u − v Sk u − Sk v, u − v = i=1 k (βi /β˜k ) Si u − Si v ≥ i=1 k (βi /β˜k )(Si u − Si v) ≥ i=1 = Sk u − Sk v ∀u, v ∈ H k S k u := i=1 βi u → S∞ u ✈➔ β˜k → tø ✤✐➲✉ ❦✐➺♥ ✭β ✮ ❦❤✐ k → ∞✳ Sk u = (1/β˜k )S k u → S∞ u✳ ❈❤♦ k → ∞ t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ ❚❛ ❜✐➳t r➡♥❣ ❑❤✐ ✤â✱ S∞ u − S∞ v, u − v ≥ S∞ u − S∞ v , s✉② r❛ S∞ ❧➔ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t✳ ❚✐➳♣ t❤❡♦✱ t❛ ❝❤➾ r❛ I − S∞ ❧➔ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t✳ ❚❤➟t ✈➟②✱ ✈➻ (I−Si )u − (I − Si )v, u − v − (I − Si )u − (I − Si )v = Si u − Si v, u − v ✈➔ Si ❧➔ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t ♥➯♥ I − Si − Si u − Si v 2 ❝ơ♥❣ ❧➔ ❦❤ỉ♥❣ ❣✐➣♥ ❝❤➦t✳ ❉♦ ✤â✱ k (βi /β˜k ) (I − Si )u − (I − Si )v, u − v (I − Sk )u − (I − Sk )v, u − v = i=1 k ≥ (βi /β˜k ) (I − Si )u − (I − Si )v i=1 k 2 (βi /β˜k )(I − Si )u − (I − Si )v ≥ i=1 = (I − Sk )u − (I − Sk )v ❈❤♦ k t✐➳♥ ✤➳♥ ✈ỉ ❝ị♥❣ tr♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥✱ t❛ ❝â ✤÷đ❝ ❦➳t ❧✉➟♥ t❤ù ❤❛✐✳ ❇ê ✤➲ ✷✳✹✳ ❈❤♦ H1✱ H2 ✈➔ A ❣✐è♥❣ ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✷✳✶✳ ❑❤✐ õ ợ ởt số tũ ỵ ố (0, 2/( A + 2α))✱ →♥❤ ①↕ Tγ,α := I − γ(A∗(I − V )A + αI) ❧➔ ♠ët →♥❤ ①↕ ❝♦ ✈ỵ✐ ❤➺ sè − γα✱ tr♦♥❣ ✤â V ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t ✈➔ α ❧➔ ♠ët sè ❞÷ì♥❣ tr♦♥❣ (0, 1)✳ ❑❤✐ α = t❤➻ Tγ := I − γA∗(I − V )A ❧➔ ❦❤æ♥❣ ❣✐➣♥✳ ✷✷ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤➟t ✈➟②✱ tø ❝→❝ tự ữợ T, x T, y = (1 − γα)(x − y) − γ[A∗ (I − V )Ax − A∗ (I − V )Ay = (1 − γα)2 x − y 2 + γ A∗ (I − V )Ax − A∗ (I − V )Ay − 2γ(1 − γα) A∗ (I − V )Ax − A∗ (I − V )Ay, x − y γ ✈➔ ✤✐➲✉ ❦✐➺♥ ❝õ❛ Tγ,α x − Tγ,α y ✈➔ V✱ t❛ ❝â ≤ (1 − γα)2 x − y + γ A∗ (I − V )Ax − A∗ (I − V )Ay − 2γ(1 − γα) (I − V )Ax − A∗ (I − V )Ay ≤ (1 − γα)2 x − y 2 + γ A∗ (I − V )Ax − A∗ (I − V )Ay − 2γ(1 − γα) A∗ (I − V )Ax − A∗ (I − V )Ay / A 2 ≤ (1 − γα)2 x − y , ❞♦ 2γ(1 − γα)/ A Tγ ❧➔ ❦❤æ♥❣ ❣✐➣♥✳ ≥ γ 2✳ ❉♦ ✤â✱ Tγ,α ❧➔ ♠ët →♥❤ ①↕ ❝♦✳ ❘ã r➔♥❣✱ ❦❤✐ α=0 ✷✳✷✳ ❚❤✉➟t t♦→♥ ✈➔ sü ❤ë✐ tö ❚r♦♥❣ ♠ö❝ ♥➔② tæ✐ ♠ð rë♥❣ ✭✶✳✶✸✮ ❝❤♦ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✤❛ t➟♣ ✈ỵ✐ ❝→❝ ❤å ✈ỉ ❤↕♥ J1 ✈➔ J2 ✱ ❝â ♥❣❤➽❛ ❧➔ J1 = J2 = N+ ✳ ❚r♦♥❣ ✤â ❧➔ ❦➼ ❤✐➺✉ ❧➔ N+ t➜t ❝↔ ❝→❝ sè tü ♥❤✐➯♥ ❞÷ì♥❣✳ P❤÷ì♥❣ ♣❤→♣ ♥➔② ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✳ xk+1 = Uk Tγk ,αk xk , x1 ∈ H1 , ✭✷✳✶✮ tr♦♥❣ ✤â Uk = β˜k k = I − γk (A (I − Vk )A + αk I), Vk = η˜k k ∗ βi PCi , Tγk ,αk i=1 η j P Qj , j=1 ✭✷✳✷✮ β˜k = β1 + · · · + βk ✱ η˜k = η1 + · · · + ηk ✈➔ ❝→❝ t❤❛♠ sè βi ✱ ηj ✱ αk ✈➔ γk t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✭β ✮ βi > ✈ỵ✐ ♠å✐ i ∈ N+ ✈➔ ✭η ✮ ηj > ✈ỵ✐ ♠å✐ j ∈ N+ ✈➔ ✭α✮ αk ∈ (0, 1) ✭γ ✮ γk ∈ (ε0 , 2/( A ✈ỵ✐ ♠å✐ k ∈ N+ + 2)) ∞ i=1 βi = 1❀ ∞ j=1 ηj = 1❀ s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ limk→∞ αk = k ∈ N+ tr♦♥❣ ✤â ✈➔ ε0 ∞ k=1 αk =∞ ởt số ữỡ ọ ữ ỵ r ộ ữợ ữỡ sỷ tờ ❤ú✉ ❤↕♥✳ ❉♦ ✤â✱ ✈✐➺❝ t➼♥❤ t♦→♥ ð ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ❧➔ ✤ì♥ ❣✐↔♥✳ ✷✸ ❚✐➳♣ t❤❡♦✱ t❛ ❝❤➾ r❛ sü ❤ë✐ tö ♠↕♥❤ ❝õ❛ t❤✉➟t t♦→♥ ✭✷✳✶✮ ✲ ✭✷✳✷✮ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✭β ✮✱ ✭η ✮✱ ✭α✮ ✈➔ ✭γ ✮✳ ❉ü❛ ✈➔♦ ♠ö❝ ✷✳✶ t❛ ❝â ❝→❝ ❦➳t q✉↔ s❛✉✳ ✣à♥❤ ❧➼ ✷✳✶✳ ❈❤♦ H1, H2 ✈➔ A ❣✐è♥❣ ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✷✳✶✳ ❈❤♦ {Ci}i∈N ✈➔ + ❧➔ ổ t ỗ õ tr H1 ✈➔ H2✱ t÷ì♥❣ ù♥❣✳ ●✐↔ sû Γ = ∅ ✈➔ ❣✐ú ♥❣✉②➯♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ✭β ✮✱ ✭η✮✱ ✭α✮ ✈➔ ✭γ ✮✳ ❑❤✐ ✤â✱ ❞➣② {xk } ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ✭✷✳✶✮ ✈➔ ✭✷✳✶✮ ❤ë✐ tö ♠↕♥❤ ✈➲ ♥❣❤✐➺♠ ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t ❝õ❛ ✭✶✳✾✮ ❦❤✐ k → ợ J1 = J2 = N+ ự rữợ ❤➳t✱ t❛ ❝❤➾ r❛ r➡♥❣ {xk } ❧➔ ❞➣② ❜à ❝❤➦♥✳ ❚❤➟t ✈➟②✱ ❧➜② ♠ët {Qj }j∈N+ ✤✐➸♠ ❝è ✤à♥❤ ❝â p ∈ Ci p ∈ Γ✱ tø ❇ê ✤➲ ✷✳✶ ✈ỵ✐ Tj = PQj (I − γk A∗ (I − PQj )A)p = p ✈➔ p = Uk p, Tγk p = p ✈ỵ✐ ♠é✐ i, k ∈ N+ ✱ ❣✐➣♥ ❝õ❛ Uk tr♦♥❣ ✤â ✈➔ t➼♥❤ ❦❤æ♥❣ ❣✐➣♥ ❝õ❛ ✈ỵ✐ ♠å✐ ✈➔ i, j, k ∈ N+ t❛ ✈➔ ❞♦ ✤â PCi Tγk p = p, Tγk = I − γk A∗ (I − Vk )A✳ P Qj ✭✷✳✸✮ ❉♦ ✈➟②✱ tø t➼♥❤ ❦❤æ♥❣ ✈➔ ❇ê ✤➲ ✷✳✹✱ s✉② r❛ xk+1 − p = Uk Tγk ,αk xk − Uk Tγk p ≤ Tγk ,αk xk − Tγk p = Tγk ,αk xk − Tγk ,αk p − γk αk p ≤ (1 − γk αk ) xk − p + γk αk p ≤ max{ x1 − p , p } ✣✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔ ❞➣② {xk } õ tỗ t ởt sè ❞÷ì♥❣ ˜ M s❛♦ ❝❤♦ xk , xk − p , A∗ (I − Vk )Axk sup ˜ ≤M k≥1 ❚✐➳♣ t❤❡♦✱ ✈➻ U k ✱ Tγ k ❧➔ ❦❤æ♥❣ ❣✐➣♥✱ ♠✐♥❤ ❝õ❛ ❇ê ✤➲ ✷✳✸ ✈➔ xk+1 − p 2 x I − Vk ❧➔ ♠ët ❤➔♠ ỗ tứ t õ = Uk Tk ,k xk − Uk Tγk p ≤ Tγk ,αk xk − Tγk p 2 = Tγk xk − Tγk p − γk αk xk = Tγk xk − Tγk p ≤ xk − p ❧➔ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t✱ ❞ü❛ t❤❡♦ ❝❤ù♥❣ 2 + (γk αk )2 xk − 2γk αk Tγk xk − Tγk p, xk − 2γk (I − Vk )Axk − (I − Vk )Ap, Axk − Ap + γk2 A∗ (I − Vk )Axk ˜2 + (γk αk )2 M ✷✹ − 2γk αk Tγk xk − Tγk p, xk ≤ xk − p − 2γk Axk − Vk Axk + γk2 A Axk − Vk Axk ˜ − 2γk αk Tγ xk − Tγ p, xk + (γk αk )2 M k k ≤ xk − p − γk (2 − γk A ) Axk − Vk Axk ˜ + 2γk αk M ˜ + (γk αk )2 M ❍ì♥ ♥ú❛✱ Uk ✭✷✳✹✮ ❧➔ ❦❤æ♥❣ ❣✐➣♥ ♥➯♥ t❛ ❝â xk+1 − Uk xk = Uk Tγk ,αk xk − Uk xk ≤ Tγk ,αk xk − xk ˜ Axk − Vk Axk + γk αk M ≤ γk A ▼➦t ❦❤→❝✱ sû ❞ö♥❣ t➼♥❤ ❝❤➜t ❝õ❛ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ P Ci ✈ỵ✐ ✭✷✳✺✮ x = z k := Tγk ,αk xk ✱ t❛ ❝â t❤➸ ✈✐➳t z k − P Ci z k (βi /β˜k ) z k − PCi z k + P Ci z k − p ≤ zk − p 2, ❦❤✐ ✤â✱ k k x 2 ≤ zk − p i=1 i=1 ❱➻ (βi /β˜k ) PCi z k − p + ❧➔ ❤➔♠ ỗ tr H1 tứ t tự ố ũ t❛ ✤÷đ❝ k k (βi /β˜k )(z k − PCi z k ) (βi /β˜k )(PCi z k − p) + i=1 ≤ zk − p i=1 ❉♦ ✤â✱ Tγk ,αk − xk+1 + xk+1 − p ≤ Tγk ,αk xk − p = Tγk xk − Tγk p − γk αk xk = Tγ k x k − Tγ k p ≤ xk − p 2 2 ˜2 − 2γk αk Tγk xk − Tγk p, xk + (γk αk )2 M ˜ + (γk αk )2 M ˜ + 2γk αk M ✭✷✳✻✮ ▼➦t ❦❤→❝✱ Tγk ,αk xk − xk+1 = xk − xk+1 + γk αk xk + γk A∗ (I − Vk )Axk − γk αk xk + γk A∗ (I − Vk )Axk , xk − xk+1 ✭✷✳✼✮ ✷✺ ❈✉è✐ ❝ò♥❣✱ ✈➻ H1 Uk ❧➔ ❦❤æ♥❣ ❣✐➣♥✱ P Ci ❧➔ ❦❤æ♥❣ t x ỗ tr t❛ ❝â xk+1 − p = Uk Tγk ,αk xk − Uk Tγk p k (βi /β˜k ) PCi Tγk ,αk xk − PCi Tγk p ≤ i=1 ≤ Tγk ,αk xk − Tγk p, xk+1 − p = Tγk ,αk xk − Tγk ,αk p, xk+1 − p + γk αk p, p − xk+1 = (1 − γk αk ) γk (A∗ (I − Vk )A) xk − γk αk I− γk (A∗ (I − Vk )A) p, xk+1 − p +γk αk p, p − xk+1 − γk αk − I− ≤ (1 − γk αk ) xk − p xk+1 − p + γk αk p, p − xk+1 − γk αk k ≤ x − p + xk+1 − p + γk αk p, p − xk+1 , 2 I − γk [A∗ (I − Vk )A]/(1 − γk αk ) ❞♦ ✭γ ✮ ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ A∗ (I − Vk )A✳ xk+1 − p ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ t❤❡♦ ❇ê ✤➲ ✷✳✹✱ ✭α✮✱ ❱➟②✱ ≤ (1 − γk αk ) xk − p + γk αk p, p − xk+1 rữớ ủ ỗ t ởt số tỹ ữỡ k0 s❛♦ ❝❤♦ ♠å✐ k ≥ k0 ✳ ❱➟②✱ limk→∞ xk p ỡ ỳ t sỹ tỗ t xk+1 p xk p ợ tỗ t↕✐✳ limk→∞ xk − p ✈➔ ✭✷✳✹✮✱ t❛ ❝â lim sup γk (2 − γk A ) Axk − Vk Axk = ✭✷✳✾✮ k→∞ ❘ã r➔♥❣ tø ✤✐➲✉ ❦✐➺♥ ✭γ ✮✱ t❛ ❝â γk (2 − γk A ) ≥ γk (2 − (2 A /( A + 2)) ≥ 4ε0 /( A + 2) ❚ø ✤✐➲✉ ♥➔② ✈➔ ✭✷✳✾✮ s✉② r❛ lim Axk − Vk Axk k→∞ ❍ì♥ ♥ú❛✱ t❤❡♦ ✭✷✳✺✮✱ ✭✷✳✶✵✮ ✈➔ αk → = t❤➻ lim xk+1 − Uk xk = k→∞ ❉♦ limk→∞ xk − p tỗ t tứ k t❛ ❝â lim Tγk ,αk xk − xk+1 = k→∞ ✭✷✳✶✵✮ ✭✷✳✶✷✮ ✷✻ γk αk xk + γk A∗ (I − Vk )Axk ≤ γk αk xk + γk A ❱➻ ✭✷✳✶✵✮ ✈➔ ✤✐➲✉ ❦✐➺♥ tr➯♥ αk Axk − Vk Axk ♥➯♥ tø t❛ ❝â lim γk αk xk + γk A∗ (I − Vk )Axk = ✭✷✳✶✸✮ k→∞ ❉♦ ✤â✱ tø ❣✐ỵ✐ ❤↕♥ ❝õ❛ {xk } s✉② r❛ lim γk αk xk + γk A∗ (I − Vk )Axk , xk − xk+1 = ✭✷✳✶✹✮ k→∞ ❚ø ✭✷✳✼✮✱ ✭✷✳✶✷✮✱ ✭✷✳✶✸✮ ✈➔ ✭✷✳✶✹✮✱ t❛ ❝â lim xk+1 − xk = ✭✷✳✶✺✮ k→∞ ❱➻ ❝→❝ ❞➣② {xk } ✈➔ {γk } ❧➔ ❣✐ỵ✐ ♥ë✐ ♥➯♥ ❦❤ỉ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ t❛ ❝â t❤➸ ❣✐↔ sû ♠ët ❞➣② ❝♦♥ ✈➔ {kj } ❝õ❛ {k} s❛♦ ❝❤♦ {xkj } ❤ë✐ tö ②➳✉ ✤➳♥ ♠ët ♣❤➛♥ tû x˜ ∈ H1 γkj → γ ∈ [ε0 , 2/( A ✷✳✶✱ ✷✳✷ ✈➔ ✷✳✸✱ s✉② r❛ tr♦♥❣ ✤â U∞ = + 2)] ❦❤✐ j → ∞✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ x˜ ∈ ∩i∈N+ Ci ∩ F ix(T∞ ) ❤♦➦❝ x˜ ∈ Γ✳ ❚ø ❇ê ✤➲ x˜ ∈ F ix(U∞ ) ∩ F ix(T∞ )✱ ∞ i=1 βi PCi ✳ x˜ ∈ F ix(U ) rữợ t t ự r t ✈➟②✱ tø ✭✷✳✶✶✮ ✈➔ ✭✷✳✶✺✮✱ s✉② r❛ lim xk − Uk xk = ✭✷✳✶✻✮ k→∞ ❱➻ ❦❤✐ Uk x → U∞ x j → ∞✳ ❦❤✐ k→∞ ❱➻ ✈➟②✱ ợ ợ ố >0 tỗ t x ∈ H1 jε > ♥➯♥ t❛ ❝â s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ Ukj x → U∞ x j ≥ jε sup Ukj x − U∞ x < ε, x∈D tr♦♥❣ ✤â D ❧➔ t➟♣ ❝♦♥ ❣✐ỵ✐ ♥ë✐ ❝õ❛ H1 ✳ ▲➜② D = {xkj }✱ t❛ ❝â Ukj xkj − U∞ xkj < ε, ❝â ♥❣❤➽❛ ❧➔ lim Ukj xkj − U∞ xkj = ✭✷✳✶✼✮ j→∞ ❉♦ ✤â✱ tø ✭✷✳✶✻✮✱ ✭✷✳✶✼✮ ✈➔ xkj − U∞ xkj ≤ xkj − Ukj xkj + Ukj xkj − U∞ xkj s✉② r❛ xkj − U∞ xkj → 0✳ ❑❤✐ ✤â✱ ❞ü❛ ✈➔♦ ❇ê ✤➲ ✶✳✸ t❤➻ x˜ ∈ F ix(U∞ )✳ ✷✼ x˜ ∈ F ix(T∞ )✳ ❇➙② ❣✐í✱ t❛ ❝❤ù♥❣ ♠✐♥❤ ❚❤➟t ✈➟②✱ tø ✭✷✳✶✷✮ ✈➔ ✭✷✳✶✺✮✱ t❛ ❝â lim xk − Tγk ,αk xk = ✭✷✳✶✽✮ k→∞ ❇➡♥❣ ❝→❝❤ ❧➟♣ ❧✉➟♥ ♥❤÷ tr➯♥✱ t❛ ❝â lim Vkj Axkj − V∞ Axkj = 0, j→∞ ❦➳t ❤đ♣ ✈ỵ✐ ✭✷✳✶✾✮✱ t❛ ✤÷đ❝ xkj − T∞ xkj ≤ xkj − Tγkj ,αkj xkj + Tγkj ,αkj xkj − T∞ xkj ˜ ≤ xkj − Tγkj ,αkj xkj + Tγkj xkj − T∞ xkj + γkj αkj M ˜ ≤ xkj − Tγkj ,αkj xkj + |γkj − γ|M ˜ + γ Vkj Axkj − V∞ Axkj + γkj αkj M ✈➔ ❣✐↔ ✤à♥❤ s✉② r❛ r➡♥❣ xkj − T∞ xkj → 0✳ ▲↕✐ t❤❡♦ ❇ê ✤➲ ✶✳✸✱ t❛ ❝â x˜ ∈ F ix(T∞ )✳ ❘ã r➔♥❣✱ ♠å✐ ❞➣② ❝♦♥ ❤ë✐ tö ②➳✉ ❝õ❛ {xk } ❤ë✐ tö ②➳✉ ✤➳♥ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✾✮✳ ❚✐➳♣ t❤❡♦✱ t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥❤ä ♥❤➜t x∗ {xk } ❤ë✐ tö ♠↕♥❤ ✈➲ ♥❣❤✐➺♠ ❝â ❝❤✉➞♥ ❝õ❛ ✭✶✳✾✮✳ ❚❛ t❤➜② r➡♥❣ lim sup x∗ , x∗ − xk = lim x∗ , x∗ − xkl = x∗ , x∗ − x˜ ≤ 0, ✭✷✳✶✾✮ l→∞ k→∞ ❞♦ x∗ x∗ ✱ ✭✷✳✶✾✮ ✈➔ ❇ê ✤➲ ✶✳✶ t❛ ❝â ❧➔ ♥❣❤✐➺♠ ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t ❝õ❛ ✭✶✳✾✮✳ ❈✉è✐ ❝ò♥❣✱ tø ✭✷✳✽✮ t❤❛② xk − x∗ → ❦❤✐ l {mk } ⊆ N+ s❛♦ ❝❤♦ mk → ∞ {xk } k N+ ợ tỗ t↕✐ ♠ët ❞➣② ❦❤æ♥❣ t➠♥❣ ✈➔ xmk − p ≤ xmk +1 p õ tỗ t xk − p ≤ xmk +1 − p limk→∞ xmk − p tr➯♥✱ t❛ ❝â ✤➥♥❣ t❤ù❝ ✭✷✳✶✵✮✲✭✷✳✶✷✮ ✈➔ ✭✷✳✶✺✮ ✈ỵ✐ ❤↕♥ ②➳✉ xkl −p < xkl +1 −p l ∈ N+ ✳ ❑❤✐ ✤â✱ t❤❡♦ ❇ê ✤➲ ✶✳✷✱ ❞♦ ✈ỵ✐ k rữớ ủ ỗ t ♠ët ❞➣② ❝♦♥ {xk } ❝õ❛ {xk } s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ p x˜ ❝õ❛ {xmk } t❤✉ë❝ Γ✳ k ✭✷✳✷✵✮ ✈➔ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tü ♥❤÷ t❤❛② ❜ð✐ mk ✈➔ ♠é✐ ✤✐➸♠ ❣✐ỵ✐ ❍ì♥ ♥ú❛✱ lim sup x∗ , x∗ − xmk +1 = x∗ , x∗ − x˜ ≤ ✭✷✳✷✶✮ k→∞ ❚ø ✭✷✳✽✮ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ✤➛✉ t✐➯♥ tr♦♥❣ ✭✷✳✷✵✮ ✈ỵ✐ xmk − x∗ p ✤÷đ❝ t❤❛② ❜ð✐ ≤ x∗ , x∗ − xmk +1 , x∗ ✱ s✉② r❛ ✭✷✳✷✷✮ ✷✽ ✈ỵ✐ ♠é✐ k ∈ N+ ✳ ❚ø ✭✷✳✷✶✮ ✈➔ ✭✷✳✷✷✮✱ t❛ ❝â lim sup xmk − x∗ = ✭✷✳✷✸✮ k→∞ ❚ø ợ k ữủ t mk t ❝â lim xmk +1 − x∗ ≤ lim xmk +1 − xmk + lim xmk − x∗ = 0, k→∞ k→∞ k→∞ ❝ị♥❣ ✈ỵ✐ ❜➜t ✤➥♥❣ t❤ù❝ t❤ù ❤❛✐ tr♦♥❣ ✭✷✳✷✵✮ s✉② r❛ lim xk − x∗ = k→∞ ◆❤÷ ✈➟② t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣✳ ❚❛ ❝â ❝→❝ ỵ s H1, H2 A ❣✐è♥❣ ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✷✳✶✳ ❈❤♦ {Ci}Ni=1 ✈➔ t ỗ õ tr ổ ❣✐❛♥ H1 ✈➔ H2✱ t÷ì♥❣ ù♥❣✳ ●✐↔ sû Γ = ∅ ✈➔ ❣✐ú ♥❣✉②➯♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ✭η✮✱ ✭α✮✱ ✭γ ✮ ✈➔ ✭β ✮ βi > ❦❤✐ ≤ i ≤ N s❛♦ ❝❤♦ Ni=1 βi = 1✳ ❑❤✐ ✤â✱ ✈ỵ✐ k → ∞ t❤➻ ❞➣② {xk } ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ {Qj }j∈N+ N x k+1 k = U Tγk ,αk x , k ≥ 1, x ∈ H1 , U = βi PCi , i=1 ❤ë✐ tö ♠↕♥❤ ✈➲ ♥❣❤✐➺♠ ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t ❝õ❛ ✭✶✳✾✮✳ ❈❤ù♥❣ ♠✐♥❤✳ ▲➜② Ci = CN ✈ỵ✐ i > N ự ữ ỵ ✷✳✸✳ ❈❤♦ H1, H2 ✈➔ A ❣✐è♥❣ ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✷✳✶✳ ❈❤♦ {Ci}i∈N ✈➔ {Qj }M j=1 ❧➔ ❤❛✐ t ỗ õ tr H1 H2 ✱ t÷ì♥❣ ù♥❣✳ ●✐↔ sû Γ = ∅ ✈➔ ❣✐ú ♥❣✉②➯♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ✭β ✮✱ ✭α✮✱ ✭γ ✮ ✈➔ ✭η ✮ ηj > ❦❤✐ ≤ j ≤ M s❛♦ ❝❤♦ Mj=1 ηj = 1✳ ❑❤✐ ✤â✱ ợ k t {xk } ữủ ♥❣❤➽❛ ❜ð✐ + M x k+1 ∗ k = Uk (I − γk (A (I − V )A + αk I))x , k ≥ 1, x ∈ H1 , V = η j P Qj , i=1 ❤ë✐ tö ♠↕♥❤ ✈➲ ♥❣❤✐➺♠ ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t ❝õ❛ ✭✶✳✾✮✳ ✣à♥❤ ❧➼ ✷✳✹✳ ❈❤♦ H1, H2 ✈➔ A ❣✐è♥❣ ♥❤÷ tr♦♥❣ ❇ê ✤➲ ✷✳✶✳ ❈❤♦ {Ci}Mi=1 ✈➔ {Qj }N j=1 ❧➔ ỳ t ỗ õ tr♦♥❣ H1 ✈➔ H2 ✱ t÷ì♥❣ ✷✾ ù♥❣✳ ●✐↔ sû Γ = ∅ ✈➔ ❣✐ú ♥❣✉②➯♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ✭β ✮✱ ✭α✮✱ ✭γ ✮ ✈➔ ✭η ✮✳ ❑❤✐ ✤â✱ ợ k {xk } ữủ ❜ð✐ xk+1 = U (I − γk (A∗ (I − V )A + αk I))xk , x1 ∈ H1 , tr♦♥❣ ✤â U ✈➔ V ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ tr♦♥❣ ỵ tữỡ ự tử õ ọ t tr ú ỵ ✷✳✶✳ P Ci ✈ỵ✐ ✣➦t T i = PCi (I − γA∗ (I − PQi )A) I − γA∗ (I − PQi )A✱ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ✭✶✳✾✮ ✈ỵ✐ Vk k t ộ ữợ ữ ỵ t tr ❜➻♥❤ ❝õ❛ ❇ê ✤➲ ✶✳✹ ✈➔ ✷✳✶ t❛ ❝â J1 = J2 = N+ ✱ ❞ü❛ tr➯♥ k Γ = ∩i∈N+ F ix(T i )✳ t❛ ❝➛♥ ①➙② ❞ü♥❣ ❝→❝ →♥❤ ①↕ →♥❤ ①↕ ✤➛✉ t✐➯♥ ❝õ❛ Ti ✈➔ k ❉♦ ✤â✱ Wk ✱ Sk ✈➔ sè t❤ü❝ ❞÷ì♥❣✳ ✷✳✸✳ ❱➼ ❞ö sè ❚❛ ①➨t ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✤❛ t➟♣ ✈ỵ✐ C = ∩∞ i=1 Ci ✈➔ Q = ∩∞ j=1 Qj , tr♦♥❣ ✤â Ci = {x ∈ En : ai1 x1 + ai2 x2 + · · · + ain ≤ bi }, ail , bi ∈ (−∞; +∞)✱ ✈ỵ✐ 1≤l≤n ✈➔ ✭✷✳✷✹✮ i ∈ N+ ✱ m (yl − ajl )2 ≤ Rj }, Rj > 0, m Qj = {y ∈ E : ✭✷✳✷✺✮ l=1 ajl ∈ (−∞; +∞)✱ ❱➼ ❞ư ✷✳✶✳ ✤ì♥ ✈à✱ ♠å✐ ✈ỵ✐ 1≤l≤m j ∈ N+ ✈➔ A ❧➔ ♠ët ♠❛ tr➟♥ ❝ï ❚r♦♥❣ ✈➼ ❞ư ✤➛✉ t✐➯♥✱ tỉ✐ ①➨t tr÷í♥❣ ❤đ♣ ai1 = 1/i, ai2 = −1 j ≥ 1✳ ✈➔ ✈➔ bi = ✈ỵ✐ ♠å✐ M = N = 2✱ A ❧➔ ♠❛ tr➟♥ i ≥ 1✱ Rj = ❑❤✐ ✤â✱ ❦❤æ♥❣ ❦❤â ✤➸ ❦✐➸♠ tr❛ r➡♥❣ m × n✳ x∗ = (0; 0) ✈➔ aj = (1/j, 0) ✈ỵ✐ ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t A = I✱ t❤✉➟t t♦→♥ ✭✷✳✶✮ ✲ xk+1 = Uk ((1 − γk (1 + αk ))xk + γk Vk xk ) ✭✷✳✷✻✮ ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✷✹✮ ✲ ✭✷✳✷✺✮✳ ❱➻ ✭✷✳✷✮ ❝â ❞↕♥❣ ❉û ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✷✻✮ ợ 1/(1 + 0.05 + (1/k) ữợ i = ηi = 1/(i(i + 1))✱ αk = 1/k ✱ γk = ✈➔ ✤✐➸♠ ①✉➜t ♣❤→t x1 = (−3.0; 3.0)✱ t❛ ♥❤➟♥ ✤÷đ❝ ❜↔♥❣ sè ✸✵ k xk+1 xk+1 k xk+1 xk+1 ✶ ✵✳✵✷✹✸✾✵✷✹✸✾ ✵✳✸✻✺✽✺✸✻✺✽✺ ✶✵✵ ✵✳✵✵✶✷✸✾✵✺✵✺ ✵✳✵✵✽✸✾✹✺✷✺✶ ✶✵ ✵✳✵✶✵✷✺✺✸✷✼✹ ✵✳✵✻✾✹✼✾✹✾✻✽ ✺✵✵ ✵✳✵✵✵✷✻✾✺✸✹✼ ✵✳✵✵✶✽✷✻✵✽✽✽ ✷✵ ✵✳✵✵✺✺✸✹✹✾✽✷ ✵✳✵✸✼✹✾✻✵✸✼✻ ✶✵✵✵ ✵✳✵✵✵✶✸✾✹✶✾✷ ✵✳✵✵✵✾✹✹✺✻✵✻ ✸✵ ✵✳✵✵✸✽✶✽✵✹✷✽ ✵✳✵✷✺✽✻✼✶✶✶✷ ✷✵✵✵ ✵✳✵✵✵✵✼✷✵✽✷✹ ✵✳✵✵✵✹✽✽✸✺✺✽ ✹✵ ✵✳✵✵✷✾✷✹✾✽✻✷ ✵✳✵✶✾✽✶✻✻✽✷✼ ✸✵✵✵ ✵✳✵✵✵✵✹✽✾✾✾✹ ✵✳✵✵✵✵✸✸✶✾✻✾ ❱➼ ❞ö ✷✳✷✳ ❚r♦♥❣ ✈➼ ❞ư t❤ù ❤❛✐✱ ❣✐ú ♥❣✉②➯♥ ♣❤→t✱ tỉ✐ ①➨t t➟♣ ♠ỵ✐ Qj = {y ∈ E3 : y − aj ≤ 1} 1); 1/(j + 1); 1/(j + 1)) ✈ỵ✐ i = 1, 2, Ci , βi , ηj , Rj , γk , αk ✈➔ A ❧➔ ♠ët ♠❛ tr➟♥ ❝ï 3×2 tr♦♥❣ ✤â ✈➔ ✤✐➸♠ ①✉➜t aj = (1/(j + ✈ỵ✐ ❝→❝ ♣❤➛♥ tû ✈➔ ♣❤➛♥ tû ❝á♥ ❧↕✐ ❜➡♥❣ ❦❤æ♥❣✳ ❉➵ t❤➜② x∗ = (0; 0) ai1 = 1✱ ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t✳ ❑➳t q✉↔ t➼♥❤ t♦→♥ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✮ ✲ ✭✷✳✷✮ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❜↔♥❣ sè ❧✐➺✉ s❛✉✳ k xk+1 xk+1 k xk+1 xk+1 ✶ ✵✳✻✵✶✾✸✽✽✷✼✹ ✶✳✺✸✻✺✽✸✸✻✺✾ ✶✵✵ ✵✳✵✶✹✷✵✹✼✹✶✺ ✵✳✵✸✻✸✵✵✾✽✺✷ ✶✵ ✵✳✶✶✼✻✾✾✹✾✽✶ ✵✳✸✵✵✹✺✹✻✻✶✵ ✺✵✵ ✵✳✵✵✸✵✾✸✹✷✻✽ ✵✳✵✵✼✽✾✻✻✼✸✹ ✷✵ ✵✳✵✻✸✺✶✽✾✺✶✻ ✵✳✶✻✷✶✹✻✺✷✾✵ ✶✵✵✵ ✵✳✵✵✶✻✵✵✶✵✷✹ ✵✳✵✵✹✵✽✹✻✷✹✹ ✸✵ ✵✳✵✹✸✽✶✾✸✹✹✸ ✵✳✶✶✶✽✺✽✽✶✸✾ ✷✵✵✵ ✵✳✵✵✵✽✷✼✷✽✸✹ ✵✳✵✵✷✶✶✶✽✷✽✹ ✹✵ ✵✳✵✸✺✻✾✽✶✶✹✵ ✵✳✵✽✺✻✾✹✺✺✻✻ ✸✵✵✵ ✵✳✵✵✵✺✻✷✸✻✶✺ ✵✳✵✵✶✹✸✺✺✺✺✸ ◆❤÷ ✈➟②✱ tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝õ❛ ❧✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ♠ët sè ❜ê ✤➲ ❝➛♥ t❤✐➳t ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❝❤♦ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✤❛ t➟♣ ✈➔ ✤÷❛ r❛ ♠ët ✈➔✐ ✈➼ ❞ư ♠✐♥❤ ❤å❛ ❝❤♦ ♣❤÷ì♥❣ ♣❤→♣ ♥➔②✳ ❑➳t ❧✉➟♥ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ ♠ët ❝→❝❤ ❦❤→ ❝❤✐ t✐➳t ✈➔ ❤➺ t❤è♥❣ ✈➲ ❝→❝ ✈➜♥ ✤➲ s❛✉✿ • ▼ët sè ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ ❣✐↔✐ t➼❝❤ ỗ ổ rt t tỷ tr ổ rt ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥❀ • ❈→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ◆✳ ❇✉♦♥❣✱ P✳❚✳❚✳ ❍♦❛✐✱ ❑✳❚✳ ❇✐♥❤ tr♦♥❣ t➔✐ ❧✐➺✉ ❬✸❪ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❧➦♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✤❛ t➟♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ ❱➠♥ ❍✐➲♥✱ ▲➯ ❉ơ♥❣ ▼÷✉ ✭✷✵✵✸✮✱ ❞ư♥❣✱ ❱✐➺♥ ❚♦→♥ ❤å❝✱ ❍➔ ◆ë✐✳ ❬✷❪ ✣é ❱➠♥ ▲÷✉✱ ◆❣✉②➵♥ ✣ù❝ ▲↕♥❣ ổ t ỗ ự tr ❣✐↔✐ t➼❝❤ ❤➔♠✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛✱ ❍➔ ◆ë✐✳ ❚✐➳♥❣ ❆♥❤ ■t❡r❛t✐✈❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤✲ ♦❞s ❢♦r t❤❡ ▼✉❧t✐♣❧❡✲❙❡ts s♣❧✐t ❢❡❛s✐❜✐❧✐t② ♣r♦❜❧❡♠ ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✱ ❆❝t❛ ❬✸❪ ◆✳ ❇✉♦♥❣✱ P✳❚✳❚✳ ❍♦❛✐✱ ❑✳❚✳ ❇✐♥❤ ✭✷✵✶✾✮✱ ❆♣♣❧✐❝❛♥❞❛❡ ▼❛t❤✐❝❛❧❛❡✱ ❙♣r✐♥❣❡r✳ ❬✹❪ ❆❣❛r✇❛❧ ❘✳ P✳✱ ❖✬❘❡❣❛♥ ❉✳✱ ❙❛❤✉ ❉✳ ❘✳ ✭✷✵✵✾✮✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❢♦r ▲✐♣s❝❤✐t③✐❛♥✲t②♣❡ ▼❛♣♣✐♥❣s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✱ ❙♣r✐♥❣❡r✳ ❬✺❪ ❆❧❜❡r✱ ❨✳✱ ❘②❛③❛♥ts❡✈❛✱ ■✳P✳ ✭✷✵✵✻✮✱ ◆♦♥❧✐♥❡❛r ■❧❧✲P♦s❡❞ Pr♦❜❧❡♠s ♦❢ ▼♦♥♦✲ t♦♥❡ ❚②♣❡✱ ❙♣r✐♥❣❡r✳ ❬✻❪ ❇❛❦✉s❤✐♥s❦②✱ ❆✳❇✳ ✭✶✾✼✼✮✱ ✏▼❡t❤♦❞s ❢♦r s♦❧✈✐♥❣ ♠♦♥♦t♦♥✐❝ ✈❛r✐❛t✐♦♥❛❧ ✐♥✲ ❡q✉❛❧✐t✐❡s ❜❛s❡❞ ♦♥ t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ✐t❡r❛t✐✈❡ r❡❣✉❧❛r✐③❛t✐♦♥✑✱ ❛♥❞ ▼❛t❤✳ P❤②s✐❝s✳✱ ✶✼✱ ✶✷✲✷✹✳ ❬✼❪ ❇❛❦✉s❤✐♥s❦②✱ ❆✳❇✳✱ ●♦♥❝❤❛rs❦②✱ ❆✳ ✭✶✾✽✾✮✱ ❈♦♠♣✉t✳ ▼❛t❤✳ ■❧❧✲P♦s❡❞ Pr♦❜❧❡♠s✿ ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✳ ❑❧✉✇❡r ❆❝❛❞❡♠✐❝ P✉❜❧✐s❤❡rs✳ ❬✽❪ ❇②r♥❡ ❈✳ ✭✷✵✵✹✮✱ ✏❆ ✉♥✐❢✐❡❞ tr❡❛t♠❡♥t ♦❢ s♦♠❡ ✐t❡r❛t✐✈❡ ❛❧❣♦r✐t❤♠s ✐♥ s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣ ❛♥❞ ✐♠❛❣❡ r❡❝♦♥str✉❝t✐♦♥✑✱ ■♥✈❡rs❡ Pr♦❜❧❡♠s✱ ✶✽✱ ✶✵✸✲✶✷✵✳ ❬✾❪ ❇r✉❝❦✱ ❘✳ ❊✳ ✭✶✾✼✹✮✱ ✏❆ str♦♥❣ ❝♦♥✈❡r❣❡♥t ✐t❡r❛t✐✈❡ ♠❡t❤♦❞ ❢♦r t❤❡ s♦❧✉t✐♦♥ ✸✸ ∈ Ux ❢♦r ❛ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦r ❆♥❛❧✳ ❆♣♣❧✳✱ ✹✽✱ ✶✶✹✲✶✷✻✳ U ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✑✳ ❏✳ ▼❛t❤✳ ❬✶✵❪ ❈❡♥s♦r ❨✳✱ ❊❧❢✈✐♥❣ ❚✳ ✭✶✾✾✹✮✱ ✏❆ ♠✉❧t✐ ♣r♦❥❡❝t✐♦♥ ❛❧❣♦r✐t❤♠ ✉s✐♥❣ ❇r❡❣♠❛♥ ♣r♦❥❡❝t✐♦♥s ✐♥ ❛ ♣r♦❞✉❝t s♣❛❝❡✑✱ ◆✉♠❡r✳ ❆❧❣♦r✐t❤♠s✱ ✽✭✷✲✹✮✱ ✷✷✶✲✷✸✾✳ ❬✶✶❪ ❈❤✉❛♥❣✱ ❈❤✳❙❤✳ ✭✷✵✶✸✮✱ ✏❙tr♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ t❤❡♦r❡♠s ❢♦r t❤❡ s♣❧✐t ✈❛r✐❛✲ t✐♦♥❛❧ ✐♥❝❧✉s✐♦♥ ♣r♦❜❧❡♠ ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✑✱ ❝❛t✐♦♥s✱ ✷✵✶✸✱ ✸✺✵✳ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐✲ ❬✶✷❪ ❳✉✱ ❍✳❑✳ ✭✷✵✶✵✮✱ ✏■t❡r❛t✐✈❡ ♠❡t❤♦❞s ❢♦r t❤❡ s♣❧✐t ❢❡❛s✐❜✐❧✐t② ♣r♦❜❧❡♠ ✐♥ ✐♥❢✐♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡s✑✱ ■♥✈❡rs❡ Pr♦❜❧❡♠s✱ ✷✻✱ ✶✵✵✺✵✶✽✱ ✶✼✳ ❬✶✸❪ ❨✳❨❛♦✱ ❲✳ ❏✐❛♥❣✱ ❨✳❈✳▲✐♦✉ ✭✷✵✶✷✮✱ ✏❘❡❣✉❧❛r✐③❡❞ ♠❡t❤♦❞s ❢♦r t❤❡ s♣❧✐t ❢❡❛s✐✲ ❜✐❧✐t② ♣r♦❜❧❡♠✑✱ ❆❜str❛❝t ❛♥❞ ❆♣♣❧✐❡❞ ❆♥❛❧②s✐s✱ ✷✵✶✷✱ ❆rt✐❝❧❡ ✶✹✵✻✼✾✱ ❉❖■✿ ✶✵✳✶✶✺✺✴✷✵✶✷✴✶✹ ✵✻✼✾✳