Về phương pháp lặp krasnoselskii mann cho ánh xạ không giãn trong không gian hilbert và áp dụng

ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - NGUYỄN THỊ NGỌC MAI VỀ PHƯƠNG PHÁP LẶP KRASNOSELSKII–MANN CHO ÁNH XẠ KHÔNG GIÃN TRONG KHÔNG GIAN HILBERT VÀ ÁP DỤNG LUẬN VĂN THẠC SĨ TOÁN HỌC THÁI NGUYÊN - 2019 ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - NGUYỄN THỊ NGỌC MAI VỀ PHƯƠNG PHÁP LẶP KRASNOSELSKII–MANN CHO ÁNH XẠ KHÔNG GIÃN TRONG KHÔNG GIAN HILBERT VÀ ÁP DỤNG Chuyên ngành: Toán ứng dụng Mã số : 46 01 12 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC TS Trần Xuân Quý THÁI NGUYấN - 2019 ử ỵ ✤➛✉ ✷ ✶ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✺ ✶✳✶ ✶✳✷ ⑩♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✶ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✷ P❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✸ ⑩♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✱ →♥❤ ①↕ ✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✶ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✷ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✺ ✻ ✳ ✾ ✳ ✶✵ ✳ ✶✵ ✳ ✶✶ ✷ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ ❝❤♦ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✶✹ ✷✳✶ ✷✳✷ ✷✳✸ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ ❝❤♦ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ✷✳✶✳✶ ❇➔✐ t♦→♥ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✷ ❙ü ❤ë✐ tö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❦✐➸✉ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ s✉② rë♥❣ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✶ ❍ë✐ tö ②➳✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✷ ❍ë✐ tö ♠↕♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Ù♥❣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✶ Ù♥❣ ❞ư♥❣ ❝❤♦ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ ❉♦✉❣❧❛s✕❘❛❝❤❢♦r❞ ✳ ✷✳✸✳✷ Ù♥❣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❧✉➙♥ ♣❤✐➯♥ ❏♦❤♥ ✈♦♥ ◆❡✉♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✺ ✶✺ ✶✾ ✷✵ ✷✺ ✸✵ ✸✵ ✳ ✸✷ ✐✐ ❑➳t ❧✉➟♥ ✸✺ ❚➔✐ ❧✐➺✉ t ỵ H R R+ N ∀x A−1 I C[a, b] d(x, C) lim supn→∞ xn lim inf n→∞ xn xn → x0 xn x0 ❋✐①(T ) ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ t➟♣ ❝→❝ sè t❤ü❝ t➟♣ ❝→❝ sè t❤ü❝ ❦❤æ♥❣ ➙♠ t➟♣ ❝→❝ sè tü ♥❤✐➯♥ ợ x t tỷ ữủ t tỷ A t tỷ ỗ t t tử tr ✤♦↕♥ [a, b] ❦❤♦↔♥❣ ❝→❝❤ tø ♣❤➛♥ tû x ✤➳♥ t➟♣ ❤đ♣ C ❣✐ỵ✐ ❤↕♥ tr➯♥ ❝õ❛ ❞➣② sè {xn } ợ ữợ số {xn } {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ x0 ❞➣② {xn } ❤ë✐ tö ②➳✉ ✈➲ x0 t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ✶ ▼ð ✤➛✉ ❇➔✐ t♦→♥ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❤❛② ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❧➔ ♠ët tr÷í♥❣ ❤đ♣ r✐➯♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❝❤➜♣ ỗ ởt tỷ tở rộ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ ❤❛② ✈æ ❤↕♥ ❝→❝ t➟♣ ỗ õ {Ci }iI ổ rt H ❤❛② ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✧ ✈ỵ✐ I ❧➔ t➟♣ ❝❤➾ sè✳ ❇➔✐ t♦→♥ ♥➔② ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤❛✉ ♥❤÷✿ ①û ❧➼ ↔♥❤✱ ổ t t ỵ ❑❤✐ Ci = ❋✐①(Ti )✱ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ Ti ✈ỵ✐ i = 1, 2, , N ✱ ✤➣ ❝â ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ✤÷đ❝ ✤➲ ①✉➜t t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ {Ti }N i=1 ❞ü❛ tr➯♥ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❝ê ✤✐➸♥ ♥ê✐ t✐➳♥❣ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥✱ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❍❛❧♣❡r♥✱ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ■s❤✐❦❛✇❛✱ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛♥♦s❡❧s❦✐✐✳ ✳ ✳ ❱✐➺❝ ❝↔✐ t✐➳♥ ✈➔ ♠ð rë♥❣ ❝→❝ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ❝❤♦ ❝→❝ ❧ỵ♣ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤❛♥❣ ❧➔ ✤➲ t➔✐ t❤✉ ❤ót ✤÷đ❝ sü q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ♥❤✐➲✉ t tr ữợ ữợ sỹ ữợ r ỵ tổ t ✧❱➲ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ ❝❤♦ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ →♣ ❞ư♥❣✧ ❝❤♦ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ❝õ❛ ♠➻♥❤✳ ▼ö❝ t✐➯✉ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H tr➯♥ ❝ì sð ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛s♥♦s❡❧s❦✐✐ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥✳ ◆ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✳ ❈ư t❤➸ ♥❤÷ s❛✉✿ ❈❤÷ì♥❣ ✶✳ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H ✱ tr➻♥❤ ❜➔② ✈➲ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✱ →♥❤ ①↕ ✤ì♥ ✤✐➺✉✱ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ ✷ ✸ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❝ò♥❣ ♠ët sè t➼♥❤ ❝❤➜t✱ ❣✐ỵ✐ t❤✐➺✉ ✈➲ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ✈➔ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❝ê ✤✐➸♥ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H ✳ ❈❤÷ì♥❣ ✷✳ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ ❝❤♦ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤♦ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ❚r➻♥❤ ❜➔② ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ✤à♥❤ ỵ sỹ tử ữỡ ♣❤→♣ ❝ò♥❣ ♠ët sè ✈➼ ♠✐♥❤ ❤å❛ ❝❤♦ ✤✐➲✉ ❦✐➺♥ ✤➦t r❛ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣✳▼ët ✈➔✐ ù♥❣ ❞ư♥❣ ❝õ❛ ữỡ rsss ố ợ ữỡ t sr ✈➔ ♣❤➨♣ ❝❤✐➳✉ ❧✉➟♥ ♣❤✐➯♥ ❏♦❤♥ ✈♦♥ ◆❡✉♠❛♥♥ ❝ơ♥❣ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❝❤÷ì♥❣ ♥➔②✳ ❚r♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❡♠ ❧✉ỉ♥ ♥❤➟♥ ✤÷đ❝ sü q✉❛♥ t➙♠ ❣✐ó♣ ✤ï ✈➔ ✤ë♥❣ ✈✐➯♥ ❝õ❛ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ ❇❛♥ ●✐→♠ ❤✐➺✉✱ ♣❤á♥❣ ✣➔♦ t↕♦✱ ❑❤♦❛ ❚♦→♥ ✕ ❚✐♥✳ ợ ố ữủ õ ♠ët ♣❤➛♥ ♥❤ä ❝æ♥❣ sù❝ ❝õ❛ ♠➻♥❤ ✈➔♦ ✈✐➺❝ ❣➻♥ ❣✐ú ✈➔ ♣❤→t ❤✉② ✈➫ ✤➭♣✱ sü ❤➜♣ ❞➝♥ ❝❤♦ ỳ ỵ t ố rt ✣➙② ❝ơ♥❣ ❧➔ ♠ët ❝ì ❤ë✐ ❝❤♦ ❡♠ ❣û✐ ❧í✐ tr✐ ➙♥ tỵ✐ t➟♣ t❤➸ ❝→❝ t❤➛② ❝ỉ ❣✐↔♥❣ ✈✐➯♥ ❝õ❛ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✕ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ♥â✐ ❝❤✉♥❣ ✈➔ ❑❤♦❛ ❚♦→♥ ✕ ❚✐♥ ♥â✐ r✐➯♥❣✱ ✤➣ tr✉②➲♥ t❤ö ❝❤♦ ❡♠ ♥❤✐➲✉ ❦✐➳♥ t❤ù❝ ❦❤♦❛ ❤å❝ qỵ tr tớ ữủ ❝õ❛ tr÷í♥❣✳ ❚→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉ tr÷í♥❣ ❚❍❙❈ ◗✉❛♥❣ ❚r✉♥❣✱ ❚P ❨➯♥ ❇→✐ ❝ò♥❣ t t ỗ t ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ t❤í✐ ❣✐❛♥ ✤✐ ❤å❝ ❈❛♦ ❤å❝❀ ❝↔♠ ì♥ ❝→❝ ❛♥❤ ❝❤à ❡♠ ❤å❝ ✈✐➯♥ ❧ỵ♣ ❈❛♦ ❤å❝ ❚♦→♥ ❑✶✶ ✈➔ ❜↕♥ ❜➧ ỗ tr t→❝ ❣✐↔ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❧➔♠ ❧✉➟♥ ✈➠♥ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ✣➦❝ ❜✐➺t ❡♠ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ỡ s s tợ t r ỵ ✤➣ ❧✉æ♥ q✉❛♥ t➙♠ ➙♥ ❝➛♥ ❝❤➾ ❜↔♦✱ ✤ë♥❣ ✈✐➯♥ ú ù t t õ ỵ s s➢❝ ❝❤♦ ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ❝ô♥❣ ♥❤÷ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✳ ❈❤➦♥❣ ✤÷í♥❣ ✈ø❛ q✉❛ s➩ ỳ ợ ỵ ✤è✐ ✈ỵ✐ ❝→❝ ❛♥❤ ❝❤à ❡♠ ❤å❝ ✈✐➯♥ ❧ỵ♣ ❑✶✶ ♥â✐ ❝❤✉♥❣ ✈➔ ✈ỵ✐ ❜↔♥ t❤➙♥ ❡♠ ✹ ♥â✐ r✐➯♥❣✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ t➜t ❝↔ ♥❤ú♥❣ ♥❣÷í✐ t❤➙♥ ú ù ỗ ũ tr ✤÷í♥❣ ✈ø❛ q✉❛✳ ▼ët ❧➛♥ ♥ú❛✱ ❡♠ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥✦ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✷✷ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✾ ❍å❝ ✈✐➯♥ ◆❣✉②➵♥ ❚❤à ◆❣å❝ ▼❛✐ ❈❤÷ì♥❣ ✶ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ rt ữỡ ợ t ởt số t ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✱ →♥❤ ①↕ ✤ì♥ ✤✐➺✉✱ ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❝ò♥❣ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ ✈✐➳t tr➯♥ ❝ì sð tê♥❣ ❤đ♣ ❦✐➳♥ t❤ù❝ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✷❪✱ ❬✸❪✱ ❬✺❪✱ ❬✽❪ ✈➔ ♠ët sè t➔✐ ❧✐➺✉ ✤÷đ❝ tr➼❝❤ ❞➝♥ tr♦♥❣ ✤â✳ ✶✳✶ ⑩♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❈❤♦ H ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ✈ỵ✐ t➼❝❤ ✈ỉ ữợ , tữỡ ự {xn } ❧➔ ♠ët ❞➣② tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ H ✳ ỵ xn x {xn } ❤ë✐ tö ②➳✉ ✤➳♥ x ✈➔ xn → x ♥❣❤➽❛ ❧➔ ❞➣② {xn } ❤ë✐ tö ♠↕♥❤ ✤➳♥ x✳ ✶✳✶✳✶ ởt số t t ổ rt rữợ t t❛ ♥❤➢❝ ❧↕✐ ✤à♥❤ ♥❣❤➽❛ ✈➲ sü ❤ë✐ tö ②➳✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❉➣② {xn } tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✤÷đ❝ ❣å✐ ❧➔ ❤ë✐ tö ②➳✉ ✈➲ ♣❤➛♥ tû x ∈ H ✱ ♥➳✉ lim xn , y = x, y , n→∞ ∀y ∈ H ◆❤➟♥ ①➨t ✶✳✶✳✷✳ ❚ø t➼♥❤ ❧✐➯♥ tử t ổ ữợ s r xn x✱ t❤➻ xn x✳ ❚✉② ♥❤✐➯♥✱ ✤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❦❤ỉ♥❣ ✤ó♥❣✳ ✺ ✻ ❈❤➥♥❣ ❤↕♥ ①➨t ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ∞ |xn |2 < ∞ l := {xn } ⊂ R : n=1 ✈➔ ❣✐↔ sû ❞➣② {en } ⊂ l2 ✤÷đ❝ ❝❤♦ ❜ð✐ en = (0, , 0, ✈ỵ✐ ♠å✐ n 1✳ ❑❤✐ ✤â✱ en ✤➥♥❣ t❤ù❝ ❇❡ss❡❧✱ t❛ ❝â , 0, , 0, ), ✈à tr➼ t❤ù ♥ 0✱ ❦❤✐ n → ∞✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠é✐ y ∈ H ✱ tø ❜➜t ∞ | en , y |2 < y < ∞ n=1 ❙✉② r❛ limn→∞ en , y = 0✱ tù❝ ❧➔ en ✈➲ 0✱ ✈➻ en = ✈ỵ✐ ♠å✐ n 1✳ 0✳ ❚✉② ♥❤✐➯♥✱ {en } ❦❤ỉ♥❣ ❤ë✐ tư ♠↕♥❤ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H ✤÷đ❝ tr➻♥❤ ❜➔② tr ữợ ❈❤♦ H ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✳ ❑❤✐ ✤â✿ (i) x + y x + x + y, y (ii) x + y = x + y + x, y (iii) tx + (1 − t)y = t x ✈➔ ♠å✐ x, y ∈ H ✳ + (1 − t) y ∀x, y ∈ H ✈ỵ✐ ♠å✐ x, y ∈ H ❀ − t(1 − t) x − y ✈ỵ✐ ♠å✐ t ∈ [0, 1] ▼å✐ ❞➣② ❜à ❝❤➦♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❣✐❛♥ ❍✐❧❜❡rt ✤➲✉ ❝❤ù❛ ♠ët ❞➣② ❝♦♥ ❤ë✐ tö ②➳✉✳ ❇ê ✤➲ ✶✳✶✳✹✳ ✭①❡♠ ❬✷❪✮ ✶✳✶✳✷ P❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❈❤♦ C ❧➔ ♠ët t➟♣ ỗ õ rộ tr ổ rt tỹ H ✳ ❑❤✐ ✤â ✈ỵ✐ ♠é✐ x ∈ H ✱ tỗ t t tỷ Pc x C s❛♦ ❝❤♦ x − PC x ≤ x − y ✈ỵ✐ ♠å✐ y ∈ C ✭✶✳✶✮ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤➟t ✈➟②✱ ✤➦t d = u∈C inf x − u ✳ ❑❤✐ õ tỗ t {un } C ✶✳✶✳✺✳ ✭①❡♠ ❬✷❪✮ s❛♦ ❝❤♦ x − un → d ❦❤✐ n → ∞✳ ❚ø ✤â✱ un − um = (x − un ) − (x − um ) ứ ữợ t õ lim ||xn T xn || = n ữợ ự ♠✐♥❤ sü ❤ë✐ tư ②➳✉ ❝õ❛ ❞➣② {xn } tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ✳ ❱➻ ❞➣② {xn } tỗ t {xnk } ❝õ❛ ❞➣② {xn } ❤ë✐ tö ②➳✉ ✤➳♥ ✤✐➸♠ p✳ ❉➵ ❞➔♥❣ t❤➜② p ∈ ❋✐①(T )✳ ❱➼ ❞ö ữợ ự tọ (a) (b) (c) tr ỵ tt ✭①❡♠ ❬✹❪✮✳ ▲➜② T : R → R, T x := max{0, −x}✳ ❑❤✐ ✤â T ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✈➔ ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ x = 0✳ ❳➨t ❞➣② ❧➦♣ xn+1 := αn xn + βn T xn + rn ✈ỵ✐ βn = ❦❤✐ ✤â xn+1 = 1− 1 , α = − n n2 n2 ✈➔ rn := 0, 1 x + max{0, −xn } n n2 n2 ❚❛ t❤➜② {αn }✱ {βn }✱ ✈➔ {rn } t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (b), (c) ỵ tr tt (a) ❦❤æ♥❣ t❤ä❛ ♠➣♥ ✈➻ ∞ ∞ αn βn = n=1 n=1 1 − n2 n2 ❈❤å♥ x1 = −1✱ s✉② r❛ xn = ∞ = n=1 1 − n2 n2 ∞ n=1 < ∞ n2 n ✈ỵ✐ ♠å✐ n ≥ 2✳ ❱➻ ✈➟② 2(n − 1) xn = n → = 0, 2(n − 1) ♥❣❤➽❛ ❧➔ ❞➣② {xn } ❤ë✐ tư✱ ♥❤÷♥❣ ❦❤ỉ♥❣ ♣❤↔✐ ❤ë✐ tư ✤➳♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❝õ❛ →♥❤ ①↕ T ✳ ❱➼ ❞ö t✐➳♣ t❤❡♦ ❝❤➾ r ỵ ổ ú (b) ❦❤æ♥❣ t❤ä❛ ♠➣♥ tr♦♥❣ ❦❤✐ ❝→❝ ✤✐➲✉ ❦✐➺♥ (a) ✈➔ (c) t❤ä❛ ♠➣♥✳ ❱➼ ❞ư ✷✳✷✳✸ ✭①❡♠ ❬✹❪✮✳ ❚÷ì♥❣ tü ✈➼ ❞ö tr➯♥✱ ①➨t T : R → R, T x := max{0, −x}✳ ❳→❝ ✤à♥❤ ❞➣② {xn } ❜ð✐ xn+1 := αn xn + βn T xn + rn ✈ỵ✐ βn = 1 , αn = − ✈➔ rn = n n n ❑❤✐ ✤â ❞➣② ❧➦♣ trð t❤➔♥❤ xn+1 := 1− 1 xn + max{0, −xn } + n n n ✷✹ ❉➣② {αn }✱ {βn } ✈➔ rn t❤ä❛ ♠➣♥ (a) (c) ỵ ữ n ✤✐➲✉ ❦✐➺♥ (b) ❦❤æ♥❣ t❤ä❛ ♠➣♥✳ ❈❤å♥ ✤✐➸♠ x1 = −1✱ s✉② r❛ xn = ✈ỵ✐ n−1 ♠å✐ n ≥ 2✳ ❚❛ ❝â xn → 1✳ ❚✉② ♥❤✐➯♥ ❣✐ỵ✐ ❤↕♥ ♥➔② ❦❤æ♥❣ ♣❤↔✐ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ✳ ❱➼ ❞ö t✐➳♣ t❤❡♦ ❝❤➾ r❛ ✣à♥❤ ỵ ổ ú (c) ổ tọ ♠➣♥ tr♦♥❣ ❦❤✐ ❝→❝ ✤✐➲✉ ❦✐➺♥ (a) ✈➔ (b) t❤ä❛ ♠➣♥✳ ❱➼ ❞ö ✷✳✷✳✹ ✭①❡♠ ❬✹❪✮✳ ❳➨t T : R → R ①→❝ ✤à♥❤ ❜ð✐ T x = −x + ✈ỵ✐ ♠å✐ x ∈ R✳ ❑❤✐ ✤â T ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✈➔ ❋✐①(T ) = {1}✳ ◆❣♦➔✐ r❛✱ t❛ ❝❤å♥ αn := βn := √ ❑❤✐ ✤â ∞ , n+3 ∞ αn βn = n=1 ✈➔ n=1 = ∞, n+3 ∞ ✈ỵ✐ ♠å✐ n ≥ rn := ∞ |rn | = < ∞, n=1 ∞ 1− √ (1 − αn − βn ) = n=1 n=1 n+3 = ∞ ❙✉② r❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ (a)✱ (b) t❤ä❛ ♠➣♥✱ tr♦♥❣ ❦❤✐ ✤✐➲✉ (c) ổ tọ tr ỵ t❤❡♦✱ ✈ỵ✐ ❜➜t ❦ý ✤✐➸♠ ❜❛♥ ✤➛✉ x1 ∈ R✱ t❤❡♦ ❝æ♥❣ t❤ù❝ ❧➦♣ ✭✷✳✶✵✮✱ t❛ ❝â 1 xn + √ (−xn + 2) n+3 n+3 =√ (xn − xn + 2) = √ → 0, n → ∞ n+3 n+3 xn+1 : = αn xn + βn T xn + rn = √ ◆❤÷♥❣ ∈ / ❋✐①(T )✳ ❱➻ ✈➟②✱ ❞➣② {xn } ❦❤æ♥❣ ❤ë✐ tư tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ✳ ứ ỵ t õ q s K t ỗ õ rộ ổ ❣✐❛♥ ❍✐❧❜❡rt H ✳ ●✐↔ sû T : H → K ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ s❛♦ ❝❤♦ ❋✐①(T ) ❦❤→❝ ré♥❣✳ ❳➨t ❞➣② xn tr♦♥❣ H ①→❝ ✤à♥❤ ❜ð✐✿ ❍➺ q✉↔ ✷✳✷✳✺ ✭①❡♠ ❬✹❪✮✳ x1 ∈ H, ✈ỵ✐ en ❧➔ s❛✐ sè✱ αn + βn + γn = xn+1 := αn + βn T xn + γn en ✈➔ {αn}✱ {βn}✱ {γn} ❧➔ ❝→❝ ❞➣② sè ❦❤æ♥❣ ➙♠ t❤ä❛ ♠➣♥ s❛♦ ❝❤♦ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t❤ä❛ ♠➣♥ ✷✺ ∞ αn βn = ∞; (a) n=1 ∞ (b) γn ||en || < ∞❀ n=1 ∞ (c) γn < ∞✳ n=1 ❑❤✐ ✤â ❞➣② {xn} ❤ë✐ tư ②➳✉ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ✳ ✷✳✷✳✷ ❍ë✐ tử r t tr ỵ ❤ë✐ tö ♠↕♥❤ ❝õ❛ ❞➣② ❧➦♣ ❑r❛s♥♦s❡❧s❦✐✐✕ ▼❛♥♥ tr♦♥❣ ❬✹❪✳ K t ỗ õ rộ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✱ T : H → K ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ t❤ä❛ ♠➣♥ ❋✐①(T ) ❦❤→❝ ré♥❣✳ ❳➨t ❞➣② {xn} tr♦♥❣ H ①→❝ ✤à♥❤ ❜ð✐✿ ✣à♥❤ ỵ x1 H, xn+1 = n u + αn xn + βn T xn + rn , ∀n ≥ 1, ✭✷✳✶✷✮ tr♦♥❣ ✤â u ∈ K tỡ trữợ rn ữ sè t❤ü❝ ❦❤æ♥❣ ➙♠ {αn }✱ {βn }✱ {γn } ✤÷đ❝ ❝❤å♥ s❛♦ ❝❤♦ αn + βn + σn 1✱ n ≥ 1✱ ✈➔ t❤ä❛ ♠➣♥ ∞ δn = ∞; (a) lim δn = 0, n→∞ n=1 (b) lim inf αn βn > 0; n→∞ ∞ (1 − αn − βn − δn ) < ∞; (c) n=1 ∞ ||rn || < ∞ (d) n=1 ❑❤✐ ✤â ❞➣② {xn} s✐♥❤ ❜ð✐ ✭✷✳✶✷✮ ❤ë✐ tö ♠↕♥❤ ✈➲ ✤✐➸♠ t❤✉ë❝ ❋✐①(T )✳ ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t x∗ ∈ ❋✐①(T )✳ ❑❤✐ ✤â tø ✭✷✳✶✷✮✱ t➼♥❤ ❦❤æ♥❣ ❣✐➣♥ ❝õ❛ →♥❤ ①↕ T ✈➔ αn + βn ||xn+1 − x∗ || − δn t❛ ❝â δn ||u − x∗ || + αn ||xn − x∗ || ✷✻ + βn ||T xn − x∗ || + ||rn − (1 − αn − βn − δn )x∗ || δn ||u − x∗ || + (αn + βn )||xn − x∗ || + ||rn − (1 − αn − βn − δn )x∗ || δn ||u − x∗ || + (1 − δn )||xn − x∗ || + (1 − αn − βn − δn )||x∗ || + ||rn || max{||u − x∗ ||, ||xn − x∗ ||} + (1 − αn − βn − δn )||x∗ || + ||rn || ❇➡♥❣ q✉② ♥↕♣✱ t❛ ❝â ∗ ||xn+1 −x || n ∗ max{||u−x ||, ||x1 −x ||}+ ||rk ||+||x || k=1 ✈ỵ✐ ♠å✐ n ∈ N✳ ❚ø ✤➙② t❛ ❝â ||xn+1 − x∗ || n ∗ ∗ (1−αk −βk −δk ) k=1 max{||u − x∗ ||, ||x1 − x∗ ||} ∞ ∞ ∗ ||rk || + ||x || + k=1 (1 − αk − βk − δk ) ✭✷✳✶✸✮ k=1 ✣➦❝ ❜✐➺t✱ t❤❡♦ ❝→❝ ❣✐↔ t❤✐➳t (c) ✈➔ (d) t❛ ❝â ❞➣② {xn } ❜à ❝❤➦♥ tr♦♥❣ H t z := P(T ) u tỗ t (T ) ỗ õ rộ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣ s❛✉✳ ❚r÷í♥❣ ❤đ♣ ✶✳ ●✐↔ sû tỗ t n0 N s {||xn z||}∞ n=n0 ❦❤æ♥❣ ∞ t➠♥❣✳ ❑❤✐ ✤â ❞➣② {||xn − z||}n=1 ❤ë✐ tö✱ ✈➻ ✈➟② t❛ ❝â ||xn − z|| − ||xn+1 − z|| → 0, n → ∞ ✭✷✳✶✹✮ ❑❤✐ ✤â tø ✭✷✳✶✷✮ ✈➔ ❇ê ✤➲ ✶✳✶✳✸ t❛ ❝â ||xn+1 − z||2 ||αn (xn − z) + βn (T xn − z)||2 + δn (u − z) + rn − (1 − αn − βn − δn )z, xn+1 − z = αn (αn + βn )||xn − z||2 + βn (αn + βn )||T xn − z||2 − αn βn ||xn − T xn ||2 + δn (u − z) + rn − (1 − αn − βn − δn )z, xn+1 − z (αn + βn )2 ||xn − z||2 − αn βn ||xn − T xn ||2 + δn (u − z) + rn − (1 − αn − βn − δn )z, xn+1 − z = (1 − δn )||xn − z||2 − αn βn ||xn − T xn ||2 + 2δn u − z, xn+1 − z + rn − (1 − αn − βn − δn )z, xn+1 − z ||xn − z||2 − αn βn ||xn − T xn ||2 + 2δn u − z, xn+1 − z + rn − (1 − αn − βn − δn )z, xn+1 − z ✭✷✳✶✺✮ ❙û ❞ö♥❣ t➼♥❤ ❜à ❝❤➦♥ ❝õ❛ ❞➣② {xn }✱ s✉② r❛ αn βn ||xn − T xn ||2 ≤||xn − z||2 − ||xn+1 − z||2 + δn M5 ✭✷✳✶✻✮ ✷✼ + (1 − αn − βn − δn )M6 + ||rn ||M7 ✈ỵ✐ M5 , M6 , M7 > 0✳ ❚ø ✤✐➲✉ ❦✐➺♥ (b) ổ t t tờ qt tỗ t > s❛♦ ❝❤♦ αn βn ≥ ✈ỵ✐ ♠å✐ n ≥ 1✳ ❱➻ ✈➟②✱ tø ✭✷✳✶✻✮ ✈➔ ✭✷✳✶✹✮ ✈➔ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❛✮✱ ✭❝✮✱ ✭❞✮ t❛ t❤✉ ✤÷đ❝ lim ||xn − T xn || = 0✳ ❱➻ ❞➣② {xn } ❜à ❝❤➦♥✱ n tỗ t {xnk } {xn } s❛♦ ❝❤♦ lim sup u − z, xn − z = lim u − z, xnk − z k→∞ n→∞ ✈➔ ❞➣② {xnk } ❤ë✐ tư ②➳✉ tỵ✐ p ỵ t õ t r ✤÷đ❝ p ∈ ❋✐①(T )✳ ❱➻ ✈➟②✱ t❛ tø t➼♥❤ ❝❤➜t ❝õ❛ →♥❤ ①↕ ❝❤✐➳✉ t❛ ❝â lim sup u − z, xn+1 − z = lim sup u − z, xn − z n→∞ n→∞ = lim sup u − z, xnk − z k→∞ = u − z, p − z ❚ø ✭✷✳✶✺✮ t❛ ❝â ||xn+1 − z||2 (1 − δn )||xn − z||2 − αn βn ||xn − T xn ||2 + 2δn u − z, xn+1 − z + rn − (1 − αn − βn − δn )z, xn+1 − z (1 − δn )||xn − z||2 + 2δn u − z, xn+1 − z ✭✷✳✶✼✮ + (1 − αn − βn − δn )M6 + ||rn ||M7 ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✶✳✷ ✈➔ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❛✮✱ ✭❝✮✱ ✭❞✮✱ t❛ ❝â lim ||xn − z|| = 0✳ n→∞ ❱➻ ✈➟②✱ xn → z = P❋✐①(T ) u ❦❤✐ n → ∞✳ rữớ ủ sỷ ổ tỗ t n0 N t❤ä❛ ♠➣♥ {||xn − z||}∞ n=n0 ✤ì♥ ✤✐➺✉ t➠♥❣✳ ✣➦t Γn = ||xn − z|| ✈ỵ✐ ♠å✐ n ≥ ✈➔ ①➨t τ : N → N ❧➔ →♥❤ ①↕ ①→❝ ✤à♥❤ ✈ỵ✐ ♠å✐ n ≥ n0 ✭✈ỵ✐ n0 ✤õ ❧ỵ♥✮ tø τ (n) := max{k ∈ N : k n, Γk Γk+1 }, ♥❣❤➽❛ ❧➔ τ (n) ❧➔ sè ❧ỵ♥ ♥❤➜t k ∈ {1, , n} t❤ä❛ ♠➣♥ k t➠♥❣ t↕✐ k = τ (n) ú ỵ r tr rữớ ủ ợ (n) ❤♦➔♥ t♦➔♥ ①→❝ ✤à♥❤ ✈ỵ✐ n ✤õ ❧ỵ♥✳ ❘ã r➔♥❣✱ τ ❧➔ ❞➣② ❦❤æ♥❣ ❣✐↔♠ t❤ä❛ ♠➣♥ τ (n) → ∞ ❦❤✐ n → ∞ ✈➔ Γτ (n) Γτ (n)+1 , ∀n ≥ n0 ❚❛ ❝â ||xτ (n) − T xτ (n) || → 0✳ ◆❣♦➔✐ r❛✱ tø t➼♥❤ ❜à ❝❤➦♥ ❝õ❛ ❞➣② {xn } ✈➔ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❛✮✱ ✭❝✮✱ ✭❞✮✱ t❛ ❝â ||xτ (n)+1 − xτ (n) δτ (n) ||u − xτ (n) || + βτ (n) ||T xτ (n) − xτ (n) || ✷✽ + ||rτ (n) − (1 − ατ (n) − βτ (n) − δτ (n) )xτ (n) || → 0, ✭✷✳✶✽✮ n → ∞ ❱➻ ❞➣② {xτ (n) } ❜à ❝❤➦♥✱ tỗ t {x (n) } tử tợ p (T ) ữỡ tỹ ữ tr ❚r÷í♥❣ ❤đ♣ ✶ ✈➔ ✭✷✳✶✾✮✱ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ lim sup u − z, xτ (n) − z n→∞ ❚❤❡♦ ✭✷✳✶✼✮ t❛ ❝â ||xτ (n)+1 − z||2 (1 − δτ (n) )||xτ (n) − z||2 ) + 2δτ (n) u − z, xτ (n)+1 − z + (1 − ατ (n) − βτ (n) − δτ (n) )M6 + ||rτ (n) ||M7 ✭✷✳✶✾✮ ❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✷ ✈➔ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❛✮✱ ✭❝✮✱ ✭❞✮✱ tø ✭✷✳✶✾✮ t❛ ❝â lim ||xτ (n) − z|| = n→∞ s✉② r❛ lim ||xτ (n)+1 − z|| = ❤❛② t÷ì♥❣ ✤÷ì♥❣ ✈ỵ✐✱ lim Γτ (n)+1 = 0✳ ◆❣♦➔✐ n→∞ n→∞ r❛✱ ✈ỵ✐ n ≥ n0 ✱ t❛ t❤➜② r➡♥❣ Γn Γτ (n)+1 ú ỵ (n) n ợ n n0 ✱ ✈➔ t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ ✈ỵ✐ τ (n) = n✱ tr♦♥❣ ❦❤➼ ✤ó♥❣ ✈ỵ✐ τ (n) < n ✈➻ t❤❡♦ ✤à♥❤ ♥❣❤➽❛✱ Γk ❧➔ ❤➔♠ ✤ì♥ ✤✐➺✉ t➠♥❣ ✈ỵ✐ τ (n) < k n)✳ ❉♦ ✤â✱ ✈ỵ✐ n ✤õ ❧ỵ♥ t❤➻ Γn Γτ (n)+1 ✳ ❱➻ ✈➟② lim Γn = 0✳ ◆➯♥ ❞➣② {xn } ❤ë✐ tö ♠↕♥❤ tỵ✐ z ✳ n→∞ ❙❛✉ ✤➙② ❧➔ ♠ët ✈➔✐ ✈➼ ❞ö ❝❤➾ r❛ r➡♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ (a)✕(d) tr♦♥❣ ỵ tt ❬✹❪✮✳ ❈❤♦ H ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✳ ❈❤å♥ ♣❤➛♥ tû ❜➜t ❦ý y ∈ H s❛♦ ❝❤♦ y = ✈➔ ||y|| = 1✱ ✈➔ ①→❝ ✤à♥❤ t➟♣ ❝♦♥ K ❝õ❛ H ①→❝ ✤à♥❤ ♥❤÷ s❛✉ K := {x ∈ H : x = λy, λ ∈ [0, 1]} t K t ỗ õ rộ ❝õ❛ H ✳ ❑❤✐ ✤â →♥❤ ①↕ T : H → K ①→❝ ✤à♥❤ ❜ð✐ T x = ✈ỵ✐ ♠å✐ x ∈ K ✳ ❘ã r➔♥❣✱ T ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✈➔ ❋✐①(T ) = {0}✳ ❳➨t ❞➣② ❧➦♣✿ δn = , n αn = 1− n 1− , n2 βn = 1 1− , rn = ✈ỵ✐ ♠å✐ n ≥ n2 n ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❛✮✱ ✭❝✮✱ ✈➔ ✭❞✮ t❤ä❛ ♠➣♥ ♥❤÷♥❣ ✤✐➲✉ ❦✐➺♥ ✭❜✮ ❦❤ỉ♥❣ t❤ä❛ ♠➣♥✳ ✣➦t u := y ∈ K ✱ ①➨t x1 := u✳ ❱➻ rn = 0✱ αn +βn +δn = 1, x1 ∈ K, u ∈ K ✷✾ ✈➔ T : H → K ✱ ①→❝ ✤à♥❤ ❜ð✐ ❞➣② {xn } t❤❡♦ ❝æ♥❣ t❤ù❝ ✭✷✳✶✷✮ t❤✉ë❝ ✈➔♦ K ✳ ❉♦ ✤â t❛ ❝â ❜✐➸✉ ❞✐➵♥ xn = λn y ✈ỵ✐ ♠é✐ n ∈ N ✈ỵ✐ λn ∈ [0, 1]✳ ❚❛ ❝â t❤➸ ❜✐➸✉ ❞✐➵♥ ✭✷✳✶✷✮ ♥❤÷ s❛✉ xn+1 = 1 y+ 1− n n 1− 1 y + − x = n n2 n n 1− λn y n2 ●✐↔ sû xn < ✱ ✈➔ xn ❦❤æ♥❣ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣✳ ❑❤✐ ✤â t❛ ❝â < λn = ||xn || < ❱➻ ✈➟② ||xn+1 || = = = = > 1 + 1− − λn y n n n 1 + 1− − λn ||y|| n n n 1 1 + 1− − λn ≥ + n n n n 1 + − + λn = + λn − n n n n + λn − λn > λn = ||xn || n n 1− n 1− λn n λn + λn n n ❉♦ ✤â✱ ❞➣② {xn } ❦❤ỉ♥❣ ❤ë✐ tư tỵ✐ 0✱ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❝õ❛ →♥❤ ①↕ T ✳ ❱➼ ❞ư ✷✳✷✳✽ ✭①❡♠ ❬✹❪✮✳ ❳➨t H ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt tũ ỵ t y, K T ✤à♥❤ ♥❤÷ tr♦♥❣ ✈➼ ❞ư tr➯♥✳ ❈❤å♥ x1 := u := y ✳ ❳➨t ❝ỉ♥❣ t❤ù❝ ✭✷✳✶✷✮ ✈ỵ✐ 1 δn = , αn = − , βn = , rn = 0, ∀n ≥ 2n n 2n ∞ ❑❤✐ ✤â δn → 0, ∞ δn = ∞, αn + βn + δn = 1✱ ✈➔ n=1 αn βn = ∞✳ ❉♦ ✤â✱ ✤✐➲✉ n=1 ❦✐➺♥ ✭❛✮✱ ✭❝✮✱ ✈➔ ✭❞✮ ❧➔ t❤ä❛ ♠➣♥✱ tr♦♥❣ ❦❤✐ ✤✐➲✉ ❦✐➺♥ ✭❜✮ ❦❤æ♥❣ t❤ä❛ ♠➣♥✱ ✈➔ ❞➣② ❧➦♣ ✭✷✳✶✷✮ trð t❤➔♥❤ xn+1 = δn u + αn xn + βn T xn + rn = 1 u+ 1− xn 2n n ✸✵ ❚❛ ❝â xn ∈ K ✈ỵ✐ ♠å✐ n ∈ N✱ ✈➻ ✈➟② t❛ ❝â t❤➸ ✈✐➳t xn = λn y ✈ỵ✐ λn ∈ [0, 1]✱ ✈➔ 1 u+ 1− xn = 2n n 1 + 1− λn = 2n n ||xn+1 || = ●✐↔ sû ||xn || < 1 + 1− λn ||y|| 2n n 1 ✱ ❦❤✐ ✤â λn = ||xn || < ✳ ❱➻ ✈➟② 2 ||xn+1 || = 1 + 1− λn > λn = ||xn || 2n n ❙✉② r❛ ||xn+1 || > ||xn || ✈ỵ✐ ❜➜t ❦ý ||xn || < ✳ ❱➻ ✈➟② ❞➣② {xn } ❦❤ỉ♥❣ ❤ë✐ tư tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ✵ ❝õ❛ →♥❤ ①↕ T ✳ ✷✳✸ ✷✳✸✳✶ Ù♥❣ ❞ö♥❣ Ù♥❣ ❞ư♥❣ ❝❤♦ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ ❉♦✉❣❧❛s✕❘❛❝❤❢♦r❞ ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët ù♥❣ ❞ư♥❣ ❝❤♦ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ ❉♦✉❣❧❛s✕❘❛❝❤❢♦r❞ t➻♠ ❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ t♦→♥ tû T ✈ỵ✐ T ❧➔ tê♥❣ ❝õ❛ ❤❛✐ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✱ tù❝ ❧➔ T = A + B ✈ỵ✐ A, B : H → 2H ❧➔ ❝→❝ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ✤❛ trà tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H t > số ữỡ trữợ ✣➦t JγA := (I + γA)−1 ✈➔ JγB := (I + γB)−1 t÷ì♥❣ ù♥❣ ❧➔ ❣✐↔✐ t❤ù❝ ✭t♦→♥ tû ❣✐↔✐✮ ❝õ❛ t♦→♥ tû A ✈➔ B ✳ ❚❛ ❝â JγA , JγB ❧➔ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥✳ ✣➦t RγA := 2JγA − I ✈➔ RγB := 2JγB − I ❧➔ t♦→♥ tû ❈❛②❧❡② ❝ơ♥❣ ❧➔ t♦→♥ tû ❦❤ỉ♥❣ ❣✐➣♥✳ ❱➻ ∈ T x ✈ỵ✐ T = A + B ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = JγB(y) ✱ tr♦♥❣ ✤â y ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ RγA RγB ✱ ♥➯♥ ✈➜♥ ✤➲ t➻♠ ❦❤æ♥❣ ✤✐➸♠ ❝õ❛ t♦→♥ tû T = A + B ❧➔ ✈➟♥ ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥ ❝❤♦ t♦→♥ tû ♥➔②✱ ❞➣② ❧➦♣ yn+1 := (1 − λn )yn + λn RγA RγB yn , n ≥ 1, ✭✷✳✷✵✮ ✸✶ ✤÷❛ r❛ ♠ët ①➜♣ ①➾ tr♦♥❣ ❝→❝ ❜✐➳♥ ❜❛♥ ✤➛✉ ❜➡♥❣ ❝→❝❤ ✤➦t xn := JγB (yn )✳ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ①→❝ ✤à♥❤ tr♦♥❣ ❞➣② ❧➦♣ ✭✷✳✷✵✮ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ ❉♦✉❣❧❛s✕❘❛❝❤❢♦r❞✱ ✤➦❝ ❜✐➺t ✈ỵ✐ λn = ∀n ∈ N t❛ t❤✉ ✤÷đ❝ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ P❡❛❝❡♠❛♥✕❘❛❝❤❢♦r❞✳ ❙û ❞ư♥❣ ✤à♥❤ ♥❣❤➽❛ t♦→♥ tû ❈❛②❧❡②✱ t❛ ✈✐➳t ❧↕✐ ❞➣② ❧➦♣ ①→❝ ✤à♥❤ tr♦♥❣ ✭✷✳✷✵✮ ♥❤÷ s❛✉ yn+1 : = (1 − λn )yn + λn (2JγA (2JγByn − yn ) − 2JγByn + y − n) = yn + 2λn (JγA (2JγByn −yn ) − JγByn ) ✭✷✳✷✶✮ ❚r♦♥❣ ✭✷✳✷✶✮ t❤❛② t❤➳ − λn ✈➔ λn t÷ì♥❣ ù♥❣ ❜ð✐ αn ✈➔ βn ✳ ❚❛ t❤✉ ✤÷đ❝ ❞➣② ❧➦♣ ①→❝ ✤à♥❤ ♥❤÷ s❛✉ yn+1 : = αn yn + βn (2JγA (2JγByn ) − 2JγByn +yn ) = (αn + βn )yn + 2βn (JγA (2JγByn − yn ) − JγByn ) ✭✷✳✷✷✮ ✭✷✳✷✸✮ ❚❤❡♦ ❦➳t q✉↔ ❝õ❛ ❈♦♠❜❡tt❡s ✭✷✵✵✹✮✱ t❛ õ s số a, bn tữỡ ự ợ t♦→♥ tû JγA ✈➔ JγB ✱ t❛ t❤✉ ✤÷đ❝ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ ❉♦✉❣❧❛s✕❘❛❝❤❢♦r❞ s✉② rë♥❣ ♥❤÷ s❛✉✿ yn+1 := (αn + βn )yn + 2βn (JγA (2(JγByn + bn ) − yn ) + an − (JγByn + bn ) ❑➳t q✉↔ t✐➳♣ t❤❡♦ ❝❤➾ r❛ sü ❤ë✐ tö ②➳✉ ữỡ tr ỵ ✷✳✸✳✶✳ ✭①❡♠ ❬✹❪✮ ●✐↔ sû H ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ❳➨t γ ∈ (0, ∞)✱ ✈➔ {αn }, {βn } ❧➔ ❝→❝ ❞➣② sè t❤ü❝ tr♦♥❣ [0, 1] s❛♦ ❝❤♦ αn + βn ✈ỵ✐ ♠å✐ n ≥ 1✳ ●✐↔ sû {an} ✈➔ {bn} ❧➔ ❤❛✐ ❞➣② ❝→❝ ♣❤➛♥ tû tr♦♥❣ H ✳ ●✐↔ sû ∈ ran(A + B)✳ ❳➨t ❞➣② {yn} ∈ H ✈ỵ✐ y1 ∈ H ①→❝ ✤à♥❤ ♥❤÷ s❛✉ yn+1 := αn yn + 2βn (JγA (2(JγByn + bn ) − yn ) + an ) −2βn (JγByn + bn ) + βn yn ✈ỵ✐ ♠å✐ n ≥ 1✳ ●✐↔ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t❤ä❛ ♠➣♥ ∞ αn βn = ∞; (a) n=1 ∞ βn (||an || + ||bn ||) < ∞; (b) n=1 ✭✷✳✷✹✮ ✸✷ ∞ (1 − αn − βn ) < ∞ (c) n=1 ❑❤✐ ✤â ❞➣② {yn} ❤ë✐ tư ②➳✉ tỵ✐ y ∈ H t❤ä❛ ♠➣♥ JγBy ∈ (A + B)−1(0)✳ ❚ø ỵ tr t ữủ q s H ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ✈➔ C ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❛❢✐♥ ✤â♥❣ ❝õ❛ H ✳ ❳➨t B : H → 2H ❧➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✳ ❳➨t γ ∈ (0, ∞) ✈➔ {αn}, {βn} ❧➔ ❝→❝ ❞➣② sè t❤ü❝ tr♦♥❣ ✤♦↕♥ [0, 1] s❛♦ ❝❤♦ αn + βn ✈ỵ✐ ♠å✐ n ≥ 1✳ ●✐↔ sû {an }, {bn } ⊂ H ✱ ∈ ran(NC + B)✳ ❳➨t ❞➣② {yn} tr♦♥❣ H ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿ y1 ∈ H ✱ ❍➺ q✉↔ ✷✳✸✳✷✳ ✭①❡♠ ❬✹❪✮ yn+1 := αn yn + 2βn (PC (2(JγByn + bn ) − yn ) + an ) − 2βn (JγByn + bn ) + βn yn ✭✷✳✷✺✮ ✈ỵ✐ ♠å✐ n ≥ 1✳ ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t❤ä❛ ♠➣♥ ∞ αn βn = ∞; (a) n=1 ∞ βn (||an || + ||bn ||) < ∞; (b) n=1 ∞ (c) (1 − αn − βn ) < ∞✳ n=1 ❑❤✐ ✤â ❞➣② {PC yn} ❤ë✐ tư ②➳✉ tỵ✐ JγBy ✱ tr♦♥❣ ✤â y ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ t♦→♥ tû RN RγB C ✷✳✸✳✷ Ù♥❣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❧✉➙♥ ♣❤✐➯♥ ❏♦❤♥ ✈♦♥ ◆❡✉✲ ♠❛♥♥ ❳➨t ❤❛✐ t➟♣ ❦❤→❝ ré♥❣ A, B H t ỗ õ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✱ ✈➔ ❣✐↔ sû A ∩ B = ∅✳ ❚❛ ❝â ❝→❝ t♦→♥ tû ❝❤✐➳✉ PA ✈➔ PB ❧➔ ❦❤æ♥❣ ❣✐➣♥✱ ✈➔ t♦→♥ tû T := PA PB ❦❤æ♥❣ ❣✐➣♥✳ ✣✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ t♦→♥ tû T ❧➔ ♣❤➛♥ tû ♥➡♠ tr♦♥❣ t➟♣ A ∩ B ữỡ ự ợ {xn } ❜ð✐ ♣❤➦♣ ❧➦♣ P✐❝❛r❞ xn+1 := T xn ✤÷đ❝ ❣å✐ ❧➔ ♣❤➨♣ ❝❤✐➳✉ ❧✉➙♥ ♣❤✐➯♥ ❝õ❛ ❏♦❤♥ ✈♦♥ ◆❡✉♠❛♥♥✳ ✣➸ t❤✉ ✤÷đ❝ sü ❤ë✐ tư t♦➔♥ ❝ư❝ ❦❤→❝✱ t❛ sû ❞ö♥❣ ♣❤➨♣ ❧➦♣ s❛✉ xn+1 := (1 − λn )xn + λn T xn = (1 − λn )xn + λn PA (PB xn ) ✭✷✳✷✻✮ ✸✸ ❚ø ❝æ♥❣ t❤ù❝ ✭✷✳✷✻✮✱ t÷ì♥❣ ù♥❣ t❛ t❤❛② − λn ✈➔ λn ❜ð✐ αn ✈➔ βn ✈➔ ❝→❝ s❛✐ sè an ❛♥❞ bn tữỡ ự ợ t tỷ PA PB ✳ ❚❛ t❤✉ ✤÷đ❝ ♣❤➨♣ ❧➦♣ s❛✉ xn+1 := αn xn + βn (PA (PB xn + bn ) + an ), n ứ ỵ ✷✳✷✳✶ t❛ t❤✉ ✤÷đ❝ ❦➳t q✉↔ s❛✉ ✈➲ sü ❤ë✐ tư ②➳✉✳ ●✐↔ sû H ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✱ ✈➔ ❝→❝ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ A, B ⊆ H ỗ õ õ tọ A B = ❳➨t ❝→❝ ❞➣② sè t❤ü❝ {αn} ✈➔ {βn} ♥➡♠ tr♦♥❣ ✤♦↕♥ [0, 1] s❛♦ ❝❤♦ αn + βn ✈ỵ✐ ♠å✐ n ≥ 1✳ ❱➔ ①➨t ❝→❝ ❞➣② ♣❤➛♥ tû {an } ✈➔ {bn } tr♦♥❣ H ✳ ❉➣② ❧➦♣ {xn } H ữ ợ tû ✤➛✉ t✐➯♥ x1 ∈ H ❜➜t ❦ý✳ ●✐↔ sû s tọ ỵ ❬✹❪✮ ∞ αn βn = ∞; (a) n=1 ∞ βn (||an || + ||bn ||) < ∞; (b) n=1 ∞ (c) (1 − αn − βn ) < ∞✳ n=1 ❑❤✐ ✤â ❞➣② {xn} ❤ë✐ tö ②➳✉ ✈➲ ♣❤➛♥ tû ♥➠♠ tr♦♥❣ t➟♣ A ∩ B ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ T := PAPB ✳ ❚❛ ✈✐➳t ❧↕✐ ♣❤➨♣ ❧➦♣ ✭✷✳✷✼✮ ♥❤÷ s❛✉ xn+1 = αn xn + βn (PA (PB xn + bn ) + an ) = αn xn + βn PA (PB xn ) + βn (PA (PB xn + bn ) − PA (PB xn ) + an ) = αn xn + βn T xn + rn ✈ỵ✐ rn := βn (PA (PB xn + bn ) − PA (PB xn ) + an ) ❙û ❞ư♥❣ t➼♥❤ ❦❤ỉ♥❣ ❣✐➣♥ ❝õ❛ t♦→♥ tû ❝❤✐➳✉✱ t❛ t❤✉ ✤÷đ❝ ||rn || βn (||PA (PB xn + bn ) − PA (PB xn )|| + ||an ||) βn (||PB xn + bn − PB xn || + ||an ||) = βn (||bn || + ||an ||) ❚ø ✤✐➲✉ ❦✐➺♥ (b)✱ t❛ ❝â ∞ ∞ ||rn || ≤ n=1 βn (||an || + ||bn ||) < ∞ n=1 ❱➻ t♦→♥ tỷ T ổ tứ ỵ s r❛ ❞➣② {xn } ❤ë✐ tư ②➳✉ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ t♦→♥ tû T ❤❛② ❤ë✐ tư tỵ✐ ♣❤➛♥ tû t❤✉ë❝ A ∩ B ✳ ✸✹ ❚÷ì♥❣ tü →♣ ỵ t t ữủ t q s ✈➲ sü ❤ë✐ tư ♠↕♥❤✿ ●✐↔ sû H ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✱ ✈➔ ❝→❝ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ A, B H ỗ õ õ tọ A B = ∅✳ ❳➨t ❝→❝ ❞➣② sè t❤ü❝ {αn}✱ {βn} ✈➔ {δn} ♥➡♠ tr♦♥❣ ✤♦↕♥ [0, 1] s❛♦ ❝❤♦ αn + βn + δn ✈ỵ✐ ♠å✐ n ≥ 1✳ ❱➔ ①➨t ❝→❝ ❞➣② ♣❤➛♥ tû {an} ✈➔ {bn} tr♦♥❣ H ✳ ❉➣② ❧➦♣ {xn} ⊂ H ①→❝ ✤à♥❤ ❜ð✐ ỵ xn+1 := n u + αn xn + βn (PA (PB xn + bn ) + an ), n ≥ 1, ✭✷✳✷✽✮ ✈ỵ✐ x1 ∈ H t ý tr õ u H trữợ ●✐↔ sû ❝â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ∞ δn = ∞; (a) lim δn = 0, n→∞ n=1 (b) lim inf αn βn > 0; n→∞ ∞ βn (||an || + ||bn ||) < ∞ (c) n=1 ∞ (1 − αn − βn − δn ) < ∞ (d) n=1 ❑❤✐ ✤â ❞➣② {xn} ❤ë✐ tö ♠↕♥❤ ✈➲ ♣❤➛♥ tû ❝õ❛ t➟♣ A ∩ B ✳ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ♥❤ú♥❣ ✈➜♥ ✤➲ s❛✉✿ • ❚r➻♥❤ ❜➔② ❧÷đ❝ ✈➲ ❦❤→✐ ♥✐➺♠ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H ✱ ❦❤→✐ ♥✐➺♠ ✈➲ →♥❤ ①↕ ✤ì♥ ✤✐➺✉✱ ❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③✱ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t✳ ●✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ✈➔ ♥➯✉ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❝ê ✤✐➸♥ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ r ữỡ rsss ❜➜t ✤ë♥❣ ❝❤♦ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ rt r ữỡ rsss s rở ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤♦ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ❝ò♥❣ ỵ tử ỵ tử ữỡ ố ợ s t rsss ✈➔ ♠ët sè ù♥❣ ❞ö♥❣✳ ✸✺ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ✣é ❱➠♥ ▲÷✉ ✭✷✵✵✵✮✱ ❬✷❪ ❍♦➔♥❣ ❚ư② tt t ỗ ố ❍➔ ◆ë✐✳ ❍➔♠ t❤ü❝ ✈➔ ●✐↔✐ t➼❝❤ ❤➔♠✱ ◆❳❇ ❑❤♦❛ ❤å❝ ❑ÿ ❚✐➳♥❣ ❆♥❤ ❬✸❪ ❆❣❛r✇❛❧ ❘✳P✳✱ ❖✬❘❡❣❛♥ ❉✳✱ ❙❛❤✉ ❉✳❘✳ ✭✷✵✵✾✮✱ ❋✐①❡❞ P♦✐♥t ▲✐♣s❝❤✐t③✐❛♥✲t②♣❡ ▼❛♣♣✐♥❣s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✱ ❙♣r✐♥❣❡r✳ ❚❤❡♦r② ❢♦r ❬✹❪ ❑❛♥③♦✇ ❈✳✱ ❙❤❡❤✉ ❨✳ ✭✷✵✶✼✮✱ ✏●❡♥❡r❛❧✐③❡❞ ❑r❛s♥♦s❡❧s❦✐✐✕▼❛♥♥✲t②♣❡ ✐t✲ ❡r❛t✐♦♥s ❢♦r ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣s ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✑✱ ❈♦♠♣✉t✳ ❖♣t✐♠✳ ❆♣♣❧✳✱ ✻✼✱ ♣♣✳ ✺✾✺✕✻✷✵✳ ❬✺❪ ❑❛③♠✐ ❑✳❘✳✱ ❘❡❤❛♥ ❆✳✱ ▼♦❤❞ ❋✳ ✭✷✵✶✽✮✱ ✏❑r❛s♥♦s❡❧s❦✐✕▼❛♥♥ t②♣❡ ✐t❡r✲ ❛t✐✈❡ ♠❡t❤♦❞ ❢♦r ❤✐❡r❛r❝❤✐❝❛❧ ❢✐①❡❞ ♣♦✐♥t ♣r♦❜❧❡♠ ❛♥❞ s♣❧✐t ♠✐①❡❞ ❡q✉✐✲ ❧✐❜r✐✉♠ ♣r♦❜❧❡♠✑✱ ◆✉♠❡r✳ ❆❧❣♦r✐t❤♠s✱ ✼✼✭✶✮✱ ♣♣✳ ✷✽✾✕✸✵✽✳ ❬✻❪ ▼♦✉❞❛❢✐ ❆✳ ✭✷✵✵✼✮✱ ✏❑r❛s♥♦s❡❧s❦✐✕▼❛♥♥ ✐t❡r❛t✐♦♥ ❢♦r ❤✐❡r❛r❝❤✐❝❛❧ ❢✐①❡❞✲ ♣♦✐♥t ♣r♦❜❧❡♠s✑✱ ■♥✈❡rs❡ Pr♦❜❧✳ ✷✸✱ ♣♣✳ ✶✻✸✺✕✶✻✹✵✳ ❬✼❪ ◗✐❛♦✲▲✐ ❉♦♥❣ ✱ ❑❛③♠✐❑✳ ❘✳✱ ❘❡❤❛♥ ❆❧✐✱ ❳✐❛♦✲❍✉❛♥ ▲✐ ✭✷✵✶✾✮✱ ✏■♥❡rt✐❛❧ ❑r❛s♥♦s❡❧✬s❦✐✞✐ ✲▼❛♥♥ t②♣❡ ❤②❜r✐❞ ❛❧❣♦r✐t❤♠s ❢♦r s♦❧✈✐♥❣ ❤✐❡r❛r❝❤✐❝❛❧ ❢✐①❡❞ ♣♦✐♥t ♣r♦❜❧❡♠s✑✱ ❏✳ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❆♣♣❧✳✱ ❤tt♣s✿✴✴❞♦✐✳♦r❣✴✶✵✳✶✵✵✼✴s✶✶✼✽✹✲✵✶✾✲✵✻✾✾✲✻✳ ❬✽❪ ❨❡❦✐♥✐ ❙✳ ✭✷✵✶✽✮✱ ✏❈♦♥✈❡r❣❡♥❝❡ r❛t❡ ❛♥❛❧②s✐s ♦❢ ✐♥❡rt✐❛❧ ❑r❛s♥♦s❡❧✲ s❦✐✐✕▼❛♥♥ t②♣❡ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s✑✱ ◆✉♠❡r✐❝❛❧ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧✲ ②s✐s ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥✱ ✸✾✭✶✵✮✱ ♣♣✳ ✶✵✼✼✲✶✵✾✶✳ ✸✻ ✸✼ ◆♦♥❧✐♥❡❛r ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s ■■■✳ ❱❛r✐❛t✐♦♥❛❧ ▼❡t❤♦❞s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✳ ❬✾❪ ❩❡✐❞❧❡r ❊✳ ✭✶✾✽✺✮✱ ...  - NGUYỄN THỊ NGỌC MAI VỀ PHƯƠNG PHÁP LẶP KRASNOSELSKII–MANN CHO ÁNH XẠ KHÔNG GIÃN TRONG KHƠNG GIAN HILBERT VÀ ÁP DỤNG Chun ngành: Tốn ứng dụng Mã số : 46 01 12 LUẬN VĂN THẠC SĨ