ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - NGUYỄN VĂN NGA PHƯƠNG PHÁP LẶP ISHIKAWA CHO MỘT HỌ VÔ HẠN CÁC ÁNH XẠ KHƠNG GIÃN LUẬN VĂN THẠC SĨ TỐN HỌC THÁI NGUN - 2019 ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - NGUYỄN VĂN NGA PHƯƠNG PHÁP LẶP ISHIKAWA CHO MỘT HỌ VÔ HẠN CÁC ÁNH XẠ KHƠNG GIÃN Chun ngành: Tốn ứng dụng Mã số : 46 01 12 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC GS.TS Nguyễn Bường THI NGUYấN - 2019 ử ỵ ▼ð ✤➛✉ ✶ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ✶✳✶ ✶✳✷ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ✶ ✷ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ổ ỗ trỡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✷ ⑩♥❤ ①↕ ✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✼ ✣✐➸♠ ❜➜t ✤ë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷✳✶ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷✳✷ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✶✺ ✷✳✶ ✣✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✳ ✳ ✳ ✶✺ ✷✳✶✳✶ ✣✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✳ ✳ ✶✺ ✷✳✶✳✷ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✷ ❈↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷✳✶ ▼ỉ t↔ ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷✳✷ ❙ü ❤ë✐ tö ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ H ổ ❍✐❧❜❡rt t❤ü❝ E ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E∗ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝õ❛ E R t➟♣ ❝→❝ sè t❤ü❝ ∅ t➟♣ rộ x ợ x I t tỷ ỗ t lp , ≤ p < ∞ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② sè ❦❤↔ tê♥❣ ❜➟❝ p l∞ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② sè ❜à ❝❤➦♥ Lp [a, b], ≤ p < ∞ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ t➼❝❤ ❜➟❝ p tr➯♥ ✤♦↕♥ [a, b] lim supn→∞ xn ❣✐ỵ✐ ❤↕♥ tr➯♥ ❝õ❛ ❞➣② sè {xn } lim inf n→∞ xn ❣✐ỵ✐ ữợ số {xn } xn x0 ❞➣② {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ x0 xn ❞➣② {xn } ❤ë✐ tö ②➳✉ ✈➲ x0 x0 J →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ❋✐①(T ) t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ✷ ▼ð ✤➛✉ ❇➔✐ t♦→♥ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❤❛② ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❧➔ ♠ët tr÷í♥❣ ❤đ♣ r✐➯♥❣ ❝õ❛ ❜➔✐ t ỗ ởt tỷ tở ❦❤→❝ ré♥❣ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ ❤❛② ✈æ ❤↕♥ t ỗ õ {Ci }iI ổ ❣✐❛♥ ❍✐❧❜❡rt H ❤❛② ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✧✳ ❇➔✐ t♦→♥ ♥➔② ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤❛✉ ♥❤÷✿ ❳û ❧➼ ↔♥❤✱ ❦❤ỉ✐ ♣❤ư❝ t➼♥ ❤✐➺✉✱ t ỵ C = (T )✱ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ T ✱ t❤➻ ✤➣ ❝â ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ✤÷đ❝ ✤➲ ①✉➜t ❞ü❛ tr➯♥ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❝ê ✤✐➸♥ ♥ê✐ t✐➳♥❣✳ ✣â ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥ ❬✷❪✱ ■s❤✐❦❛✇❛ ❬✺❪✱ ❍❛❧♣❡r♥ ❬✹❪✱ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ♠➲♠ ❬✻❪✳ ◆❤➻♥ ❝❤✉♥❣✱ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❝❤➾ ❝â sü ❤ë✐ tư ②➳✉✳ ❱➼ ❞ư✱ ❙✳ ❘❡✐❝❤ ❝❤➾ r❛ r➡♥❣ ♥➳✉ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ỗ õ rt ✈➔ ♥➳✉ ❞➣② {αn } t❤ä❛ ♠➣♥ ∞ n=0 αn (1 − αn ) = ∞ t❤➻ ❞➣② {xn } ✤÷đ❝ t↕♦ r❛ tø ♣❤÷ì♥❣ ♣❤→♣ ▼❛♥♥ ❤ë✐ tư ②➳✉ ✤➳♥ ♠ët ♣❤➛♥ tû ❝õ❛ ❋✐①(T )✳ ❱➻ ✈➟②✱ r➜t ♥❤✐➲✉ t→❝ ❣✐↔ ✤➣ ❝↔✐ t✐➳♥ ♣❤÷ì♥❣ ♣❤→♣ ▼❛♥♥ ✈➔ ■s❤✐❦❛✇❛ ✤➸ ❝â ✤÷đ❝ sü ❤ë✐ tư ♠↕♥❤ ❝❤♦ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ❈❤♦ ✤➳♥ ♥❛② ✤➣ ❝â ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ✤÷đ❝ ✤÷❛ r❛ ❞ü❛ tr➯♥ sü ❝↔✐ ❜✐➯♥ ữỡ ợ t ❧✐➯♥ q✉❛♥✳ ▼ö❝ t✐➯✉ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ❧➔ t➻♠ ❤✐➸✉ ✈➔ tr➻♥❤ ❜➔② ❧↕✐ ♠ët ❝↔✐ t✐➳♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ tr ổ ỗ t trỡ tr♦♥❣ ❜➔✐ ❜→♦ ❬✽❪ ❝æ♥❣ ❜è ♥➠♠ ✷✵✶✷✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✳ ✸ ❈❤÷ì♥❣ ✶✳ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ổ ỗ t trỡ ởt số t t P tự ữỡ ợ t❤✐➺✉ ✈➲ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣✱ tr➻♥❤ ❜➔② ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥✱ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ ❝ò♥❣ ♠ët sè ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣✳ ❈❤÷ì♥❣ ✷✳ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ❤å →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❧✉➟♥ ✈➠♥ t➟♣ tr✉♥❣ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t ❦➳t q✉↔ ❝õ❛ ❜➔✐ ❜→♦ ❬✽❪ ✈➲ ♠ët ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ❚r♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ✤➸ t→❝ ❣✐↔ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉✳ ❚→❝ ❣✐↔ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❝→❝ t❤➛②✱ ❝ỉ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✲ ❚✐♥✱ tr♦♥❣ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ✣➦❝ ❜✐➺t✱ t→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ●❙✳❚❙✳ ữớ ữớ t t ữợ t ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚→❝ ❣✐↔ ❝ô♥❣ ①✐♥ ữủ ỷ ỡ tợ trữớ ❚❍P❚ ❈❤✉②➯♥ ❇➢❝ ◆✐♥❤✱ ❇➢❝ ◆✐♥❤ ✈➔ t➟♣ t❤➸ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr♦♥❣ tê ❚♦→♥✲ ❚✐♥ ❝õ❛ ❚r÷í♥❣ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ t❤í✐ ❣✐❛♥ t→❝ ❣✐↔ t❤❛♠ ❣✐❛ ❤å❝ ❝❛♦ ❤å❝✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✵✹ ♥➠♠ ✷✵✶✾ ❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥ ◆❣✉②➵♥ ❱➠♥ ◆❣❛ ✹ ❈❤÷ì♥❣ ✶ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ổ ỗ t trỡ ởt số t t P tự ữỡ ợ t ✈➲ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣✱ tr➻♥❤ ❜➔② ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥✱ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ ❝ò♥❣ ♠ët sè ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ tê♥❣ ❤đ♣ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✲❬✽❪✳ ✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ✶✳✶✳✶ ổ ỗ trỡ E ởt ổ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ E ∗ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝õ❛ E ✳ ✣➸ ❝❤♦ ✤ì♥ ❣✐↔♥ ✈➔ t❤✉➟♥ t✐➺♥✱ t❛ sû ❞ö♥❣ ❦➼ ❤✐➺✉ ✤➸ ❝❤➾ ❝❤✉➞♥ tr➯♥ E ✈➔ E ∗ ✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ t❛ sỷ t t ữợ ổ ♣❤↔♥ ①↕✳ ▼➺♥❤ ✤➲ ✶✳✶✳✶ ✭①❡♠ ❬✷❪✱ tr❛♥❣ ✹✶✮ ❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❑❤✐ ✤â✱ ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ✭❛✮ E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕✳ ✭❜✮ ▼å✐ ❞➣② ❜à ❝❤➦♥ tr♦♥❣ E ✤➲✉ ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tö ②➳✉✳ ❙❛✉ ✤➙② ❧➔ ❦❤→✐ ♥✐➺♠ ✈➔ ♠ët sè ❝➜✉ tró❝ ❤➻♥❤ ❤å❝ ❝→❝ ổ ữ t ỗ t trỡ ổ ỗ ♠ỉ✤✉♥ trì♥✳ ✺ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ữủ ỗ t ợ x, y ∈ E, x = y ♠➔ x = 1, y = t õ x+y < ú ỵ ỏ õ t t ữợ ❝→❝ ❞↕♥❣ t÷ì♥❣ ✤÷ì♥❣ s❛✉✿ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✤÷đ❝ ỗ t x, y SE t❤ä❛ ♠➣♥ x+y = 1✱ s✉② r❛ x = y ❤♦➦❝ ✈ỵ✐ ♠å✐ x, y ∈ SE ✈➔ x = y t❛ ❝â tx + (1 − t)y < ✈ỵ✐ ♠å✐ t ∈ (0, 1)✱ tr♦♥❣ ✤â SE = {x ∈ E : x = 1} ✣à♥❤ ♥❣❤➽❛ ổ E ữủ ỗ > tỗ t () > s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ x, y ∈ E ♠➔ x = 1, y = 1, x − y ≥ ε t❛ ❧✉æ♥ ❝â x+y ≤ − δ(ε) ❉➵ t r E ởt ổ ỗ t õ ổ ỗ t ữủ ổ ú t ỗ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✱ ♥❣÷í✐ t❛ ✤÷❛ ✈➔♦ ổ ỗ ổ E ✿ δE (ε) = inf − x+y : x ≤ 1, y ≤ 1, x − y ≥ ε t ổ ỗ ổ ❣✐❛♥ ❇❛♥❛❝❤ E ❧➔ ❤➔♠ sè ①→❝ ✤à♥❤✱ ❧✐➯♥ tö❝ ✈➔ t➠♥❣ tr➯♥ ✤♦↕♥ [0; 2]✳ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ỗ t E (2) = r ổ E ỗ ✈➔ ❝❤➾ ❦❤✐ δE (ε) > 0, ∀ε > 0✳ ổ ỗ ✤➲✉ ❜➜t ❦➻ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✼ ❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥✳ ❈❤✉➞♥ tr➯♥ E ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ●➙t❡❛✉① t↕✐ ✤✐➸♠ x ∈ SE ♥➳✉ ✈ỵ✐ ♠é✐ y ∈ SE tỗ t ợ x + ty x d ( x + ty )t=0 = lim t→0 dt t ✭✶✳✶✮ ✻ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✽ ❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥✳ ❑❤✐ ✤â✿ ✭❛✮ ❈❤✉➞♥ tr➯♥ E ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ●➙t❡❛✉① ♥➳✉ ♥â ❦❤↔ ✈✐ ●➙t❡❛✉① t↕✐ ♠♦✐ x ∈ SE ✳ ✭❜✮ ❈❤✉➞♥ tr➯♥ E ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉ ♥➳✉ ✈ỵ✐ ♠♦✐ y ∈ SE ❣✐ỵ✐ ❤↕♥ tỗ t ợ x SE ✭❝✮ ❈❤✉➞♥ tr➯♥ E ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ợ x SE ợ tỗ t↕✐ ✤➲✉ ✈ỵ✐ ♠å✐ y ∈ SE ✳ ✭❞✮ ❈❤✉➞♥ tr➯♥ E ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ✤➲✉ ♥➳✉ ợ tỗ t ợ x, y SE ỵ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❑❤✐ ✤â✱ t❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉✿ ✭❛✮ ◆➳✉ E ∗ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ỗ t t E ổ trỡ E ∗ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ trì♥ t❤➻ E ❧➔ ❦❤ỉ♥❣ ỗ t ổ trỡ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ❧➔ ❤➔♠ sè ①→❝ ✤à♥❤ ❜ð✐ ρE (τ ) = sup{2−1 ( x + y + x − y ) − : x = 1, y = τ } ◆❤➟♥ ①➨t ✶✳✶✳✶✶ ▼ỉ ✤✉♥ trì♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ❧➔ ❤➔♠ sè ①→❝ ✤à♥❤✱ tử t tr [0; +) ỵ ữợ t t ố ỳ ♠ỉ ✤✉♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✈ỵ✐ ♠ỉ ✤✉♥ ỗ E ữủ ỵ ✭①❡♠ ❬✷❪✮ ❈❤♦ E ❧➔ ♠æt ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❑❤✐ ✤â t❛ ❝â τε − δX (ε) : ε ∈ [0, 2] , τ > 0✳ τε ∗ − δX (ε) : ε ∈ [0, 2] , τ > 0✳ ✭❜✮ ρE (τ ) = sup ✭❛✮ ρE ∗ (τ ) = sup ◆❤➟♥ ①➨t ✶✳✶✳✶✸ ❚ø ✣à♥❤ ỵ s r (E) = (E ) ε0 (E) ✈➔ ρ0 (E ∗ ) = , 2 tr♦♥❣ ✤â ε0 (E) = sup ε : δE (ε) = , ρ0 (E) = limτ →0 ρEτ(τ ) ✼ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✹ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✤÷đ❝ ❣å✐ ❧➔ trì♥ ✤➲✉ ♥➳✉ ρE (τ ) = τ →0 τ lim ❚ø ◆❤➟♥ ①➨t ✶✳✶✳✶✸✱ t õ ỵ ữợ ỵ ❬✷❪✮ ❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❑❤✐ ✤â t❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉✿ ✭❛✮ ◆➳✉ E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ trì♥ ✤➲✉ t❤➻ E ∗ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ỗ E ổ ỗ t❤➻ E ∗ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ trì♥ ✤➲✉✳ ✶✳✶✳✷ ⑩♥❤ ①↕ ✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✻ ⑩♥❤ ①↕ J : E → 2E ✭♥â✐ ❝❤✉♥❣ ❧➔ ✤❛ trà✮ ①→❝ ✤à♥❤ ∗ ❜ð✐ Jx = {u ∈ E ∗ : x, u = x u , u = x }, ✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✳ ❱➼ ❞ư ✶✳✶✳✶✼ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✱ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ❧➔ →♥❤ ①↕ ✤ì♥ ✈à I ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✽ ⑩♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ J : E → 2E ∗ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✤÷đ❝ ❣å✐ ❧➔ ✭✐✮ ❧✐➯♥ tư❝ ②➳✉ t❤❡♦ ❞➣② ♥➳✉ J ✤ì♥ trà ✈➔ ✈ỵ✐ ♠å✐ ❞➣② {xn } ❤ë✐ tö ②➳✉ ✤➳♥ x t❤➻ Jxn ❤ë✐ tö ②➳✉ ✤➳♥ Jx t❤❡♦ tæ♣æ ②➳✉∗ tr♦♥❣ E ∗ ✳ ✭✐✐✮ ❧✐➯♥ tư❝ ♠↕♥❤✲②➳✉ t❤❡♦ ❞➣② ♥➳✉ J ✤ì♥ trà ✈➔ ✈ỵ✐ ♠å✐ ❞➣② {xn } ❤ë✐ tö ♠↕♥❤ ✤➳♥ x t❤➻ Jxn ❤ë✐ tư ②➳✉ ✤➳♥ Jx t❤❡♦ tỉ♣ỉ ②➳✉∗ tr♦♥❣ E ∗ ✳ ◆❤➟♥ ①➨t ✶✳✶✳✶✾ ❑❤æ♥❣ ❣✐❛♥ lp, < p < ∞ ❝â →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ❧✐➯♥ tö❝ ②➳✉ t❤❡♦ ❞➣②✳ ⑩♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Lp [a, b], < p < ∞ ❦❤æ♥❣ t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t ♥➔②✳ ❚➼♥❤ ✤ì♥ trà ❝õ❛ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ❝â ♠è✐ ❧✐➯♥ ❤➺ ✈ỵ✐ t➼♥❤ ❦❤↔ ✈✐ ❝õ❛ ❝❤✉➞♥ ❝õ❛ ổ ữ tr ỵ s ỵ E ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈ỵ✐ →♥❤ ①↕ ✤è✐ ∗ ♥❣➝✉ ❝❤✉➞♥ t➢❝ J : E → 2E ✳ ❑❤✐ ✤â ❝→❝ s tữỡ ữỡ ỵ ✭①❡♠ ❬✸❪✮ ❈❤♦ H ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ C t rộ ỗ õ ❝õ❛ H ✳ ❈❤♦ {T1 , , Tr } ❧➔ ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tø C ✈➔♦ C ✈ỵ✐ F = ∩ri=1 ❋✐①(Ti ) = ∅ ✈➔ F = ❋✐①(Tr Tr−1 T1 ) = ❋✐①(T1 Tr T2 ) = · · · = ❋✐①(Tr−1 T1 Tr ) ❈❤♦ {λn } ❧➔ ♠ët ❞➣② sè t❤ü❝ tr♦♥❣ ✤♦↕♥ [0, 1) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ∞ λn = ∞, lim λn = 0, n→∞ ∞ n=0 |λn+1 − λn | < ∞ n=0 ◆➳✉ y, x0 ∈ C ✱ ❞➣② {xn } ①→❝ ✤à♥❤ ❜ð✐ xn+1 = λn y + (1 − λn )Tn+1 xn , tr♦♥❣ ✤â Tn = Tn( mod r) ✱ n ≥ 0, ✭✷✳✷✮ t❤➻ {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ PF u✳ ◆➠♠ ✷✵✵✸✱ ❏✳ ●✳ ❖✬ ❍❛r❛✱ P✳ P✐❧❧❛② ✈➔ ❍✳ ❑✳ ❳✉ ✤➣ t❤❛② ✤✐➲✉ ❦✐➺♥ λn ∞ |λ − λ | < ∞ ❜ð✐ ✤✐➲✉ ❦✐➺♥ lim = ✈➔ ❝ơ♥❣ t❤✉ ✤÷đ❝ n+1 n n→∞ n=0 λn+r sü ❤ë✐ tö ♠↕♥❤ ❝õ❛ ❞➣② ❧➦♣ ✈➲ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ T1 , , Tr ỵ H ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ ❝❤♦ C ❧➔ t➟♣ ❝♦♥ rộ ỗ õ H {T1 , , Tr } ❧➔ ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tø C ✈➔♦ C ✈ỵ✐ F = ∩ri=1 ❋✐①(Ti ) = ∅ ✈➔ F = ❋✐①(Tr Tr−1 T1 ) = ❋✐①(T1 Tr T2 ) = · · · = ❋✐①(Tr−1 T1 Tr ) ❈❤♦ {λn } ❧➔ ♠ët ❞➣② sè t❤ü❝ tr♦♥❣ ✤♦↕♥ [0, 1) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ∞ λn = ∞, lim λn = 0, n→∞ n=0 λn = n→∞ λn+r lim ◆➳✉ y, x0 ∈ C ✱ ❞➣② {xn } ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✷✮ t❤➻ {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ PF u✳ ◆➠♠ ✷✵✵✺✱ ❏✳ ❙✳ ❏✉♥❣ ✤➣ ♠ð rë♥❣ ❦➳t q✉↔ ❝õ❛ ❏✳ ●✳ ❖✬ ❍❛r❛ tr♦♥❣ ổ ự ỵ s ỵ E ởt ổ trì♥ ✤➲✉ ✈ỵ✐ →♥❤ ①↕ ✤è✐ ♥❣➝✉ j : E → E ∗ ❧✐➯♥ tö❝ ②➳✉ t❤❡♦ ❞➣② ✈➔ ❝❤♦ C ởt t rộ ỗ õ ❝õ❛ E ✳ ❈❤♦ T1 , , TN tø C ✈➔♦ ❝❤➼♥❤ ♥â ✈ỵ✐ F = ∩N i=1 ❋✐①(Ti ) = ∅ ✈➔ F = ❋✐①(Tr Tr−1 T1 ) = ❋✐①(T1 Tr T2 ) = · · · = ❋✐①(Tr−1 T1 Tr ) ✶✽ ❈❤♦ {λn } ❧➔ ♠ët ❞➣② sè t❤ü❝ tr♦♥❣ ✤♦↕♥ [0, 1) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ∞ n→∞ λn = n→∞ λn+r λn = ∞, lim λn = 0, lim n=0 ◆➳✉ y, x0 ∈ C ✱ ❞➣② {xn } ①→❝ ✤à♥❤ ❜ð✐ xn+1 = λn y + (1 − λn )Tn+1 xn , tr♦♥❣ ✤â Tn = Tn( mod r) ✱ n ≥ 0, ✭✷✳✸✮ t❤➻ {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ QF u✱ ð ✤➙② QF ❧➔ ♠ët ❝♦ rót ❦❤ỉ♥❣ ❣✐➣♥ t❤❡♦ t✐❛ tø C ❧➯♥ F ✳ ◆➠♠ ✷✵✵✼✱ ❝→❝ t→❝ ❣✐↔ ❙✳ ❙✳ ❈❤❛♥❣✱ ❏✳ ❈✳ ❨❛♦✱ ❏✳ ❑✳ ❑✐♠ ✈➔ ▲✳ ❨❛♥❣ ✤➣ ♠ð rë♥❣ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❇❛✉s❝❤❦❡ ❬✸❪ ✈➔ ❖✬ ❍❛r❛ ✤➸ ❣✐↔✐ q✉②➳t ❜➔✐ t♦→♥ ✭✷✳✶✮✱ ❤å ✤➣ ✤➲ ①✉➜t ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ xn+1 = P (αn+1 f (xn ) + (1 − αn+1 )Tn+1 xn ), n ≥ 0, tr♦♥❣ ✤â x0 ∈ E, f : C → C ❧➔ ởt trữợ Tn = Tn( mod N ) ✈➔ P ❧➔ ♠ët ❝♦ rót ❦❤ỉ♥❣ t t tứ E C ỵ ✷✳✶✳✼ ❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕ ✈ỵ✐ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ j ❧✐➯♥ tư❝ ②➳✉ t❤❡♦ ❞➣② tø E ✈➔♦ E ∗ ✳ ❈❤♦ K ởt t ỗ õ rút ❦❤ỉ♥❣ ❣✐➣♥ t❤❡♦ t✐❛ ❝õ❛ E ✈ỵ✐ P ❧➔ →♥❤ ①↕ ❝♦ rót ❦❤ỉ♥❣ ❣✐➣♥ t❤❡♦ t✐❛ tø E ❧➯♥ K ✳ ❈❤♦ f : K → K ❧➔ →♥❤ ①↕ ❝♦ ✈ỵ✐ ❤➡♥❣ sè ❝♦ ❧➔ β ∈ (0, 1) ✈➔ Ti : E → E, i = 1, 2, , N ❧➔ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✭✐✮ ∩N i=1 (❋✐①(Ti ) ∩ K) = ∅✱ ✭✐✐✮ ∩N i=1 ❋(Ti ) = ❋✐①(TN TN −1 T1 ) = · · · = ❋✐①(TN −1 T1 TN ) = ❋✐①(S)✱ ✈ỵ✐ S = TN TN −1 T1 ✱ ✐✐✐✮ →♥❤ ①↕ S : K → E t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✐♥✇❛r❞ ②➳✉✳ ❱ỵ✐ x0 ∈ K ❜➜t ❦ý✱ {xn } ❧➔ ❞➣② ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✹✮✳ ◆➳✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥ ✭❛✮ limn→∞ αn = 0, ✭❜✮ ∞ n=0 |αn+1 ∞ n=0 αn = ∞✱ − αn | < ∞ ❤♦➦❝ limn→∞ αn = 1✱ αn+r t❤➻ ❞➣② {xn } ❤ë✐ tư ♠↕♥❤ tỵ✐ ♠ët ✤✐➸♠ p ∈ ∩N i=1 (❋✐①(Ti ) ∩ K)✱ ✤â ❝ô♥❣ ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ p − f (p), j(p − u) ≤ 0, ∀u ∈ N i=1 ((Ti ) K) ú ỵ ✷✳✶✳✽ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ E ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ T1, , TN ❧➔ ❝→❝ →♥❤ ổ tứ t ỗ õ C E ✈➔♦ ❝❤➼♥❤ ♥â ✈➔ f : C → C t❤ä❛ ♠➣♥ f (x) = u, ∀x ∈ C t❤➻ ❞➣② ❧➦♣ ✭✷✳✹✮ ❝❤➼♥❤ ❧➔ ❦➳t q✉↔ ❝õ❛ ❍✳ ❍✳ s r ú ỵ K ởt t ỗ õ rộ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✱ ✈ỵ✐ ♠é✐ x ∈ K t❛ ①→❝ ✤à♥❤ t➟♣ IK (x) = {x + λ(z − x), z ∈ K, λ ≥ 0} ✈➔ ❣å✐ ❧➔ t➟♣ inward ❝õ❛ x ✤è✐ ✈ỵ✐ K ✳ ▼ët →♥❤ ①↕ S : K → E ✤÷đ❝ ❣å✐ ❧➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ inward ②➳✉✱ ♥➳✉ S(x) ∈ IK (x) ợ ộ x K Pữỡ ■s❤✐❦❛✇❛ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ◆➠♠ ✷✵✵✻✱ ❙✳ P❧✉❜t✐❡♥❣ ✈➔ ❑✳ ❯♥❣❝❤✐ttr❛❦♦♦❧ ❬✼❪ ✤➣ ♠ð rë♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✭✶✳✶✸✮ ❝❤♦ ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ❈❤♦ K ởt t ỗ õ rộ rút ổ ổ ỗ E ✈➔ T1 , T2 , , TN : K → E ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ❳→❝ ✤à♥❤ ❞➣② {xn } ❜ð✐ x1 ∈ K ✈➔    x1n = P (αn1 T1 xn + βn1 xn + γn1 u1n ),     x2 = P (α2 T2 x1 + β xn + γ u2 ), n n n n n n ✳✳   ✳     x N N N N N N n+1 = xn = P (αn TN xn + βn xn + γn un ), ✭✷✳✺✮ ✈ỵ✐ n ≥ 1✱ tr♦♥❣ ✤â P ❧➔ ♠ët →♥❤ ①↕ ❝♦ rót ❦❤ỉ♥❣ ❣✐➣♥ tø E ❧➯♥ K ❀ {αn1 }, {αn2 }, , {αnN }, {βn1 }, {βn2 }, , {βnN }, {γn1 }, {γn2 }, , {γnN } ❧➔ ❝→❝ ❞➣② sè tr♦♥❣ ✤♦↕♥ [0, 1] t❤ä❛ ♠➣♥ αni +βni +γni = ✈ỵ✐ ♠å✐ i = 1, 2, , N ✈➔ ♠å✐ n ≥ ✈➔ {u1n }, {u2n }, , {uN n } ❧➔ ❝→❝ ❞➣② ❜à ❝❤➦♥ tr♦♥❣ K ✳ ▼ët ❤å ❤ú✉ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ T1 , T2 , , TN : K → E ✈ỵ✐ F = ∩N i=1 ❋(Ti ) ✤÷đ❝ ❣å✐ ❧➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭❇✮ ♥➳✉ tỗ t ởt ổ f : [0, ) → [0, ∞) t❤ä❛ ♠➣♥ f (0) = ✈➔ f (r) > ✈ỵ✐ ♠å✐ r > s❛♦ ❝❤♦ max { x − Ti x } ≥ f (d(x, F )), 1iN x K ỵ E ởt ổ ỗ K ởt t ỗ õ rộ ✈➔ ❝♦ rót ❝õ❛ E ✳ ❈❤♦ T1 , T2 , , TN : ✷✵ K → E ❧➔ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❞➣② ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✺✮ ✈ỵ✐ ∞ i n=1 γn ✭❇✮✳ ◆➳✉ {xn} ❧➔ < ∞ ✈➔ {αni } ⊂ [ε, − ε] ✈ỵ✐ ♠å✐ i = 1, 2, , N ✈➔ ε ∈ (0, 1) t❤➻ {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ T1 , T2 , , TN ✳ ✷✳✷ ❈↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ ✷✳✷✳✶ ▼ỉ t↔ ♣❤÷ì♥❣ ♣❤→♣ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❧➔ ❝→❝❤ ♣❤ê ❜✐➳♥ ✤➸ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳ ◆❤➢❝ ❧↕✐ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ tổ tữớ ữủ ợ t Pữỡ s ữủ ợ t s ♥➠♠ ✶✾✼✹✳ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ tê♥❣ q✉→t ❤ì♥ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥✳ ◆❤÷♥❣ ❝→❝ ♥❣❤✐➯♥ ❝ù✉ ❧↕✐ t➟♣ tr✉♥❣ ✈➔♦ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥ ❝â t❤➸ ❧➔ ❞♦ ❦➳t q✉↔ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ✤ì♥ ❣✐↔♥ ❤ì♥ ❦➳t q✉↔ ữỡ s ỵ sỹ ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥ ❝â t❤➸ ❞➝♥ ỵ sỹ tử ữỡ ❧➦♣ ■s❤✐❦❛✇❛✳ ❚✉② ♥❤✐➯♥✱ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ ❝ơ♥❣ ❝â t➼♥❤ ❝❤➜t r✐➯♥❣ ❝õ❛ ♥â✳ ❱➜♥ ✤➲ t❤ü❝ t➳ ❧➔ ❝â ♥❤✐➲✉ tr÷í♥❣ ❤đ♣ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥ ❝â t❤➸ ❦❤ỉ♥❣ ❤ë✐ tư ♥❤÷♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ ❧↕✐ ❤ë✐ tư✱ ❝❤➥♥❣ ❤↕♥ tr÷í♥❣ ❤đ♣ →♥❤ ①↕ ❣✐↔ ❝♦ ▲✐♣s❝❤✐t③ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ◆❤➻♥ ❝❤✉♥❣✱ ❝↔ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ ❝❤➾ ❝❤♦ sü ❤ë✐ tö ②➳✉✳ ❱➻ ✈➟②✱ r➜t ♥❤✐➲✉ t→❝ ❣✐↔ ✤➣ ❝↔✐ t✐➳♥ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥ ✈➔ ■s❤✐❦❛✇❛ ✤➸ ❝â ✤÷đ❝ sü ❤ë✐ tư ♠↕♥❤ ❝❤♦ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳ ❱➜♥ ✤➲ ①➜♣ ①➾ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ♠ët ❤å ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐ r➜t ♥❤✐➲✉ ❝→❝ t→❝ ❣✐↔✳ ◆➠♠ ✷✵✵✶✱ ❙❤✐♠♦❥✐ ✈➔ ✷✶ ❚❛❦❛❤❛s❤✐ ✤➲ ①✉➜t →♥❤ ①↕ Wn ♥❤÷ s❛✉   Un,n+1 = I       Un,n = rn Tn Un,n+1 + (1 − rn )I,       Un,n−1 = rn−1 Tn−1 Un,n + (1 − rn−1 )I,        ···   Un,k = rk Tk Un,k+1 + (1 − rk )I,      Un,k−1 = rk−1 Tk−1 Un,k + (1 − rk−1 )I,       ···       Un,2 = r2 T2 Un,3 + (1 − r2 )I,      Wn = Un,1 = r1 T1 Un,2 + (1 − r1 )I, ✭✷✳✻✮ tr♦♥❣ ✤â r1 , r2 , · · · ❧➔ ❝→❝ sè t❤ü❝ s❛♦ ❝❤♦ ≤ rn ≤ 1, T1 , T2 , , Tn ❧➔ ❤å ✈æ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tø C ✈➔♦ ❝❤➼♥❤ ♥â✳ ❇ê ✤➲ ✷✳✷✳✶ ❈❤♦ C t ỗ õ rộ ổ ỗ T1 , T2 , ❧➔ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ❝õ❛ C ✈➔♦ ❝❤➼♥❤ ♥â t❤ä❛ ♠➣♥ ∞ n=1 ❋✐①(T ) ❦❤æ♥❣ ré♥❣ ✈➔ r1 , r2 , ❧➔ ❝→❝ sè t❤ü❝ s❛♦ ❝❤♦ < rn ≤ γ < ✈ỵ✐ n ≥ ❜➜t ❦➻✳ ❑❤✐ ✤â✱ ✈ỵ✐ x ∈ C ✈➔ k ∈ N ❜➜t ❦➻ t❤➻ ❣✐ỵ✐ limn Un,k x tỗ t ỷ ✷✳✷✳✶✱ t❛ ❝â t❤➸ ✤à♥❤ ♥❣❤➽❛ →♥❤ ①↕ W tø C ✈➔♦ ❝❤➼♥❤ ♥â ♥❤÷ s❛✉✿ W x = lim Wn x = lim Un,1 x, n→∞ n→∞ ∀x ∈ C ✭✷✳✼✮ ◆❤÷ ✈➟② →♥❤ ①↕ W ✤÷đ❝ ❣å✐ ❧➔ W ✲→♥❤ ①↕ ❝❤♦ ❜ð✐ T1 , T2 , ✈➔ r1 , r2 , ✳ ❇ê ✤➲ ✷✳✷✳✷ ❈❤♦ {xn} ✈➔ {yn} ❧➔ ❝→❝ tr ổ ỗ t T1 , T2 , ❧➔ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tø C ✈➔♦ ❝❤➼♥❤ ♥â s❛♦ ❝❤♦ ∞ n=1 ❋✐①(Tn ) ❦❤æ♥❣ ré♥❣ ✈➔ r1 , r2 , ❧➔ ❝→❝ sè t❤ü❝ s❛♦ ❝❤♦ < rn ≤ γ < ✈ỵ✐ n ≥ ❜➜t ❦➻✳ ❑❤✐ ✤â ❋✐①(W ) = ∞ n=1 ❋✐①(Tn )✳ ❉ü❛ tr➯♥ →♥❤ ①↕ W ✱ t❛ ①➙② ❞ü♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ s❛✉ ✤➙② t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ✈æ ❤↕♥ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr ổ C t ỗ ✤â♥❣ ❦❤→❝ ré♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✱ Ti : C → C ❧➔ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✈ỵ✐ i ∈ N+ ✈➔ f : C → C ❧➔ →♥❤ ✷✷ ①↕ α✲❝♦✱ {αn }✱ {βn } ✈➔ {γn } ❧➔ ❝→❝ ❞➣② sè t❤ü❝ tr♦♥❣ (0, 1)✱ ợ x0 C tũ ỵ {xn } ữủ ①➙② ❞ü♥❣ ♥❤÷ s❛✉✿    z = γn Wn xn + (1 − γn )xn ,   n yn = βn Wn zn + (1 − βn )xn ,    x n+1 = αn f (xn ) + (1 − αn )yn , ✭✷✳✽✮ ∀n ≥ 0, tr♦♥❣ ✤â Wn ✤÷đ❝ ❝❤♦ ❜ð✐ ❝ỉ♥❣ t❤ù❝ ✭✷✳✻✮✳ ❇ê ✤➲ ✷✳✷✳✸ ❈❤♦ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ trỡ C t ỗ õ E ✱ T : C → C ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✈ỵ✐ ❋✐①(T ) = ∅ ✈➔ ❝❤♦ f ∈ ΠC ✳ ❑❤✐ ✤â ❞➣② {xt } ❝❤♦ ❜ð✐ xt = tf (xt ) + (1 − t)T xt ❤ë✐ tö ♠↕♥❤ ✤➳♥ ♠ët ✤✐➸♠ tr♦♥❣ ❋✐①(T )✳ ◆➳✉ t❛ ✤à♥❤ ♥❣❤➽❛ →♥❤ ①↕ Q : Πc → ❋✐①(T ) ❜ð✐ Q(f ) := lim xt , t→0 ∀f ∈ ΠC t❤➻ Q(f ) t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉✿ (I − f )Q(f ), J(Q(f ) − p) ≤ 0, f ∈ ΠC , p ∈ ❋✐①(T ) ❇ê ✤➲ ✷✳✷✳✹ ❈❤♦ {xn} ✈➔ {yn} ❧➔ ❝→❝ ❞➣② ❜à ❝❤➦♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✈➔ {βn } ❧➔ ♠ët ❞➣② tr♦♥❣ [0, 1] ✈ỵ✐ < lim inf n→∞ βn ≤ lim supn→∞ βn < 1✳ ●✐↔ sû xn+1 = (1 − βn )yn + βn xn ✈ỵ✐ ♠å✐ n ≥ ✈➔ lim sup( yn+1 − yn − xn+1 − xn ) ≤ n→∞ ❑❤✐ ✤â limn→∞ yn − xn = 0✳ ❇ê ✤➲ ✷✳✷✳✺ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✱ t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ s❛✉✿ x+y ≤ x + y, j(x + y) , ∀x, y ∈ E, tr♦♥❣ ✤â j(x + y) ∈ J(x + y)✳ ❇ê ✤➲ ✷✳✷✳✻ ●✐↔ sû r➡♥❣ {αn} ❧➔ ❞➣② ❝→❝ sè t❤ü❝ ❦❤æ♥❣ ➙♠ s❛♦ ❝❤♦ αn+1 ≤ (1 − γn )αn + δn , ∀n ≥ 0, tr♦♥❣ ✤â {γn } ❧➔ ♠ët ❞➣② tr♦♥❣ (0, 1) ✈➔ {δn } ❧➔ ❝→❝ ❞➣② t❤ä❛ ♠➣♥✿ ✷✸ ✭❛✮ ∞ n=1 γn = ∞❀ ✭❜✮ lim supn→∞ δn ≤ ❤♦➦❝ γn ∞ n=1 |δn | < ∞✳ ❑❤✐ ✤â limn→∞ αn = ✷✳✷✳✷ ❙ü ❤ë✐ tö ❙ü ❤ë✐ tö ❝õ❛ ❞➣② ❧➦♣ ✭✷✳✽✮ ữủ tr tr ỵ s ỵ ✷✳✷✳✼ ✭①❡♠ ❬✽❪✮ ❈❤♦ C ❧➔ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ õ ỗ ổ trỡ ỗ ❝❤➦t✳ ❈❤♦ Ti : C → C ❧➔ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ✈ỵ✐ i ∈ N+ ✈➔ f : C → C ❧➔ α✲❝♦✳ ●✐↔ sû r➡♥❣ F = ∩∞ i=1 ❋✐①(Ti ) = ∅✳ ❈❤♦ {αn }, {βn } ✈➔ {γn } ❧➔ ❝→❝ ❞➣② sè t❤ü❝ tr♦♥❣ (0, 1)✳ ❈❤♦ {xn } ❧➔ ❞➣② ✤÷đ❝ ❝❤♦ ❜ð✐ ✭✷✳✽✮✳ ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t❤ä❛ ♠➣♥✳ ✭❛✮ ∞ n=0 n = , limn n = ỗ t↕✐ ❝→❝ ❤➡♥❣ sè b, b ∈ (0, 1) s❛♦ ❝❤♦ < b ≤ βn ≤ b < ợ n ỗ t số a ∈ (0, b] s❛♦ ❝❤♦ γn ≤ b−a 2−b ✈ỵ✐ ♠å✐ n ≥ 0; ✭❞✮ limn→∞ |γn+1 − γn | = 0✳ ❑❤✐ ✤â ❞➣② {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ ♠ët ✤✐➸♠ tr♦♥❣ F ✳ ❈❤ù♥❣ ♠✐♥❤✳ P❤➛♥ ự s ữủ t ố ữợ s ữợ ✶✳ ❈❤➾ r❛ r➡♥❣ ❞➣② {xn} ❜à ❝❤➦♥✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ p ∈ F ✱ t❛ ❝â zn − p ≤ γn Wn xn − p + (1 − γn ) xn − p ≤ γn xn − p + (1 − γn ) xn − p = xn − p ❉♦ xn+1 − p ≤ αn f (xn ) − p + (1 − αn ) yn − p ≤ αn f (xn ) − p + (1 − αn )(βn Wn xn − p +(1 − βn ) xn − p ) ≤ αn f (xn ) − f (p) + αn f (p) − p + (1 − αn ) xn − p ≤ [1 − αn (1 − α)] xn − p + αn f (p) − p ✷✹ ❇➡♥❣ ❜✐➳♥ ✤ê✐ ✤ì♥ ❣✐↔♥✱ t❛ ❝â ❦➳t q✉↔ s❛✉ p − f (p) , x0 − p 1−α xn − p ≤ max , ∀n ≥ ◆❤÷ ✈➟② ❞➣② {xn } ❜à ❝❤➦♥✳ ❚÷ì♥❣ tü ❞➣② {yn } ✈➔ {zn } ❝ơ♥❣ ❜à ữợ ự lim xn+1 xn = n→∞ ✣➦t ln = ✭✷✳✾✮ xn+1 − (1 − βn )xn ✳ ❚❛ ❝â βn xn+1 = βn ln + (1 − βn )xn , ∀n ≥ ✭✷✳✶✵✮ ❚✐➳♣ t❤❡♦ t❛ t➼♥❤ ln+1 − ln ✳ ❱➻ ln+1 − ln = αn+1 f (xn+1 ) + (1 − αn+1 )yn+1 − (1 − βn+1 )xn+1 βn+1 αn f (xn ) + (1 − αn )yn − (1 − βn )xn − βn αn+1 (f (xn+1 ) − yn+1 ) αn (f (xn ) − yn ) = + ✭✷✳✶✶✮ βn+1 βn +Wn+1 zn+1 − Wn zn , ♥➯♥ ln+1 − ln αn+1 αn f (xn+1 ) − yn+1 + yn − f (xn ) βn+1 βn + zn+1 − zn + Wn+1 zn − Wn zn = ✭✷✳✶✷✮ ▼➦t ❦❤→❝✱ t❛ ❝â zn+1 − zn ≤ xn+1 − xn + Wn xn − xn |γn+1 − γn | + γn+1 Wn+1 xn − Wn xn ✭✷✳✶✸✮ ❉♦ Ti ✈➔ Un,i ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✱ tø ✭✷✳✻✮ t❛ ❝â Wn+1 xn − Wn xn = r1 T1 Un+1,2 xn − r1 T1 Un,2 xn ≤ γ1 Un+1,2 xn − Un,2 xn ≤ r1 r2 Un+1,3 xn − Un,3 xn ≤ ··· ≤ r1 r2 · · · rn Un+1,n+1 xn − Un,n+1 xn n ≤ M1 ri , i=1 ✭✷✳✶✹✮ ✷✺ tr♦♥❣ ✤â M1 ≥ ❧➔ ❤➡♥❣ sè t❤➼❝❤ ❤ñ♣ s❛♦ ❝❤♦ Un+1,n+1 xn − Un,n+1 xn M1 ợ n ữỡ tü n Wn+1 zn − Wn zn ≤ M2 ✭✷✳✶✺✮ ri , i=1 tr♦♥❣ ✤â M2 ≥ ❧➔ ❤➡♥❣ sè t❤➼❝❤ ❤ñ♣ s❛♦ ❝❤♦ Un+1,n+1 zn − Un,n+1 zn ≤ M2 ✈ỵ✐ ♠å✐ n ≥ 0✳ ❑➳t ❤đ♣ ✭✷✳✶✸✮ ợ t ữủ zn+1 zn xn+1 xn + Wn xn − xn |γn+1 − γn | n ✭✷✳✶✻✮ ri + γn+1 M1 i=1 ❚❤❛② ✭✷✳✶✺✮ ✈➔ ✭✷✳✶✻✮ ✈➔♦ ✭✷✳✶✷✮✱ t❛ ✤÷đ❝ ln+1 −ln − xn+1 − xn αn+1 αn ≤ f (xn+1 ) − yn+1 + yn − f (xn ) βn+1 βn n + Wn xn − xn |γn+1 − γn | + γn+1 M1 n ri + M2 i=1 ≤ αn+1 αn f (xn+1 ) − yn+1 + yn − f (xn ) βn+1 βn ri i=1 n +M3 (|γn+1 − γn | + ri ), ✭✷✳✶✼✮ i=1 tr♦♥❣ ✤â M3 ❧➔ ❤➡♥❣ sè ❞÷ì♥❣ t❤➼❝❤ ❤đ♣ t❤ä❛ ♠➣♥ M3 = max{sup{ Wn xn − xn }, M1 , M2 } n≥0 ❚ø ✤✐➲✉ ❦✐➺♥ (a)✱ (b)✱ (d) ✈➔ < rn ≤ γ < ❞➝♥ ✤➳♥ lim sup( ln+1 − ln − xn+1 − xn ) ≤ n→∞ ❙û ❞ư♥❣ ❇ê ✤➲ ✷✳✷✳✹ t❛ ✤÷đ❝ limn→∞ ln − xn = t ủ ợ s r ữủ ự ữợ ự lim Wn xn xn = n→∞ ✭✷✳✶✽✮ ✷✻ ✣➛✉ t✐➯♥✱ t❛ ✤→♥❤ ❣✐→ ✤↕✐ ❧÷đ♥❣ xn − W n xn ≤ xn − xn+1 + |xn+1 − yn + |yn − Wn zn + |Wn zn − Wn xn ≤ xn − xn+1 + αn f (xn ) − yn + (1 − βn ) xn − Wn zn + |zn − xn ≤ xn − xn+1 + αn f (xn ) − yn + (1 − βn ) xn − Wn xn + (2 − βn )|zn − xn ≤ xn − xn+1 + αn f (xn ) − yn + (1 − βn ) xn − Wn xn + (2 − βn )γn |Wn xn − xn ❚ø ✤✐➲✉ t ữủ ữợ ự xn x n rữợ t t ❝❤➾ r❛ lim sup x∗ − f (x∗ ), J(x∗ − p) ≤ 0, n→∞ tr♦♥❣ ✤â x∗ = limt→0 xt ✈ỵ✐ xt ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❝♦ x → tf (x) + (1 − t)W x ❱➻ ✈➟② xt ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤✐➸♠ ❜➜t ✤ë♥❣ xt = tf (xt ) + (1 − t)W xt ❱➟② t❛ ❝â xt − xn = (1 − t)(W xt − xn ) + t(f (xt ) − xn ) ▼➦t ❦❤→❝✱ ✈ỵ✐ ❜➜t ❦➻ t ∈ (0, 1)✱ t❛ t❤➜② r➡♥❣ xt − xn = (1 − t)( W xt − W xn , J(xt − xn ) + W xn − xn , J(xt − xn ) ) +t f (xt ) − xt , J(xt − xn ) + t xt − xn , J(xt − xn ) ≤ (1 − t)( xt − xn + W xn − x n xt − xn ) +t f (xt ) − xt , J(xt − xn ) + t xt − xn ≤ xt − xn + W xn − x n xt − xn + t f (xt ) − xt , J(xt − xn ) ❉♦ ✤â✱ xt − f (xt ), J(xt − xn ) ≤ W xn − xn t xt − xn , ∀t ∈ (0, 1) ❚❤❡♦ ✭✷✳✶✽✮✱ lim sup xt − f (xt ), J(xt − xn ) ≤ n→∞ ✭✷✳✶✾✮ ✷✼ ∗ ❱➻ t❤ü❝ t➳ J ❧➔ ❜à ❝❤➦♥ ❧✐➯♥ t✐➳♣ ✤➲✉ tø ♠↕♥❤ tỵ✐ ②➳✉ tr♦♥❣ t➟♣ ❝♦♥ ❝õ❛ E ✱ t❛ t❤➜② | f (x∗ ) − x∗ , J(xn − x∗ ) − xt − f (xt ), J(xt − xn ) | ≤ | f (x∗ ) − x∗ , J(xn − x∗ ) − J(xn − xt ) | + f (x∗ ) − x∗ + xt − f (xt ), J(xn − xt ) ≤ f (x∗ ) − x∗ J(xn − x∗ ) − J(xn − xt ) + f (x∗ ) − x∗ + xt − f (xt ) xn − xt → ❦❤✐ x → ❱➻ ✈➟②✱ ✈ỵ✐ > tỗ t > s ∀t ∈ (0, δ) t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ f (x∗ ) − x∗ , J(xn − x∗ ) ≤ xt − f (xt ), J(xt − xn ) + ε ❚ù❝ ❧➔ lim sup f (x∗ ) − x∗ , J(xn − x∗ ) ≤ lim sup xt − f (xt ), J(xt − xn ) + ε n n ứ tũ ỵ t t❤➜② r➡♥❣ lim sup f (x∗ ) − x∗ , J(xn − x∗ ) ≤ ✭✷✳✷✵✮ n→∞ ❚❤❡♦ ❇ê ✤➲ ✷✳✷✳✺ xn+1 − x∗ = (1 − αn )(yn − x∗ ) + αn (f (xn ) − x∗ ) ≤ (1 − αn )2 yn − x∗ + 2αn f (xn ) − x∗ , J(xn+1 − x∗ ) ≤ (1 − αn )2 xn − x∗ + 2αn f (xn ) − f (x∗ ), J(xn+1 − x∗ ) + 2αn f (x∗ ) − x∗ , J(xn+1 − x∗ ) ≤ (1 − αn )2 xn − x∗ + αn α( xn − x∗ + xn+1 − x∗ ) + 2αn f (x∗ ) − x∗ , J(xn+1 − x∗ ) , ❈â ♥❣❤➽❛ ❧➔ ∗ xn+1 − x ≤ + ≤ + + (1 − αn )2 + αn α xn − x∗ − αn α 2αn f (x∗ ) − x∗ , J(xn+1 − x∗ ) − αn α 2αn (1 − α) [1 − ] xn − x∗ − αn α 2αn (1 − α) [ f (x∗ ) − x∗ , J(xn+1 − x∗ ) − αn α − α αn M4 ], ✭✷✳✷✶✮ 2(1 − α) ✷✽ tr♦♥❣ ✤â M4 ❧➔ ❤➡♥❣ sè t❤➼❝❤ ❤ñ♣ t❤ä❛ ♠➣♥ M4 ≥ supn≥1 { xn − x∗ }✳ ✣➦t jn = ✈➔ tn = 2αn (1 − α) − αn α αn f (x∗ ) − x∗ , J(xn+1 − x∗ ) + M4 1−α 2(1 − α) ❙✉② r❛ xn+1 − x∗ ≤ (1 − jn )|xn − x∗ + jn tn , ∀n ≥ ✭✷✳✷✷✮ ❚ø ✤✐➲✉ ❦✐➺♥ (a)✱ (b) ✈➔ ✭✷✳✷✵✮ t❛ ❝â ∞ jn = ∞, lim sup tn ≤ lim jn = 0, n→∞ n→∞ n=0 ❚ø ❇ê ✤➲ ✷✳✷✳✻✱ t❛ t❤➜② xn → x∗ ❦❤✐ n → ∞✳ ❱➟② t❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ①♦♥❣ ❦➳t q✉↔ tr➯♥✳ ❚÷ì♥❣ tü ❤➺ q✉↔ ❝õ❛ ✣à♥❤ ❧➼ ✷✳✷✳✼✱ t❛ ❝â ❦➳t q✉↔ s❛✉✳ ▲➜② Ti = I ✱ ❧➔ ♠ët →♥❤ ①↕✱ t❛ t❤➜② r➡♥❣ Wn = I ✈ỵ✐ ∀n tt ỗ t E tr♦♥❣ ✣à♥❤ ❧➼ ✷✳✷✳✼ ❧➔ ❦❤æ♥❣ ❝➛♥ t❤✐➳t✳ ❍➺ q✉↔ ✷✳✷✳✽ ❈❤♦ C ❧➔ t➟♣ ❝♦♥ ❦❤æ♥❣ ré♥❣ ✤â♥❣ ✈➔ ỗ ổ trỡ E T : C → C ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ s❛♦ ❝❤♦ ❋✐①(T ) = ∅ ✈➔ f : C → C ❧➔ α−❝♦✳ ❈❤♦ {αn }, {βn } ✈➔ {γn } ❧➔ ❝→❝ ❞➣② sè t❤ü❝ tr♦♥❣ (0, 1)✳ ❈❤♦ {xn } ❧➔ ❞➣② ❝❤♦ ❜ð✐ ❝→❝❤ s❛✉✿ x0 ∈ C ✈➔    z = γn T xn + (1 − γn )xn ,   n yn = βn T zn + (1 − βn )xn ,    x = α f (x ) + (1 − α )y , ∀n ≥ n+1 n n n n ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t❤ä❛ ♠➣♥✳ ✭❛✮ ∞ n=0 αn = ∞, limn→∞ αn = ỗ t số b, b ∈ (0, 1) s❛♦ ❝❤♦ < b ≤ βn ≤ b < ✈ỵ✐ ♠å✐ n ≥ 0❀ ✭❝✮ ỗ t số a (0, b] s γn ≤ b−a ✈ỵ✐ ♠å✐ n ≥ 0; 2−b ✭❞✮ limn→∞ |γn+1 − γn | = 0✳ ❑❤✐ ✤â ❞➣② {xn } ❤ë✐ tư ♠↕♥❤ tỵ✐ ♠ët sè ✤✐➸♠ tr♦♥❣ (T ) tr ỵ {n } = ✈ỵ✐ ♠å✐ n ≥ 0✱ t❛ ❝â ❦➳t q✉↔ s❛✉✳ ❍➺ q✉↔ ✷✳✷✳✾ ❈❤♦ C ❧➔ t➟♣ ỗ õ rộ tr ổ trỡ ỗ t E Ti : C → C ❧➔ ♠ët →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ✈ỵ✐ i ∈ N ✈➔ f : C → C ❧➔ α✲❝♦✳ ●✐↔ sû r➡♥❣ F = ∩∞ i=1 ❋✐①(Ti ) = ∅✳ ❈❤♦ {αn } ✈➔ {βn } ❧➔ ❝→❝ ❞➣② sè t❤ü❝ tr♦♥❣ (0, 1)✳ ❈❤♦ {xn } ❧➔ ❞➣② ❝❤♦ ❜ð✐ ❝→❝❤ s❛✉✿ x0 ∈ C ✈➔  yn = βn Wn xn + (1 − βn )xn , x n+1 = αn f (xn ) + (1 − αn )yn , ∀n ≥ 0, tr♦♥❣ ✤â Wn ❝❤♦ ❜ð✐ ✭✷✳✻✮✳ ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➲✉ t❤ä❛ ♠➣♥ ✭❛✮ ∞ n=0 αn = ∞, limn→∞ αn = ỗ t số b, b (0, 1) s❛♦ ❝❤♦ < b ≤ βn ≤ b < ✈ỵ✐ ♠å✐ n ≥ 0❀ ❑❤✐ ✤â ❞➣② {xn } ❤ë✐ tư ♠↕♥❤ tỵ✐ ♠ët sè ✤✐➸♠ tr F ú ỵ s q ợ ỵ ❝❤ó♥❣ tỉ✐ ❧➔♠ tèt t❤➯♠ ❦➳t q✉↔ ❝õ❛ ❤å tø ♠ët →♥❤ ①↕ ✤ì♥ ✤➳♥ ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝ →♥❤ ①↕✳ ✸✵ ❑➳t ❧✉➟♥ ✣➲ t➔✐ ❧✉➟♥ ✈➠♥ ✤➣ t➻♠ ❤✐➸✉ ✈➔ tr➻♥❤ ❜➔② ❧↕✐ ♠ët ❝↔✐ t✐➳♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ tr ổ ỗ t trỡ tr♦♥❣ ❜➔✐ ❜→♦ ❬✽❪ ❝ỉ♥❣ ❜è ♥➠♠ ✷✵✶✷✳ ❈ư t❤➸✿ ✭✶✮ ❚r➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ổ ỗ t trỡ ởt số t➼♥❤ ❝❤➜t✳ ●✐ỵ✐ t❤✐➺✉ ✈➲ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣✱ tr➻♥❤ ❜➔② ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥✱ ữủ s ợ t t t ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✱ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❜➔✐ t♦→♥ ♥➔②✳ ✭✸✮ ❚r➻♥❤ ❜➔② ♠ët ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr ỵ tử ữỡ ✸✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ❍♦➔♥❣ ❚ö② ✭✷✵✵✸✮✱ ❍➔♠ t❤ü❝ ✈➔ ●✐↔✐ t➼❝❤ ❤➔♠✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❚✐➳♥❣ ❆♥❤ ❬✷❪ ❘✳P✳ ❆❣❛r✇❛❧✱ ❉✳ ❖✬❘❡❣❛♥✱ ❉✳❘✳ ❙❛❤✉✱ ✭✷✵✵✾✮✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r② ❢♦r ▲✐♣s❝❤✐t③✐❛♥✲t②♣❡ ♠❛♣♣✐♥❣s ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s✱ ❙♣r✐♥❣❡r✳ ❬✸❪ ❍✳❍✳ ❇❛✉s❝❤❦❡ ✭✶✾✾✻✮✱ ✧❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❢✐①❡❞ ♣♦✐♥ts ♦❢ ❝♦♠♣♦✲s✐t✐♦♥s ♦❢ ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣s ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✧✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✷✵✷✱ ♣♣✳ ✶✺✵✕✶✺✾✳ ❬✹❪ ❇✳ ❍❛❧♣❡r♥ ✭✶✾✻✼✮✱ ✧❋✐①❡❞ ♣♦✐♥ts ♦❢ ♥♦♥❡①♣❛♥❞✐♥❣ ♠❛♣s✧✱ ❇✉❧❧✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✼✸✱ ♣♣✳ ✾✺✼✕✾✻✶✳ ❬✺❪ ❙✳ ■s❤✐❦❛✇❛ ✭✶✾✼✹✮✱ ✧❋✐①❡❞ ♣♦✐♥ts ❜② ❛ ♥❡✇ ✐t❡r❛t✐♦♥ ♠❡t❤♦❞✧✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✹✹✭✶✮✱ ♣♣✳ ✶✹✼✕✶✺✵✳ ❬✻❪ ❆✳ ▼♦✉❞❛❢✐ ✭✷✵✵✵✮✱ ✧❱✐❝♦s✐t② ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s ❢♦r ❢✐①❡❞ ♣♦✐♥t ♣r♦❜❧❡♠s✧✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✷✹✶✱ ♣♣✳ ✹✺✕✺✺✳ ❬✼❪ ❙✳ P❧✉❜t✐❡♥❣✱ ❑✳ ❯♥❣❝❤✐ttr❛❦♦♦❧ ✭✷✵✵✻✮✱ ✧❲❡❛❦ ❛♥❞ str♦♥❣ ❝♦♥✲ ✈❡r❣❡♥❝❡ ♦❢ ❢✐♥✐t❡ ❢❛♠✐❧② ✇✐t❤ ❡rr♦rs ♦❢ ♥♦♥❡①♣❛♥s✐✈❡ ♥♦♥s❡❧❢✲ ♠❛♣♣✐♥❣s✧✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥✱ ❱♦❧✳ ✷✵✵✻✱ ♣♣✳ ✶✕ ✶✷✳ ❬✽❪ ❨✉ ▲✐ ✭✷✵✶✷✮✱ ✧❈♦♥✈❡r❣❡♥❝❡ ♦❢ ♠♦❞✐❢✐❡❞ ■s❤✐❦❛✇❛ ✐t❡r❛t✐✈❡ ♣r♦❝❡ss❡s ❢♦r ❛♥ ✐♥❢✐♥✐t❡ ❢❛♠✐❧② ♦❢ ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣s✧✱ ❋✐①❡❞ P♦✐♥t ❚❤❡✲ ♦r②✱ ✶✸✭✶✮✱ ♣♣✳ ✸✵✼✕✸✶✼✳ ...ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - NGUYỄN VĂN NGA PHƯƠNG PHÁP LẶP ISHIKAWA CHO MỘT HỌ VÔ HẠN CÁC ÁNH XẠ KHƠNG GIÃN Chun ngành: Tốn ứng dụng... 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