❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ✯✯✯✯✯✯✯✯✯✯✯✯✯ P❍❸▼ ❚❍➚ ▲❆◆ ⑨ ▼❐❚ ❙➮ ❚❍❯❾❚ ❚❖⑩◆ ▲➄P ❚➐▼ ✣■➎▼ ❇❻❚ ✣❐◆● ❈ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ ❈❤✉②➯♥ ♥❣➔♥❤✿ ●✐↔✐ t➼❝❤ ❍➔ ◆ë✐ ✲ ✷✵✶✹ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ✯✯✯✯✯✯✯✯✯✯✯✯✯ P❍❸▼ ❚❍➚ ▲❆◆ ❱⑨ ▼❐❚ ❙➮ ❚❍❯❾❚ ❚❖⑩◆ ▲➄P ❚➐▼ ✣■➎▼ ❇❻❚ ✣❐◆● ❈ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ t ữớ ữợ ◆●❯❨➍◆ ❱❿◆ ❚❯❨➊◆ ❍➔ ◆ë✐ ✲ ✷✵✶✹ ▲❮■ ❈❷▼ ❒◆ ữủ ỷ ỡ tợ t ❝ỉ ❣✐→♦ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷✱ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ❦❤♦❛ ❚♦→♥ ✤➣ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ tr÷í♥❣ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ ❤♦➔♥ t❤➔♥❤ ✤➲ t➔✐ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳ ✣➦❝ ❜✐➺t ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ t❤➛② ❚✉②➯♥ ◆❣✉②➵♥ ❱➠♥ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉✱ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât ✈➔ ❤↕♥ ữủ sỹ õ õ ỵ ❝õ❛ ❝→❝ t❤➛② ❣✐→♦✱ ❝æ ❣✐→♦ ✈➔ t♦➔♥ t❤➸ ❜↕♥ ✤å❝ ✤➸ ✤➲ t➔✐ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ✦ ❍➔ ◆ë✐✱ ♥❣➔② ✷ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✹ ❙✐♥❤ ✈✐➯♥ P❤↕♠ ❚❤à ▲❛♥ ▲❮■ ❈❆▼ ữợ sỹ ữợ t❤➛② ❣✐→♦ ❚✉②➯♥ ◆❣✉②➵♥ ❱➠♥ ❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❦❤ỉ♥❣ trò♥❣ ✈ỵ✐ ❜➜t ❦➻ ✤➲ t➔✐ ♥➔♦ ❦❤→❝✳ ❚r♦♥❣ ❦❤✐ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ❡♠ ✤➣ sû ❞ö♥❣ ✈➔ t❤❛♠ ❦❤↔♦ ❝→❝ t❤➔♥❤ tü✉ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈ỵ✐ ❧á♥❣ ❜✐➳t ì♥ tr➙♥ trå♥❣✳ ❍➔ ◆ë✐✱ ♥❣➔② ✷ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✹ ❙✐♥❤ ❱✐➯♥ P❤↕♠ ❚❤à ▲❛♥ ✐✐ ▼ư❝ ❧ư❝ ▼ð ✤➛✉ ✶ ▲í✐ ♠ð ✤➛✉ ✶ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✸ ✶✳✶✳ ✶✳✷✳ ✶✳✸✳ ❑❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✶✳ ❑➼ ❤✐➺✉ ✈➔ ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t tự ỗ ♥❤➜t t❤ù❝ ❝ì ❜↔♥ ✳ ✳ ✳ ✺ ✶✳✶✳✸✳ ❚♦→♥ tû t✉②➳♥ t➼♥❤ ✈➔ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✻ ✶✳✶✳✹✳ ❚æ♣æ ♠↕♥❤ ✈➔ ②➳✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✶✳✺✳ ❙ü ❤ë✐ tö ②➳✉ ❝õ❛ ❞➣② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✷✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✈➔ ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✷✳✷✳ ❚➼♥❤ ①➜♣ ①➾ tèt ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✷✳✸✳ ❚➼♥❤ ❝❤➜t tæ♣æ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ỵ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỗ t ổ ✐✐✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷ ✶✳✸✳✶✳ ❈→❝ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ P t ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✸✳✸✳ ❈→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥✳ ✶✳✸✳✹✳ ❈→❝ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ tr✉♥❣ ❜➻♥❤ ✳ ✳ ✳ ✷✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ❚➼♥❤ ✤ì♥ ✤✐➺✉ ❋❡❥➨r ✈➔ ♠ët sè t❤✉➟t t♦→♥ ❧➦♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ ✸✼ ✷✳✶✳ ❉➣② ✤ì♥ ✤✐➺✉ ❋❡❥➨r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✷✳ P❤➨♣ ❧➦♣ ❑r❛s♥♦s❡❧✬s❦✐ ✲ ▼❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✸✳ ❈→❝ t❤✉➟t t♦→♥ ❝õ❛ ❝→❝ t♦→♥ tû tr✉♥❣ ❜➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ❑➳t ❧✉➟♥ ✺✵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✺✵ ✐✈ ▲í✐ ỵ tt t tỷ ỡ r ữủ t tr sợ õ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tø ♥❤ú♥❣ t❤➟♣ ♥✐➯♥ ✻✵ ❝õ❛ ❚❤➳ ❦✛ ✷✵ ❜ð✐ ♥❤➔ t♦→♥ ❤å❝ ♥❣÷í✐ ◆❣❛ t➯♥ ❧➔ ❊r❡♠✐♥ ❬✹❪✳ ❈❤♦ ré♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ♠ët ❞➣② ✤ì♥ ✤✐➺✉ ❋❡❥➨r tr♦♥❣ H C ✈➔ (xn ) ⊂ H✳ C ❧➔ ♠ët t➟♣ ❝♦♥ ❦❤→❝ ❑❤✐ ✤â✱ (xn ) ✤÷đ❝ ❣å✐ ❧➔ ♥➳✉ ∀x ∈ C, ∀n ∈ N, xn+1 − x ≤ xn − x ❚➼♥❤ ✤ì♥ ✤✐➺✉ ❋❡❥➨r ❧➔ ♠ët ❦❤→✐ ♥✐➺♠ ♥ỵ✐ ❧ä♥❣ ❝õ❛ t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ♠ët ❞➣②✳ ◆❤÷ t❛ ✤➣ ❜✐➳t✱ ♠ët ❞➣② ✤ì♥ ✤✐➺✉ t❤➻ ✤ì♥ ✤✐➺✉ ❋❡❥➨r✳ ❚➼♥❤ ✤ì♥ ✤✐➺✉ ❋❡❥➨r ❝â ♠ët ✈❛✐ trá r➜t q✉❛♥ trå♥❣ tr♦♥❣ t♦→♥ ❤å❝ ù♥❣ ❞ư♥❣✳ P❤÷ì♥❣ ♣❤→♣ ❞➣② ❧➦♣ ❋❡❥➨r ❣✐ó♣ ❝❤ó♥❣ t❛ t➻♠ ❝→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ t♦→♥ tû✳ ▼ët ✤✐➲✉ ✤➦❝ ❜✐➺t t❤ó ✈à ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❞➠② ❧➦♣ ❋❡❥➨r ✤â ❧➔ ♥â ❝â ♣❤↕♠ ✈✐ ù♥❣ ❞ư♥❣ r➜t rë♥❣ ❞➣✐✳ P❤÷ì♥❣ ♣❤→♣ ♥➔② ✤➦❝ ❜✐➺t q✉❛♥ trå♥❣ tr♦♥❣ ❧➽♥❤ ✈ü❝ t➼♥❤ t♦→♥ ✈➔ ❦❤ỉ✐ ♣❤ư❝ ❤➻♥❤ ợ ỵ q trồ tr ữủ sỹ ữợ t tổ ❝❤å♥ ✤➲ t➔✐ ✏ ❚➼♥❤ ✤ì♥ ✤✐➺✉ ❋❡❥➨r ✈➔ ♠ët sè t❤✉➟t t♦→♥ ❧➦♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✑ ❧➔♠ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳ ❑❤â❛ ❧✉➟♥ ỗ ữỡ tr ởt số tự ỡ s t ỗ r ữỡ ♥➔② ❝â tr➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ổ rt t ỗ ỗ ởt sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ❈❤÷ì♥❣ ữỡ tr t ỡ ❋❡❥➨r ✈➔ ♠ët sè t❤✉➟t t♦→♥ ❧➦♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ t♦→♥ tû ❦❤ỉ♥❣ ❣✐➣♥✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ✶ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♠ët ❞➣② ✤ì♥ ✤✐➺✉ ❋❡❥➨r✳ ❚✐➳♣ t❤❡♦ ✤â✱ ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② t❤✉➟t t♦→♥ ❧➦♣ ❝õ❛ ❑r❛s♥♦s❡❧✬s❦✐✐✕▼❛♥♥ ✈➔ t❤✉➟t t♦→♥ ❧➦♣ ✤è✐ ✈ỵ✐ ❝→❝ t♦→♥ tû tr✉♥❣ ❜➻♥❤ ✤➸ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥✳ ✷ ❈❤÷ì♥❣ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶✳ ❑❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ◆ë✐ ❞✉♥❣ ❝õ❛ ❦❤♦→ ❧✉➟♥ ♥➔② ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✱ t❛ ❦➼ ❤✐➺✉ ❦❤æ♥❣ ❣✐❛♥ ♥➔② ❧➔ · d✱ ❧➔ ❝❤✉➞♥ tr➯♥ H H ã|ã ợ t ổ ữợ tữỡ ự ợ t ổ ữợ ợ tự (x H), (y H), t õ tỷ ỗ ♥❤➜t t❤ù❝ tr➯♥ x|x x = H ❦➼ ❤✐➺✉ ❧➔ ✈➔ d(x, y) = x − y ✭✶✳✶✮ Id✳ Ð ♣❤➛♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ❣✐ỵ✐ t❤✐➺✉ ❝→❝ ✤➥♥❣ t❤ù❝✱ ❜➜t ✤➥♥❣ t❤ù❝ ✈➔ tr➻♥❤ ❜➔② ❝→❝ ❦➳t q✉↔ ❝ì ❜↔♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ✶✳✶✳✶✳ ❑➼ ❤✐➺✉ ✈➔ ✈➼ ❞ư P❤➛♥ ❜ò trü❝ ❣✐❛♦ ❝õ❛ t➟♣ ❝♦♥ C tr♦♥❣ H✱ ❦➼ ❤✐➺✉ ❧➔ C ⊥✱ tù❝ ❧➔ C ⊥ = {u ∈ H | (∀x ∈ C) x | u = 0} ▼ët t➟♣ trü❝ ❣✐❛♦ spanC = H✳ C ⊂H ❑❤ỉ♥❣ ❣✐❛♥ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❝ì sð trü❝ ❣✐❛♦ ❝õ❛ H ✭✶✳✷✮ H ♥➳✉ ❧➔ t→❝❤ ✤÷đ❝ ♥➳✉ ♥â ❝❤ù❛ ♠ët ❝ì sð trü❝ ✸ ❣✐❛♦ ✤➳♠ ✤÷đ❝✳ ❈❤♦ (xi )i∈I ❧➔ ♠ët ❤å ❝→❝ ✈❡❝tì t❤✉ë❝ t➟♣ ❝♦♥ ❤ú✉ ❤↕♥ ❦❤→❝ ré♥❣ ❝õ❛ ⊂✳ ❑❤✐ ✤â✱ ❤ë✐ tư tỵ✐ (xi )i∈I x✱ H ✈➔ I ❧➔ ợ I ữủ s tự tỹ q tờ tỗ t xH t❤ä❛ ♠➣♥ i∈J (xi )J∈I tù❝ ❧➔ ⇒ x− (∀ε ∈ R++ )(∃K ∈ I)(∀J ∈ I) J ⊃ K (xi ) ε ✭✶✳✸✮ i∈J ❑❤✐ ✤â✱ t❛ ✈✐➳t x= i∈I (xi ) ❍å (αi )i∈I (αi ) = sup J∈I i∈I ❱➼ ❞ö ✶✳✶✳ (H, · i )i∈I [0, +∞)✱ ①→❝ ✤à♥❤ tr➯♥ (αi ) t❛ ❝â ✭✶✳✹✮ i∈J ❚ê♥❣ trü❝ t✐➳♣ ❝õ❛ ♠ët ❤å ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✳ Hi = x = (xi )i∈I ∈ ×i∈I Hi | i∈I i xi < + iI ữủ tr ợ ❝ë♥❣ (α, x) → (αxi )i∈I , (x, y) → (xi + yi )iI , ổ ữợ t ổ ữợ (x, y) xi | yi i ✭✶✳✻✮ i∈I ❑❤✐ I ❧➔ ❤ú✉ ❤↕♥ t❛ ✈✐➳t ●✐↔ sû ✈ỵ✐ ∀i ∈ I inf i∈I fi ≥ 0✳ ✱ i∈I l2 (I) = t❤❛② ❝❤♦ i∈I fi : Hi → (−∞, +∞) Hi ✳ ✈➔ ♥➳✉ I ❧➔ ❤ú✉ ❤↕♥ t❤➻ ❑❤✐ ✤â Hi → (−∞, +∞) : (xi )i∈I → fi : ❱➼ ❞ư ✶✳✷✳ ×i∈I Hi i∈I ◆➳✉ ♠é✐ i∈I R fi (xi ) ✭✶✳✼✮ i∈I Hi ❧➔ ♠ët ✤÷í♥❣ ❒❝❧✐t tr♦♥❣ ❱➼ ❞ư ✶✳✶ t ữủ ợ t ổ ữợ tỡ ỡ ❝❤✉➞♥ (ei )i∈I ❝õ❛ (x, y) = ((ξi )i∈I , (ηi )i∈I ) → l2 (I) (∀i ∈ I) ei : I → R : j → ✹ i∈I ξi ηi ✳ ✤÷đ❝ ①→❝ ✤à♥❤ 1 ♥➳✉ 0 ♥➳✉ i = j, ✭✶✳✽✮ i = j ❈❤÷ì♥❣ ✷ ❚➼♥❤ ✤ì♥ ✤✐➺✉ ❋❡❥➨r ✈➔ ♠ët sè t❤✉➟t t♦→♥ ❧➦♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ ✷✳✶✳ ❉➣② ✤ì♥ ✤✐➺✉ ❋❡❥➨r✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✈➲ ❝→❝ ❞➣② ✤ì♥ ✤✐➺✉ ❋❡❥➨r ✈➔ →♣ ❞ư♥❣ ♥â ✈➔♦ ✈✐➺❝ t❤✐➳t ❧➟♣ ❝→❝ ❦➳t q✉↔ ✈➲ sü ❤ë✐ tö ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❝ê ✤✐➸♥✳ ✣➦❝ ❜✐➺t ❧➔ ✈✐➺❝ ①➙② ❞ü♥❣ ❝→❝ ❞➣② ❧➦♣ ❤ë✐ tö ✤➳♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳ ❍✐❧❜❡rt H ✈➔ ❈❤♦ C ❧➔ ♠ët t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ (xn )n∈N ⊂ H✳ ❋❡❥➨r t÷ì♥❣ ự ợ C õ (xn )nN ữủ ❞➣② ✤ì♥ ✤✐➺✉ ♥➳✉ ∀x ∈ C, ∀n ∈ N, xn+1 − x ≤ xn − x ❱➼ ❞ö ✷✳✶✳ ❈❤♦ (xn )n∈N ❧➔ ❞➣② ❜à ❝❤➦♥ tr♦♥❣ t➠♥❣ ✭t÷ì♥❣ ù♥❣ ❣✐↔♠✮ t❤➻ ✈ỵ✐ (xn )n∈N [sup {xn }n∈N , +)(tữỡ ự ợ R (xn )n∈N ❧➔ ❞➣② ✤ì♥ ✤✐➺✉ ❋❡❥➨r t÷ì♥❣ ù♥❣ (−∞, inf {xn }n∈N ]) ❱➼ ❞ö ✷✳✷✳ H✳ ❈❤♦ D ❧➔ ♠ët t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt T :D→D T xn = xn+1 ù♥❣ ✈ỵ✐ ❈❤♦ ✈ỵ✐ ♠å✐ ❧➔ ❦❤ỉ♥❣ ❣✐➣♥✱ ✈ỵ✐ n ∈ N✳ ❑❤✐ ✤â✱ F ixT = ∅ (xn )n∈N ✈➔ ✤➦t x0 ∈ D ✳ ✣➦t ❧➔ ❞➣② ✤ì♥ ✤✐➺✉ ❋❡❥➨r t÷ì♥❣ F ixT rữợ t ú t tr ởt số t ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❞➣② ✤ì♥ ✤✐➺✉ ❋❡❥➨r✳ ❈❤♦ (xn)n∈N ❧➔ ♠ët ❞➣② tr♦♥❣ H ✈➔ ∅ = C ⊂ H ✳ ●✐↔ sû (xn )n∈N ❧➔ ✤ì♥ ✤✐➺✉ ❋❡❥➨r tữỡ ự ợ t õ t õ ♠➺♥❤ ✤➲ t÷ì♥❣ ✤÷ì♥❣✿ (i) (xn )n∈N ❧➔ ❜à ❝❤➦♥✳ (ii) ∀x ∈ C, ( xn − x )n∈N ❤ë✐ tö✳ (iii) (dC xn )n∈N ❧➔ ❣✐↔♠ ✈➔ ❤ë✐ tö✳ ▼➺♥❤ ✤➲ ✷✳✶✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮✿ ▲➜② x ∈ C ✳ ❚ø ✭✷✳✶✮ t❛ ❝â (xn)n∈N ⊂ B (x; x0 − x ) ✭✐✐✮✿ ❙✉② r❛ tø ✭✷✳✶✮✳ ✭✐✐✐✮✿ ❚ø ✭✷✳✶✮✱ ♥➳✉ x∈C t❤➻ ∀n ∈ N, dC xn+1 ≤ dC xn ❑➳t q✉↔ t✐➳♣ t❤❡♦ t❤✐➳t ❧➟♣ sỹ tử ỵ (xn )nN ❧➔ ❞➣② tr♦♥❣ H ✈➔ C ❧➔ ♠ët t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ tr♦♥❣ H ✳●✐↔ sû (xn)n∈N ❧➔ ✤ì♥ ✤✐➺✉ r tữỡ ự ợ t tử ②➳✉ t❤❡♦ ❞➣② ❝õ❛ (xn)n∈N ✤➲✉ t❤✉ë❝ C ✳ ❑❤✐ ✤â✱ (xn)n∈N ❤ë✐ tư ②➳✉ tỵ✐ ♠ët ✤✐➸♠ t❤✉ë❝ C ✳ ❱➼ ❞ö ✷✳✸✳ tr♦♥❣ H✳ ●✐↔ sû ❑❤✐ ✤â✱ H ❧➔ ✈æ ❤↕♥ ❝❤✐➲✉ ✈➔ (xn )n∈N (xn )n∈N ❧➔ ✤ì♥ ✤✐➺✉ ❋❡❥➨r ✈ỵ✐ t➟♣ ▼➺♥❤ ✤➲ ✷✳✷✳ ❈❤♦(xn )n∈N ❧➔ ❞➣② tr♦♥❣ H ✈➔ C ❧➔ ♠ët ❞➣② trỹ {0} ởt t ỗ õ ❦❤→❝ ré♥❣ tr♦♥❣ H ✳●✐↔ sû✱ ❞➣② (xn)n∈N ❧➔ ✤ì♥ r tữỡ ự ợ t C ✤â✱ ❞➣② ❤➻♥❤ ❝❤✐➳✉ (PC xn)n∈N ❤ë✐ tö ♠↕♥❤ ✤➳♥ ♠ët ✤✐➸♠ t❤✉ë❝ C ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠å✐ n ∈ N t❛ ❝â✿ PC xn − PC xn+m = PC xn − xn+m + xn+m − PC xn+m + PC xn − xn+m |xn+m − PC xn+m ≤ PC xn + d2C (xn+m ) + PC xn − PC xn+m |xn+m − PC xn+m + PC xn+m − xn+m |xn+m − PC xn+m ≤ d2C (xn ) − d2C (xn+m ) ✭✷✳✷✮ dC (xn ) ❤ë✐ tö ♥➯♥ ❞➣② (PC xn )n∈N ❧➔ ♠ët ❞➣② ❈❛✉❝❤② tr♦♥❣ ❦❤æ♥❣ ❚ø ❞➣② ❣✐❛♥ ✤õ C ❉♦ ✤â✱ t❛ ❝â ❞➣② (PC xn )n∈N ❤ë✐ tö✳ ❈❤♦ (xn)n∈N ❧➔ ♠ët ❞➣② tr♦♥❣ H✳ ❈❤♦ C ❧➔ t ỗ õ rộ H x ∈ C ✳ ●✐↔ sû (xn)n∈N ❧➔ ✤ì♥ ✤✐➺✉ r tữỡ ự ợ t C xn x ❑❤✐ ✤â✱ PC xn → x ❍➺ q✉↔ ✷✳✶✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â (PC xn) n ∈ N x − PC x n → x − y ✈➔ xn − PC xn → x − y ❱➻ ✈➟②✱ ❚ø ✤â s✉② r❛ ≥ x − PC xn | xn − PC xn → x − y ❉♦ ✤â✱ y ∈ C ❤ë✐ tư ♠↕♥❤ tỵ✐ ♠é✐ x = y✳ ▼➺♥❤ ✤➲ ✷✳✸✳ ❈❤♦ (xn )n∈N ❧➔ ♠ët ❞➣② tr♦♥❣ H ✈➔ C ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❛❢❢✐♥❡ ✤â♥❣ tr♦♥❣ H✳ ●✐↔ sû (xn)n∈N ✤ì♥ ✤✐➺✉ ❋❡❥➨r t÷ì♥❣ ù♥❣ ✈ỵ✐ t➟♣ C ✳ ❑❤✐ ✤â✱ t❛ ❝â ❝→❝ ♠➺♥❤ s❛✉✿ (i) ∀n ∈ N, PC xn = PC x0 (ii) ●✐↔ sû t➜t ❝↔ ❝→❝ ✤✐➸♠ tö ②➳✉ t❤❡♦ ❞➣② ❝õ❛ (x)n , n ∈ N ∈ C ✳ ❑❤✐ ✤â✱ xn → PC x0 ✸✾ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮✿ ❈è ✤à♥❤ n ∈ N, α ∈ R ✈➔ ✤➦t yα = αPC x0+(1 − α) PC xn ❑❤✐ ✤â✱ α2 C yα ∈ C ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❛❢❢✐♥❡ ✈➔ PC x n − PC x = ||PC xn − yα ||2 ≤ x n − PC x n ≤ x0 − y α + (2α − 1) PC x n − PC x PC xn − yα = x0 − PC x0 = d2C x0 + (1 − α)2 ❙✉② r❛ ❑❤✐ ✤â✱ ✭✷✳✶✮ trð t❤➔♥❤✿ 2 = x n − yα + PC x − y α PC xn − PC x0 2 ✭✷✳✸✮ ≤ d2C x0 ❑❤✐ α → +∞ t❛ ✤÷đ❝ PC xn = PC x ✭✐✐✮✿ ❉➵ ❞➔♥❣ s✉② r❛ tø ✭✐✮✳ ❈❤♦ (xn)n∈N ❧➔ ♠ët ❞➣② tr♦♥❣ H ✈➔ C ⊂ H, intC = ∅✳ ●✐↔ sû (xn)nN ỡ r tữỡ ự ợ C ❑❤✐ ✤â✱ (xn)n∈N ❤ë✐ tư ♠↕♥❤ tỵ✐ ♠ët ✤✐➸♠ tr♦♥❣ H✳ ▼➺♥❤ ✤➲ ✷✳✹✳ ❈❤ù♥❣ ♠✐♥❤✳ ▲➜② x ∈ intC ✈➔ ρ ∈ R++ s❛♦ ❝❤♦ B(x; ρ) ⊂ C ✣à♥❤ ♥❣❤➽❛ ❞➣② (zn )n∈N B(x; ρ) x tr♦♥❣ ∀n ∈ N, zn = ♥❤÷ s❛✉✿ x − ρ(xn+1 − xn )/( xn+1 − xn ) ❚ø ✭✷✳✶✮ ❝â (∀n ∈ N) xn+1 − zn ≤ xn − zn ♥➳✉ xn+1 = xn ♥➳✉ xn+1 = xn ✈➔ s❛✉ ❦❤✐ ❦❤❛✐ tr✐➸♥ t❛ ✤÷đ❝✿ ∀n ∈ N xn+1 − x ≤ xn − x xn+1 − xn ≤ x0 − x ❙✉② r❛ 2 /2ρ −2ρ ✈➔ xn+1 − xn (xn )n∈N ✭✷✳✹✮ ❧➔ ❞➣② ❈❛✉❝❤②✳ n∈N ❈❤♦ (xn)n∈N ❧➔ ♠ët tr H C ởt t ỗ ✤â♥❣ ✈➔ ❦❤→❝ ré♥❣ tr♦♥❣ H✳ ●✐↔ sû (xn)n∈N ❧➔ ỡ r tữỡ ự ợ C õ tữỡ ữỡ ỵ tư ♠↕♥❤ tỵ✐ ♠ët ✤✐➸♠ tr♦♥❣ ❈✳ ✭✐✐✮ (xn )n∈N ❝❤ù❛ ♠ët ✤✐➸♠ ❤ë✐ tö ♠↕♥❤ tr♦♥❣ ❈✳ ✭✐✮ (xn )n∈N ✭✐✐✐✮ lim dC (xn ) = ❈❤ù♥❣ ♠✐♥❤✳ (i) ⇒ (ii)✿ ❍✐➸♥ ♥❤✐➯♥✳ (ii) ⇒ (iii)✿ ●✐↔ sû xkn → x ∈ C ✳ ❑❤✐ ✤â✿ dC (xkn ) ≤ xkn − x → (iii) ⇒ (i)✿ ❉➵ t❤➜② dC (xn ) → 0✳ ❱➻ ✈➟②✱ xn − dC (xn ) → ✈➔ tø ✤â t❛ ❝â ✭✐✮✳ ❈❤♦ (xn)n∈N ❧➔ ♠ët ❞➣② tr♦♥❣ H ✈➔ C C ởt t ỗ õ ré♥❣ tr♦♥❣ H✳ ●✐↔ sû✱ (xn)n∈N ❧➔ ✤ì♥ ✤✐➺✉ ❋❡❥➨r tữỡ ự ợ C tỗ t k [0, 1] ỵ n N, dC (xn+1 ) ≤ kdC xn ✭✷✳✺✮ ❑❤✐ ✤â✱ (xn)n∈N ❤ë✐ tö t✉②➳♥ t➼♥❤ tỵ✐ ♠ët ✤✐➸♠ x ∈ C ❀ ❤ì♥ ♥ú❛ ∀n ∈ N, xn − x ≤ 2k n dC (x0 ) ự ứ ỵ ✈➔ ✭✷✳✸✮ s✉② r❛ (xn)n∈N ♠ët ✤✐➸♠ x ∈ C ▼➦t ❦❤→❝✱ tø ✭✷✳✶✮ t❛ ❝â xn − xn+m ≤ xn − PC xn ❈❤♦ m → +∞ t❛ ✤÷đ❝ + ✭✷✳✻✮ ❤ë✐ tư ♠↕♥❤ tỵ✐ ∀n, m ∈ N xn+m − PC xn ≤ 2dC (xn ) ✭✷✳✼✮ xn − x ≤ 2dC (xn ) ✷✳✷✳ P❤➨♣ ❧➦♣ ❑r❛s♥♦s❡❧✬s❦✐ ✲ ▼❛♥♥ ❈❤♦ ♠ët t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ ❚✱ ❞➣② ✤÷đ❝ s✐♥❤ r❛ ❜ð✐ ♣❤➨♣ ❧➦♣ ❇❛♥❛❝❤ ✲ P✐❝❛r❞ ❝õ❛ T✳ xn+1 = T xn ❝â t❤➸ ❦❤ỉ♥❣ ❤ë✐ tư ✤➳♥ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ▼ët ✈➼ ❞ư ✤ì♥ ❣✐↔♥ ❝õ❛ tr÷í♥❣ ❤đ♣ ♥➔② ❧➔ ✹✶ T = −Id ✈➔ x0 = ✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ t➼♥❤ ❝❤➼♥❤ q✉② t✐➺♠ ❝➟♥ xn − T xn → ❦❤æ♥❣ ✤÷đ❝ ✤↔♠ ❜↔♦✳ ❇➙② ❣✐í✱ t❛ s➩ ❝❤➾ r❛ t➼♥❤ ❝❤➜t ♥➔② ❝❤➼♥❤ ❧➔ ♠ët t✐➯✉ ❝❤✉➞♥ ✤➸ ❞➣② ❧➦♣ ❇❛♥❛❝❤ ✲ P✐❝❛r❞ ❤ë✐ tö ✤➳♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T D t ỗ õ ré♥❣ ❝õ❛ H✱ ❝❤♦ T : D → D ❧➔ ♠ët t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ t❤ä❛ ♠➣♥ F ixT = x0 D t ỵ (n ∈ N), xn+1 = T xn ✭✷✳✽✮ ✈➔ ❣✐↔ sû r➡♥❣ xn − T xn → 0✳ ❑❤✐ ✤â✱ t❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉✿ (i) (xn )n∈N ❤ë✐ tö ②➳✉ tỵ✐ ♠ët ✤✐➸♠ tr♦♥❣ F ixT (ii) ●✐↔ sû D = −D ✈➔ T ❧➔ t♦→♥ tû ❧➫✱ tù❝ ❧➔ (∀x ∈ D) T (−x) = −T x✳ ❑❤✐ ✤â✱ (xn )n∈N ❤ë✐ tư ♠↕♥❤ tỵ✐ ♠ët ✤✐➸♠ t❤✉ë❝ F ixT ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â (xn)n∈N ❧➔ ✤ì♥ ✤✐➺✉ r tữỡ ự ợ F ixT (i) x T xkn − xkn → 0✱ (ii)✿ ❑❤✐ ❧➔ ✤✐➸♠ tö ②➳✉ t❤❡♦ ❞➣② ❝õ❛ s✉② r❛ D = −D x F ixT ỗ l R+ xn → l ✳ s❛♦ ❝❤♦ tù❝ ❧➔ xkn x✳ ❚ø ❑❤➥♥❣ ✤à♥❤ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ 0∈D ✈➔ tø ❱➻ ✈➟②✱ tø t➼♥❤ ✤ì♥ ✤✐➺✉ ❋❡❥➨r t❛ ❝â✱(∀n t↕✐ (xn )n∈N T ❧➔ t♦→♥ tû ❧➫✱ ∈ F ixT ✳ N) xn+1 xn õ tỗ ❣✐í✱ ❧➜② m ∈ N✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ n∈N t❛ ❝â xn+1+m + xn+1 = T xn+m − T (−xn ) ≤ xn+m + xn , ✭✷✳✾✮ ✈➔ ❞♦ ✤➥♥❣ t❤ù❝ ❤➻♥❤ ❜➻♥❤ ❤➔♥❤ t❛ ❝â xn+m + xn ❚✉② ♥❤✐➯♥✱ tø l = 2( xn+m + xm ) − xn+m − xn ✭✷✳✶✵✮ T xn − xn → 0✱ t❛ ❝â lim xn+m − xn = 0✳ ❱➻ ✈➟②✱ xn → ❚ø ✭✷✳✾✮ ✈➔ ✭✷✳✶✵✮ t❤✉ ✤÷đ❝ ❦❤→❝✱ ✭✷✳✶✵✮ ❦➨♦ t❤❡♦ xn+m − xn xn+m + xn → 2l ≤ 2( xn+m ✹✷ ❦❤✐ n → +∞ ▼➦t + xm ) − 4l2 → ❦❤✐ n, m → +∞ ❚ø xn+1 → x ❑❤✐ ✤â✱ ✈➔ (xn )n∈N ❧➔ ♠ët ❞➣② ❈❛✉❝❤② ✈➔ xn+1 = T xn → T x✱ t❛ ❝â xn → x ✈ỵ✐ x ∈ D✳ x F ixT ỵ t t rsss D t ỗ õ ✈➔ ❦❤→❝ ré♥❣ ❝õ❛ H✱ ❝❤♦ T : D → D ❧➔ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ t❤ä❛ ♠➣♥ F ixT = ∅✱ ❝❤♦ (λn)n∈N ❧➔ ❞➣② tr♦♥❣ [0, 1] t❤ä❛ ♠➣♥ λn(1−λn) = n∈N +∞ ✈➔ ❝❤♦ x0 ∈ D✳ ✣➦t (∀n ∈ N) xn+1 = xn + λn (T xn − xn ) ✭✷✳✶✶✮ ❑❤✐ ✤â✱ t❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉✿ (i) (xn )n∈N ❧➔ ✤ì♥ ✤✐➺✉ ❋❡❥➨r tữỡ ự ợ F ixT (ii) (T xn xn )n∈N ❤ë✐ tư ♠↕♥❤ tỵ✐ ✵✳ (iii) (xn )n∈N ❤ë✐ tư ②➳✉ tỵ✐ ♠ët ✤✐➸♠ t❤✉ë❝ F ixT ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø x0 ∈ D ✈➔ D ❧➔ t➟♣ ỗ s r tốt (i) : ∀y ∈ F ixT, n ∈ N❀ ❚❛ ❝â xn+1 − y (1 − λn ) (xn − y) + λn T xn − T y) − λn (1 − λn ) T xn − xn ≤ xn − y ❱➻ ✈➟②✱ (xn )n∈N (ii) : − λn (1 − λn) T xn − xn n∈N t❛ ❝â lim T xn − xn = n∈N ✹✸ ✭✷✳✶✷✮ F ixT ✳ λn (1 − λn ) T xn − xn ❚ø ✭✷✳✶✷✮ t❛ ❝â 2 ỡ r tữỡ ự ợ n (1 − λn ) = +∞✱ ❚ø = (1 − λn )(xn − y) + λn (T xn − y) ≤ x0 − y ❚✉② ♥❤✐➯♥✱ ✈ỵ✐ ♠å✐ n∈N t❛ ❝â✿ T xn+1 − xn+1 = T xn+1 − T xn + (1 − λn )(T xn − xn ) ≤ xn+1 − xn + (1 − λn ) T xn − xn = T xn − xn ❙✉② r❛ ( T xn − xn )n∈N ✭✷✳✶✸✮ ❤ë✐ tö ✈➔ t❛ ❝â T xn − xn → (iii) : ❈❤♦ x ❧➔ ✤✐➸♠ ②➳✉ t❤❡♦ ❞➣② ❝õ❛ (xn )n∈N ✱ tù❝ ❧➔ xkn ✤â✱ x F ixT x ỵ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❧➔ t♦→♥ tû αi ✲ tr✉♥❣ ❜➻♥❤ t❤ä❛ ♠➣♥ F ixT = ∅✳ ❈❤♦ (λn)n∈N ❧➔ ❞➣② tr♦♥❣ [0, 1/α] t❤ä❛ ♠➣♥ λn (1 − αλn ) = +∞ ✈➔ ❝❤♦ x0 ∈ H✳ ✣➦t ▼➺♥❤ ✤➲ ✷✳✺✳ ❈❤♦ α ∈ [0, 1] ✈➔ T : H → H n∈N (∀n ∈ N)xn+1 = xn + λn (T xn − xn ) ✭✷✳✶✹✮ ❑❤✐ ✤â✱ t❛ ❝â ❦❤➥♥❣ ✤à♥❤ s❛✉✿ (i) (xn )n∈N ❧➔ ỡ r tữỡ ự ợ F ixT (ii) (T xn − xn )n∈N ❤ë✐ tư ♠↕♥❤ tỵ✐ ✵✳ (iii) (xn )n∈N ❤ë✐ tư ②➳✉ tỵ✐ ♠ët ✤✐➸♠ t❤✉ë❝ F ixT ✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t R = (1 − 1/α)Id + (1/α)T ❚❛ ❝â F ixR = F ixT xn + µ(Rxn − xn ) ❚ø ✈➔ R ✈➔ (∀n ∈ N) µn = αλn ❧➔ ❦❤ỉ♥❣ ❣✐➣♥✳ ❚❛ ✈✐➳t✿ (µn )n∈N ⊂ [0, 1]✱ (∀n ∈ N)xn+1 = µn (1 − µn ) = +∞ ✈➔ ✣à♥❤ ỵ nN s r ự q✉↔ ✷✳✷✳ ❈❤♦ T : H → H ❧➔ ♠ët t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ ❝❤➢❝ t❤ä❛ ♠➣♥ F ixT = ∅✱ ❝❤♦ (λn )n∈N ❧➔ ❞➣② t❤✉ë❝ [0, 2] t❤ä❛ ♠➣♥ λn (2 − λn ) = +∞ n∈N ✈➔ ❝❤♦ x0 ∈ H✳ ✣➦t (∀n ∈ N)xn+1 = xn + λn(T xn − xn) ❑❤✐ ✤â✱ t❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉✿ ✹✹ ❧➔ ✤ì♥ ✤✐➺✉ ❋❡❥➨r ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ tr♦♥❣ F ixT ✳ (ii) (T xn − xn )n∈N ❤ë✐ tư ♠↕♥❤ tỵ✐ ✵✳ (iii) (xn )n∈N ❤ë✐ tư ②➳✉ tỵ✐ ♠ët ✤✐➸♠ t❤✉ë❝ F ixT ✳ (i) (xn )n∈N ❱➼ ❞ö ✷✳✹✳ F ixT = ∅✱ T : H → H ❈❤♦ x0 ∈ H✳ ❝❤♦ ❈❤♦ (Ti )i∈I F ixTi = ∅ t❤ä❛ ♠➣♥ i∈I i ∈ I, Ti ♠å✐ αi ❧➔ k ∈ {1, , p}✱ mk ✈➔ ❣✐↔ sû p ❑❤✐ ✤â✱ (xn )n∈N ❧➔ ♠ët ❤å ❝→❝ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ tø ✈➔ ❤ë✐ F ixT (αi )i∈I ❧➔ ❝→❝ sè t❤ü❝ t❤✉ë❝ [0, 1] H tỵ✐ H s❛♦ ❝❤♦ ✈ỵ✐ tr✉♥❣ ❜➻♥❤✳ p ởt số ữỡ ợ ộ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣ ✈➔ wk ❧➔ ♠ët sè t❤ü❝ ❞÷ì♥❣ i : {(k, l) | k ∈ {1, , p} , l ∈ {1, , mk }} → I wk = ♠➣♥ ✲ (∀n ∈ N)xn+1 = T xn ✣➦t tư ②➳✉ tỵ✐ ♠ët ✤✐➸♠ t❤✉ë❝ ❍➺ q✉↔ ✷✳✸✳ ❧➔ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ ❝❤➢❝ t❤ä❛ ♠➣♥ k ∈ {1, , p}✱ ❱ỵ✐ ♠é✐ ✤➦t ❧➔ t♦➔♥ →♥❤ t❤ä❛ Ik = {i(k, 1), , i(k, mk )} ✈➔ k=1 ✤➦t α = max pk ✱ 1≤k≤p ð ✤â ✈ỵ✐ ♠é✐ ρk = k ∈ {1, , p} mk mk − + , α i max ✭✷✳✶✺✮ i∈Ik ✈➔ ❝❤♦ (λn )n∈N ❧➔ ❞➣② t❤✉ë❝ [0, 1] λn (1 − αλn ) = +∞ t❤ä❛ ♠➣♥ n∈N ◆❣♦➔✐ r❛✱ ❝❤♦ x0 ∈ H ✈➔ ✤➦t✿ p (∀n ∈ N) xn+1 = xn + λn wk Ti(k,1) Ti(k,mk ) xn − xn ✭✷✳✶✻✮ k=1 ❑❤✐ ✤â✱ (xn )n∈N ❤ë✐ tư ②➳✉ tỵ✐ ♠ët ✤✐➸♠ t❤✉ë❝ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t T = i∈I F ixTi p wk R k ✈ỵ✐ k=1 ❚ø ✭✷✳✶✻✮ ✈➔ ✭✷✳✺✮ s✉② r❛ T ❧➔ α ✲ k ∈ {1, , p} , Rk = Ti(k,1) Ti(k,mk ) ✳ tr✉♥❣ ❜➻♥❤ ✈➔ F ixT = F ixTi i∈I ✹✺ ❱ỵ✐ ♠é✐ k ∈ {1, , p}✱ tø ✭✷✳✶✺✮ t❤✉ ✤÷đ❝ Rk ❧➔ ρk ✲tr✉♥❣ ❜➻♥❤ ✈➔ F ixRk = F ixTi ✳ ❚ø ✭✷✳✶✺✮ ✤÷đ❝ T ❧➔ α tr✉♥❣ ❜➻♥❤ ✈➔ ✲ i∈Ik p p F ixRk = F ixT = k=1 i∈Ik k=1 F ixTi F ixTi = i∈I ◆❤➟♥ ①➨t ✷✳✶✳ ❍➺ q✉↔ ✷✳✸ ✤÷đ❝ →♣ ❞ư♥❣ tø ❝→❝ t♦→♥ tû ❦❤ỉ♥❣ ❣✐➣♥ ❝❤➢❝ ✈➔ ❞♦ ✤â ❝â t❤➸ →♣ ❞ư♥❣ ❝❤♦ ❝→❝ t♦→♥ tû ❝❤✐➳✉✳ ✷✳✸✳ ❈→❝ t❤✉➟t t♦→♥ ❝õ❛ t tỷ tr ỵ D ởt t ỗ õ t õ ỗ H m ởt sè ♥❣✉②➯♥✱ ✤➦t I = {1, , m}✱ ❝❤♦ (Ti)i∈I ❧➔ ♠ët ❤å ❝→❝ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ tø D tỵ✐ D t❤ä❛ ♠➣♥ F ix(T1 Tm) = ∅ ✈➔ ❝❤♦ (αi)i∈I ❧➔ ❝→❝ sè t❤ü❝ t❤✉ë❝ [0, 1] s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ i ∈ I, Ti ❧➔ αi ✲ tr✉♥❣ ❜➻♥❤✳ ❈❤♦ x0 ∈ D ✈➔ ✤➦t (∀n ∈ N) xn+1 = T1 Tm xn ✭✷✳✶✼✮ ❑❤✐ ✤â✱ xn T1 Tmxn tỗ t ✤✐➸♠ y1 ∈ F ixT1 Tm, y2 ∈ F ixT2 T mT1 , , ym ∈ F ixTm T1 Tm−1 t❤ä❛ ♠➣♥✿ xn Tm xn y1 = T1 y2 ✭✷✳✶✽✮ ym = Tm y1 ✭✷✳✶✾✮ ym−1 = Tm−1 ym ✭✷✳✷✵✮ T3 Tm xn y3 = T3 y4 ✭✷✳✷✶✮ T2 Tm xn y2 = T2 y3 ✭✷✳✷✷✮ Tm−1 Tm xn ✳✳ ✹✻ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t T = T1 Tm ✈➔ (∀i ∈ I), βi = (1 − αi)/αi✳ ❇➙② ❣✐í✱ ❧➜② y ∈ F ixT ✳ (i) ⇔ (ii) ❉♦ xn+1 − y s✉② r❛ = T xn − T y ≤ T2 Tm xn − T2 Tm y 2 − β1 (Id − T1 )T2 Tm xn − (Id − T1 )T2 Tm y ≤ xn − y − βm−1 (Id − Tm−1 )Tm xn − (Id − Tm−1 )Tm y − β2 (Id − T2 )T3 Tm xn − (Id − T2 )T3 Tm y − β1 (Id − T1 )T2 Tm xn − (T2 Tm y − y) ❱➻ ✈➟②✱ (xn )n∈N − βm (Id − Tm )xn − (Id − Tm )y ỡ r tữỡ ự ợ F ixT ✭✷✳✷✸✮ ✈➔ (Id − Tm )xn − (Id − Tm )y → ✭✷✳✷✹✮ (Id − Tm−1 )Tm xn − (Id − Tm−1 )Tm y → ✭✷✳✷✺✮ ✳✳ ✳ (Id − T2 )T3 Tm xn − (Id − T2 )T3 Tm y → ✭✷✳✷✻✮ (Id − T1 )T2 Tm xn − (T2 Tm y − y) → ✭✷✳✷✼✮ ❚ø ✭✷✳✷✹✮ ✲ ✭✷✳✷✼✮ t❤✉ ✤÷đ❝ xn −T xn → ❱➻ ✈➟②✱ tø T ❧➔ ❦❤æ♥❣ ❣✐➣♥ s➩ ❧➔ ❤đ♣ ❝õ❛ ❝→❝ t♦→♥ tû ❦❤ỉ♥❣ ❣✐➣♥✱ t❛ t❤✉ ✤÷đ❝ (xn )n∈N y1 ∈ F ixT Tm xn −xn → Tm y1 −y1 ❇ð✐ ✈➟② ❝â Tm xn ♥➔♦ ✤â✳ ▼➦t ❦❤→❝✱ tø ✭✷✳✷✹✮ s✉② r❛ Tm y1 = ym ✈➔ ❝❤ó♥❣ t❛ t❤✉ ✤÷đ❝ ✭✷✳✶✾✮✳ ❚÷ì♥❣ tü✱ tø ✭✷✳✷✺✮ Tm−1 Tm xn − Tm xn → Tm−1 ym − ym Tm−1 ym = ym−1 ✳ tử tợ t ữủ Tm1 Tm xn − Tm xn ❚✐➳♣ tư❝ q✉→ tr➻♥❤ t❛ ✤÷đ❝ ✭✷✳✷✵✮✳ q t t tr t ỗ sè ♥❣✉②➯♥ ❞÷ì♥❣✱ ✤➦t I = {1, , m}✱ ✹✼ ❝❤♦ (Ci )i∈I ❈❤♦ m ❧➔ ♠ët ❧➔ ❤å ❝→❝ t ỗ õ rộ x0 H✳ H✱ ❝❤♦ ●✐↔ sû r➡♥❣ (Pi )i∈I ❧➔ ❝→❝ ♣❤➨♣ t÷ì♥❣ ù♥❣ ❝õ❛ ❝❤ó♥❣ ✈➔ F ix(P1 Pm ) = ∅ ✈➔ ✤➦t (∀n ∈ N), xn+1 = P1 Pm xn õ tỗ t (y1 , , ym ) ∈ C1 .Cm ym = Pm y1 , Pm−1 Pm xn P3 y4 , P2 Pm xn ym−1 = Pm−1 ym , , P3 Pm xn i ∈ I, Cj y3 = ❧➔ ❜à ❝❤➦♥✳ ❚❤➟t ✈➟②✱ ①➨t ❤ñ♣ ✈á♥❣ q✉❛♥❤ ❝õ❛ T = Pj Pm P1 Pj1 ỡ ỳ T tứ t ỗ õ ré♥❣ ❜à ❝❤➦♥ t❤ä❛ ♠➣♥ y1 = P1 y2 , Pm xn ❚ø ❍➺ q✉↔ ✷✳✹✱ ❣✐↔ sû r➡♥❣ ✈ỵ✐ ♠å✐ F ix(P1 Pm ) = ∅✳ ❝❤✐➳✉ ❝❤♦ ❜ð✐ xn y2 = P2 y3 ◆❤➟♥ ①➨t ✷✳✷✳ ❑❤✐ ✤â✱ s❛♦ ❝❤♦ ✭✷✳✷✽✮ Cj m ♣❤➨♣ ❧➔ t♦→♥ tû ổ õ tỗ t x Cj T x = x ❈❤♦ m ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣✱ ✤➦t I = {1, , m}✱ ❝❤♦ (Ci )iI t ỗ õ H tọ ♠➣♥ C = Ci = ∅✱ ❝❤♦ (Pi )i∈I i∈I ❧➔ ♣❤➨♣ ❝❤✐➳✉ t÷ì♥❣ ù♥❣ ❝õ❛ ❝❤ó♥❣ ✈➔ ❝❤♦ x0 ∈ H✳ ✣➦t ❍➺ q✉↔ ✷✳✺✳ (∀n ∈ N), xn+1 = P1 Pm xn ❑❤✐ ✤â✱ (xn)n∈N ❤ë✐ tö ②➳✉ tỵ✐ ♠ët ✤✐➸♠ tr♦♥❣ ❈✳ ▼➺♥❤ ✤➲ ✷✳✻✳ ❈❤♦ T ∈ B(H) ❧➔ ❦❤æ♥❣ ❣✐➣♥ ✈➔ x0 ∈ H✳ ✣➦t V = F ixT ✈➔ ∀n ∈ N, xn+1 = T xn ❑❤✐ ✤â✿ xn → PV x0 ⇔ xn − xn+1 → ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ xn → PV x0 t❤➻ xn − xn+1 → PV x0 − PV x0 = ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû r➡♥❣ ✤â s✉② r xn xn+1 õ tỗ t υ∈V s❛♦ ❝❤♦ xn → υ ✳ ❚ø υ = P V x0 ❍➺ q✉↔ ✷✳✻✳ ✭✈♦♥ ◆❡✉♠❛♥♥✕❍❛❧♣❡r✐♥✮ ❞÷ì♥❣✱ ✤➦t I = {1, , m}✱ ❝❤♦ (Ci )i∈I ✹✽ ❈❤♦ m ❧➔ ♠ët sè ♥❣✉②➯♥ ❧➔ ❤å ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❛♣❤✐♥ ✤â♥❣ ❝õ❛ H s❛♦ ❝❤♦ Ci = ∅✱ C = ❝❤♦ (Pi )i∈I ❧➔ ♣❤➨♣ ❝❤✐➳✉ t÷ì♥❣ ù♥❣ ❝õ❛ i∈I ❝❤ó♥❣ ✈➔ x0 ∈ H✳ ✣➦t (∀n ∈ N), xn+1 = P1 Pm xn ❑❤✐ ✤â✱ xn → PC x0 ✹✾ ✭✷✳✷✾✮ ❑➌❚ ▲❯❾◆ ❑❤â❛ ❧✉➟♥ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ♥❤➜t ✈➲ ❝→❝ tr t ỗ ữ t ỗ ỗ tr ổ rt ổ ởt số ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❝→❝ t♦→♥ tû ❦❤ỉ♥❣ ❣✐➣♥ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ t♦→♥ tû ♥➔② ❝ơ♥❣ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❦❤â❛ ❧✉➟♥✳ ❚✐➳♣ t❤❡♦ ✤â✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♠ët ❞➣② ✤ì♥ ✤✐➺✉ ❋❡❥➨r✱ t❤✉➟t t♦→♥ ❧➦♣ ❝õ❛ ❑r❛s♥♦s❡❧✬s❦✐✐✕▼❛♥♥ ✈➔ t❤✉➟t t♦→♥ ❧➦♣ ✤è✐ ✈ỵ✐ ❝→❝ t♦→♥ tû tr✉♥❣ ❜➻♥❤ ✤➸ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥✳ ✺✵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬❆❪ ❚➔✐ ❧✐➺✉ t✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ P❤ö ❍②✱ ❬✷❪ ❍♦➔♥❣ ❚✉✢✱ ●✐↔✐ t➼❝❤ ❤➔♠✱ ◆❳❇ ❑❍ ✫ ❑❚✱ ♥➠♠ ✷✵✵✺ ❍➔♠ t❤ü❝ ✈➔ ●✐↔✐ t➼❝❤ ❤➔♠✱ ◆❳❇ ✣❍◗● ❍➔ ◆ë✐✱ ♥➠♠ ✷✵✵✸ ❬❇❪ ❚➔✐ ❧✐➺✉ t✐➳♥❣ ❆♥❤ ❈♦♥✈❡① ❆♥❛❧②s✐s ❛♥❞ ▼♦♥♦✲ t♦♥❡ ❖♣❡r❛t♦r ❚❤❡♦r② ✐♥ ❍✐❧❜❡rt ❙♣❛❝❡s✱ ❙♣r✐♥❣❡r ✷✵✶✵✳ ❬✸❪ ❍✳ ❍✳ ❇❛✉s❝❤❦❡ ❛♥❞ P✳ ▲✳ ❈♦♠❜❡tt❡s✱ ❬✹❪ ■✳ ■✳ ❊r❡♠✐♥✱ ❚❤❡ r❡❧❛①❛t✐♦♥ ♠❡t❤♦❞ ♦❢ s♦❧✈✐♥❣ s②st❡♠s ♦❢ ✐♥❡q✉❛❧✐✲ t✐❡s ✇✐t❤ ❝♦♥✈❡① ❢✉♥❝t✐♦♥s ♦♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡s✱ ❉♦❦❧✳ ❆❦❛❞✳ ◆❛✉❦ ❙❙❙❘ ✶✻✵✱ ✾✾✹✲✾✾✻ ✭✶✾✻✺✮✳ ❬✺❪ ❆✳ ❘✉s③❝③②♥✬s❦✐✱ ◆♦♥❧✐♥❡❛r ❖♣t✐♠✐③❛t✐♦♥✱ Pr❡ss✱ Pr✐♥❝❡t♦♥ ✷✵✵✻✳ ✺✶ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② ... t♦→♥ ❧➦♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✑ ❧➔♠ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳ ❑❤â❛ ỗ ữỡ tr ởt số tự ỡ s t ỗ r ❝❤÷ì♥❣ ♥➔② ❝â tr➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❝ì ổ rt t ỗ ỗ ♠ët sè ❦✐➳♥ t❤ù❝... ②➳✉✳ ❇ê ✤➲ ✶✳✽✳ ❈❤♦ T : H → K ❧✐➯♥ tö❝ ②➳✉✳ ❧➔ ♠ët t♦→♥ tû ❛❢❢✐♥❡ ❧✐➯♥ tö❝✳ ❑❤✐ ✤â✱ T H ỷ tử ữợ tự ợ ữợ (xa )aA H ∀x ∈ H✱ t❛ ❝â xa x ⇒ x ≤ lim xa ✶✵ tr♦♥❣ ✭✶✳✷✶✮ ❈❤♦ (xa)a∈A ✈➔ (ua)a∈A... ✣à♥❤ ❧➼ ✷✳✻ t❛ t❤➜② ❧➔ ❝♦♠♣❛❝t ②➳✉✱ tø ❇ê ✤➲ ✶✳✺ t❤➜② ✈➔ ❦❤→❝ ré♥❣✳ ❉♦ ✤â✱ f D C ❧➔ ✤â♥❣ t ỷ tử ữợ t õ tỗ t ởt trà ♥❤ä ♥❤➜t ❝õ❛ f tr➯♥ D✳ ❍➺ q✉↔ ✶✳✹✳ ●✐↔ sû H ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt